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squares line</phrase><phrase>line of best fit</phrase><phrase>best fit, line of</phrase><phrase>confidence intervals</phrase><phrase>intervals, confidence</phrase><phrase>mean square error</phrase><phrase>error, mean square</phrase><phrase>standard error</phrase><phrase>mean square regression</phrase><phrase>regression, mean square</phrase><phrase>linear regression</phrase></indexes></region><region region-id="1060" left="66" top="97.5" width="48.75" height="12.75" align-x="66" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">line(vx,vy)</p></text></region><region region-id="1054" left="144" top="97.5" width="54.75" height="12.75" align-x="144" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">slope(vx,vy)</p></text></region><region region-id="1061" left="66" top="115.5" width="59.25" height="12.75" align-x="66" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">stderr(vx,vy)</p></text></region><region region-id="1055" left="144" top="115.5" width="72.75" height="12.75" align-x="144" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">intercept(vx,vy)</p></text></region><region region-id="927" left="36" top="140.25" width="275.25" height="84" align-x="36" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">line</c></inlineAttr></b></f>, <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">slope</c></inlineAttr></b></f>, and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">intercept</c></inlineAttr></b></f> functions find the least-squares line-of-best-fit through a set of x-y data. The <b><inlineAttr line-through="false"><c val="#000"><f family="Verdana" charset="0">stderr</f></c></inlineAttr></b> function calculates the error in fitted parameters. Least-squares regression (fitting) uses the idea that the "line of best fit" is the line that minimizes the distances (errors) between each point in the data set and the final fitted line.</p></text><indexes><phrase>line function</phrase><phrase>slope function</phrase><phrase>intercept function</phrase><phrase>stderr function</phrase></indexes></region><region region-id="935" left="36" top="230.25" width="283.5" height="24" align-x="36" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Here, for example, is a set of distances in miles, and how long it takes, in minutes, to drive these distances.</p></text></region><region region-id="1064" left="66" top="261.75" width="159" height="18" align-x="87" align-y="276" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">dist</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:matrix rows="1" cols="9"><ml:real>8</ml:real><ml:real>3</ml:real><ml:real>4</ml:real><ml:real>2</ml:real><ml:real>5</ml:real><ml:real>4</ml:real><ml:real>12</ml:real><ml:real>18</ml:real><ml:real>3</ml:real></ml:matrix></ml:apply><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:define></math><rendering item-idref="4" /></region><region region-id="1065" left="66" top="279.75" width="200.25" height="18" align-x="91.5" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Time</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:matrix rows="1" cols="9"><ml:real>30</ml:real><ml:real>15</ml:real><ml:real>30</ml:real><ml:real>10</ml:real><ml:real>16</ml:real><ml:real>15</ml:real><ml:real>30</ml:real><ml:real>60</ml:real><ml:real>15</ml:real></ml:matrix></ml:apply><ml:id xml:space="preserve">min</ml:id></ml:apply></ml:define></math><rendering item-idref="5" /></region><region region-id="1194" left="66" top="303" width="81.75" height="12.75" align-x="92.25" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Npts</ml:id><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">dist</ml:id></ml:apply></ml:define></math><rendering item-idref="6" /></region><region region-id="1195" left="228" top="303" width="63.75" height="12.75" align-x="236.25" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:minus /><ml:id xml:space="preserve">Npts</ml:id><ml:real>1</ml:real></ml:apply></ml:range></ml:define></math><rendering item-idref="7" /></region><region region-id="1193" left="36" top="326.25" width="267" height="84" align-x="36" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">We would like to create a linear function estimating how long it should take to drive any distances not specifically given in this data, but of the same general order of magnitude. We don't care whether the points are under or over the line, so we square the distances to get rid of any negative signs, then add them up to get a measure of total distance.</p></text></region><region region-id="1190" left="36" top="427.5" width="88.5" height="12.75" align-x="36" align-y="438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Slope and intercept</p></text><indexes><phrase>slope</phrase><phrase>intercept</phrase></indexes></region><region region-id="1113" left="66" top="447" width="88.5" height="12.75" align-x="78.75" align-y="456" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id xml:space="preserve">m</ml:id><ml:apply><ml:id xml:space="preserve">slope</ml:id><ml:sequence><ml:id xml:space="preserve">dist</ml:id><ml:id xml:space="preserve">Time</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="8" /></region><region region-id="1592" left="222" top="447" width="100.5" height="12.75" align-x="233.25" align-y="456" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:id xml:space="preserve">intercept</ml:id><ml:sequence><ml:id xml:space="preserve">dist</ml:id><ml:id xml:space="preserve">Time</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="9" /></region><region region-id="1216" left="36" top="476.25" width="291.75" height="48" align-x="36" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When computing <b>slope</b><b>(x, y)</b> and <b>intercept</b><b>(x, y)</b>, y must represent the quantity you wish to estimate. The order of the arguments to <b>slope</b> and <b>intercept</b> does matter! The least-squares line is given by</p></text></region><region region-id="1116" left="66" top="537" width="63.75" height="12.75" align-x="90" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">ti</ml:id><ml:boundVars><ml:id xml:space="preserve">d</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">d</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:define></math><rendering item-idref="10" /></region><region region-id="1125" left="144" top="530.25" width="57.75" height="27.75" align-x="156" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">m</ml:id><ml:unitOverride><ml:apply><ml:div /><ml:id xml:space="preserve">min</ml:id><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>2.6813676907829529</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="mile" power-numerator="-1" /><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="11" /></region><region region-id="1593" left="222" top="537" width="53.25" height="12.75" align-x="232.5" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">b</ml:id><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>6.9777006937562227</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="12" /></region><region region-id="1199" left="36" top="560.25" width="288.75" height="48" align-x="36" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">which minimizes the sum of the squares of the distances from each point to the line, that is the difference between the actual Y values and the values calculated by the prediction function for each x-value.</p></text></region><region region-id="1217" left="66" top="613.5" width="168.75" height="43.5" align-x="178.5" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:summation /><ml:lambda><ml:boundVars><ml:id xml:space="preserve">i</ml:id></ml:boundVars><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:indexer /><ml:id xml:space="preserve">Time</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">ti</ml:id><ml:apply><ml:indexer /><ml:id xml:space="preserve">dist</ml:id><ml:id xml:space="preserve">i</ml:id></ml:apply></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:lambda><ml:bounds><ml:real>0</ml:real><ml:apply><ml:minus /><ml:id xml:space="preserve">Npts</ml:id><ml:real>1</ml:real></ml:apply></ml:bounds></ml:apply><ml:unitOverride><ml:apply><ml:pow /><ml:id xml:space="preserve">min</ml:id><ml:real>2</ml:real></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>292.1243805748266</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" power-numerator="2" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="13" /></region><region region-id="1627" left="0" top="660" width="6000" height="6" align-x="0" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1136" left="36" top="674.25" width="283.5" height="24" align-x="36" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The "Error bar" trace type is used to show the vertical distances from the original points to the line.</p></text><indexes><phrase>Error bar trace</phrase></indexes></region><region region-id="1657" left="30" top="702" width="271.5" height="261.75" align-x="30" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="14" /><rendering item-idref="15" /></region><region region-id="1200" left="36" top="961.5" width="21" height="12.75" align-x="36" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Line</p></text><indexes><phrase>line function</phrase></indexes></region><region region-id="1152" left="36" top="980.25" width="315" height="24" align-x="36" align-y="990" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">As a convenience, the slope and intercept functions are also combined to form the single built-in function, <b>line</b>.</p></text></region><region region-id="1818" left="66" top="1010.25" width="40.5" height="27.75" align-x="81" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vx</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">dist</ml:id><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:define></math><rendering item-idref="16" /></region><region region-id="1819" left="138" top="1010.25" width="46.5" height="27.75" align-x="154.5" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">vy</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">Time</ml:id><ml:id xml:space="preserve">min</ml:id></ml:apply></ml:define></math><rendering item-idref="17" /></region><region region-id="1141" left="66" top="1053" width="87.75" height="12.75" align-x="100.5" align-y="1062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">params</ml:id><ml:apply><ml:id xml:space="preserve">line</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="1258" left="222" top="1083" width="53.25" height="12.75" align-x="232.5" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">b</ml:id><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>6.9777006937562227</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="19" /></region><region region-id="1203" left="156" top="1088.25" width="45" height="12" align-x="156" align-y="1098" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">intercept</p></text></region><region region-id="1202" left="66" top="1086.75" width="81" height="29.25" align-x="99.75" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">params</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="1"><ml:real>6.9777006937561943</ml:real><ml:real>2.6813676907829533</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="20" /></region><region region-id="1204" left="156" top="1106.25" width="25.5" height="12" align-x="156" align-y="1116" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">slope</p></text></region><region region-id="1260" left="222" top="1100.25" width="57.75" height="27.75" align-x="234" align-y="1116" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">m</ml:id><ml:unitOverride><ml:apply><ml:div /><ml:id xml:space="preserve">min</ml:id><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>2.6813676907829529</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="mile" power-numerator="-1" /><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="21" /></region><region region-id="1821" left="36" top="1136.25" width="299.25" height="24" align-x="36" align-y="1146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that this produces the same result as the familiar linear algebra expression</p></text></region><region region-id="1935" left="60" top="1173" width="94.5" height="18.75" align-x="73.5" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">X</ml:id><ml:apply><ml:id xml:space="preserve">augment</ml:id><ml:sequence><ml:apply><ml:pow /><ml:id xml:space="preserve">vx</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:pow /><ml:id xml:space="preserve">vx</ml:id><ml:real>1</ml:real></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1936" left="180" top="1167.75" width="132" height="32.25" align-x="264.75" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:mult style="default" /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">X</ml:id></ml:apply><ml:id xml:space="preserve">X</ml:id></ml:apply></ml:parens><ml:real>-1</ml:real></ml:apply><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">X</ml:id></ml:apply><ml:id xml:space="preserve">vy</ml:id></ml:apply></ml:parens></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="1"><ml:real>6.9777006937561907</ml:real><ml:real>2.6813676907829533</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="23" /></region><region region-id="1212" left="36" top="1208.25" width="318" height="60" align-x="36" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">and also explains why the arguments to the line function must be dimensionless, since slope and intercept will have different dimensions with dimensioned x and y, but packed into the same vector, results must have the same dimension. You can use whichever representation is best suited to your application.</p></text></region><region region-id="1213" left="0" top="1284" width="6000" height="6" align-x="0" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1377" left="36" top="1291.5" width="78" height="12.75" align-x="36" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using the results</p></text></region><region region-id="1378" left="36" top="1316.25" width="289.5" height="60" align-x="36" align-y="1326" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The resulting parametrized equation from linear or any other type of regression should be used only for values near the original observed data. For example, note that this equation predicts that at a distance of 0 miles, it will take </p></text></region><region region-id="1379" left="66" top="1389" width="77.25" height="12.75" align-x="100.5" align-y="1398" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">ti</ml:id><ml:apply><ml:mult style="default" /><ml:real>0</ml:real><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>6.9777006937562227</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="24" /></region><region region-id="1380" left="36" top="1412.25" width="300.75" height="72" align-x="36" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">minutes to drive that distance, which doesn't make sense if the measured time is strictly travel time at constant velocity. In some cases, the measurements may include something mundane, such as waiting time at traffic lights, which must be accounted for in any interpretation of a linear regression.</p></text></region><region region-id="1154" left="36" top="1495.5" width="89.25" height="12.75" align-x="36" align-y="1506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The Standard Error</p></text><indexes><phrase>standard error</phrase></indexes></region><region region-id="1155" left="36" top="1514.25" width="284.25" height="36" align-x="36" align-y="1524" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">As a measure of how good the linear fit of a data is, the <f family="Verdana" charset="0"><i><inlineAttr line-through="false"><c val="#000">standard error in the estimate</c></inlineAttr></i></f> or the standard error in y given x is defined to be</p></text></region><region region-id="1434" left="66" top="1563" width="32.25" height="12.75" align-x="75.75" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">dist</ml:id></ml:define></math><rendering item-idref="25" /></region><region region-id="1435" left="144" top="1563" width="38.25" height="12.75" align-x="155.25" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">y</ml:id><ml:id xml:space="preserve">Time</ml:id></ml:define></math><rendering item-idref="26" /></region><region region-id="2141" left="66" top="1584.75" width="147" height="32.25" align-x="122.25" align-y="1602" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:sqrt /><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">y</ml:id><ml:parens><ml:apply><ml:plus /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:id xml:space="preserve">b</ml:id></ml:apply></ml:parens></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></math><rendering item-idref="27" /></region><region region-id="1158" left="36" top="1628.25" width="315" height="122.25" align-x="36" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">where x and y are the paired data vectors being fitted, and m and b are the slope and intercept of the linear fit, respectively. The <region region-id="1159" left="126" top="1653" width="427.5" height="12.75" align-x="141" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">df</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="28" /></region> expression represents the degrees of freedom in the problem. This is the number of data points minus the number of fitted parameters. So the standard error is really the sum-of-squared-distances, the quantity which we minimized to get the fitting equation, divided by the degrees of freedom in the data. This says that the more data you fit with the fewer parameters, the smaller your error is likely to be.</p></text></region><region region-id="1162" left="36" top="1766.25" width="260.25" height="12" align-x="36" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">In the case of the data above, the standard error is</p></text></region><region region-id="2144" left="66" top="1794.75" width="124.5" height="32.25" align-x="152.25" align-y="1812" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:sqrt /><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:id xml:space="preserve">df</ml:id></ml:apply><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">y</ml:id><ml:apply><ml:id xml:space="preserve">ti</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>6.4600351676931584</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="29" /></region><region region-id="1215" left="36" top="1844.25" width="199.5" height="12" align-x="36" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">or, as computed by the built-in function</p></text></region><region region-id="1852" left="66" top="1869" width="113.25" height="12.75" align-x="141" align-y="1878" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">dist</ml:id><ml:id xml:space="preserve">Time</ml:id></ml:sequence></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>6.4600351676931584</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="30" /></region><region region-id="1854" left="36" top="1883.25" width="309.75" height="15" align-x="36" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This is the square root of the mean squared error, MSE, or <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">s</c></inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sup>2</sup></c></inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">.</c></inlineAttr></f></p></text></region><region region-id="1634" left="0" top="1908" width="6000" height="6" align-x="0" align-y="1908" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1635" left="36" top="1922.25" width="211.5" height="12" align-x="36" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The standard error in the slope is given by</p></text></region><region region-id="1636" left="66" top="1946.25" width="134.25" height="46.5" align-x="109.5" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">slope_err</ml:id><ml:apply><ml:div /><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="31" /></region><region region-id="1637" left="222" top="1946.25" width="88.5" height="27.75" align-x="264.75" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">slope_err</ml:id><ml:unitOverride><ml:apply><ml:div /><ml:id xml:space="preserve">min</ml:id><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>0.43141531278605427</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="mile" power-numerator="-1" /><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="32" /></region><region region-id="2000" left="36" top="2006.25" width="250.5" height="12" align-x="36" align-y="2016" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">and the standard error in the intercept is given by</p></text></region><region region-id="2146" left="66" top="2022.75" width="222" height="52.5" align-x="98.25" align-y="2046" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">int_err</ml:id><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:div /><ml:apply><ml:pow /><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>2</ml:real></ml:apply><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="2147" left="66" top="2079" width="74.25" height="12.75" align-x="97.5" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">int_err</ml:id><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>3.5546341295857506</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="34" /></region><region region-id="1944" left="36" top="2108.25" width="293.25" height="24" align-x="36" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can get the standard deviation (error) in both variables at once using</p></text></region><region region-id="2148" left="66" top="2139.75" width="63" height="24" align-x="78.75" align-y="2160" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:id xml:space="preserve">H</ml:id><ml:apply><ml:pow /><ml:parens><ml:apply><ml:mult style="default" /><ml:apply><ml:transpose /><ml:id xml:space="preserve">X</ml:id></ml:apply><ml:id xml:space="preserve">X</ml:id></ml:apply></ml:parens><ml:real>-1</ml:real></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="1956" left="66" top="2181" width="80.25" height="12.75" align-x="76.5" align-y="2190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">k</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id xml:space="preserve">last</ml:id><ml:id xml:space="preserve">params</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="36" /></region><region region-id="2151" left="174" top="2179.5" width="112.5" height="19.5" align-x="194.25" align-y="2190" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id xml:space="preserve">sβ</ml:id><ml:id xml:space="preserve">k</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id></ml:sequence></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:indexer /><ml:id xml:space="preserve">H</ml:id><ml:sequence><ml:id xml:space="preserve">k</ml:id><ml:id xml:space="preserve">k</ml:id></ml:sequence></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="37" /></region><region region-id="1967" left="138" top="2204.25" width="207" height="36" align-x="138" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">where the first entry is the error in the intercept, and the second is the error in slope.</p></text></region><region region-id="1962" left="66" top="2208.75" width="62.25" height="29.25" align-x="81" align-y="2226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">sβ</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="1"><ml:real>3.5546341295857498</ml:real><ml:real>0.43141531278605427</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="38" /></region><region region-id="1968" left="36" top="2251.5" width="95.25" height="12.75" align-x="36" align-y="2262" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Confidence intervals</p></text><indexes><phrase>confidence intervals</phrase></indexes></region><region region-id="1417" left="36" top="2270.25" width="294" height="48" align-x="36" align-y="2280" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">We can use these estimates, together with percentile points from the Student's </c><f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">t</c></inlineAttr></f><c val="#000"> distribution with the same </c><c val="#000">degrees of freedom as the regression to form a </c>confidence interval for the estimate of the slope.</p></text></region><region region-id="1444" left="36" top="2330.25" width="7.5" height="12" align-x="36" align-y="2340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If</p></text></region><region region-id="1445" left="66" top="2330.25" width="36" height="13.5" align-x="78.75" align-y="2340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">α</ml:id><ml:real>0.02</ml:real></ml:define></math><rendering item-idref="39" /></region><region region-id="1899" left="138" top="2323.5" width="75.75" height="28.5" align-x="147" align-y="2340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">t</ml:id><ml:apply><ml:id xml:space="preserve">qt</ml:id><ml:sequence><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:div /><ml:id xml:space="preserve">α</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:id xml:space="preserve">df</ml:id></ml:sequence></ml:apply></ml:define></math><rendering item-idref="40" /></region><region region-id="2206" left="234" top="2331" width="41.25" height="12.75" align-x="242.25" align-y="2340" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">t</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>2.9979515668685264</ml:real></result></ml:eval></math><rendering item-idref="41" /></region><region region-id="2005" left="36" top="2360.25" width="275.25" height="54.75" align-x="36" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">we are <region region-id="2004" left="73.5" top="2360.25" width="60.75" height="13.5" align-x="108.75" align-y="2370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:parens><ml:apply><ml:minus /><ml:real>1</ml:real><ml:id xml:space="preserve">α</ml:id></ml:apply></ml:parens><ml:unitOverride><ml:id xml:space="preserve">%</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>98</ml:real></result></ml:eval></math><rendering item-idref="42" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sp /></c></inlineAttr></f>confident that the correct value of m for this data lies on the interval between <region region-id="1971" left="36" top="2386.5" width="112.5" height="27.75" align-x="102.75" align-y="2402.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:minus /><ml:id xml:space="preserve">m</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">slope_err</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:apply><ml:div /><ml:id xml:space="preserve">min</ml:id><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>1.388005477844926</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="mile" power-numerator="-1" /><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="43" /></region> and <region region-id="1972" left="174" top="2386.5" width="427.5" height="27.75" align-x="241.5" align-y="2402.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:plus /><ml:id xml:space="preserve">m</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">slope_err</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:apply><ml:div /><ml:id xml:space="preserve">min</ml:id><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>3.97472990372098</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="mile" power-numerator="-1" /><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="44" /></region>.</p></text></region><region region-id="2008" left="36" top="2432.25" width="306.75" height="54" align-x="36" align-y="2442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Likewise, we are <region region-id="2152" left="121.5" top="2432.25" width="60.75" height="13.5" align-x="156.75" align-y="2442" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:parens><ml:apply><ml:minus /><ml:real>1</ml:real><ml:id xml:space="preserve">α</ml:id></ml:apply></ml:parens><ml:unitOverride><ml:id xml:space="preserve">%</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>98</ml:real></result></ml:eval></math><rendering item-idref="45" /></region> confident that the true intercept lies between <region region-id="2009" left="150.75" top="2447.25" width="102" height="12.75" align-x="204.75" align-y="2456.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:minus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">int_err</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>-3.6789202646797188</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="46" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sp /></c></inlineAttr></f>and <region region-id="2012" left="36" top="2460.75" width="427.5" height="12.75" align-x="90.75" align-y="2469.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:plus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:id xml:space="preserve">int_err</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>17.634321652192163</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="47" /></region><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">.The wide range on this value indicates the high level of scatter in the data.</c></inlineAttr></f></p></text></region><region region-id="2199" left="36" top="2498.25" width="296.25" height="36" align-x="36" align-y="2508" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This calculation is wrapped up in the <b>confidence</b> function that will be desribed in detail in the section on <b><u><link href="./quality.xmcd" title="Links to section" popup="false">Quality of Fit</link></u></b>.</p></text></region><region region-id="2219" left="0" top="2544" width="6000" height="6" align-x="0" align-y="2544" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2215" left="66" top="2547" width="87" height="12.75" align-x="114.75" align-y="2556" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">lsf</ml:id><ml:boundVars><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">b</ml:id><ml:id xml:space="preserve">m</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="48" /></region><region region-id="2216" left="66" top="2570.25" width="178.5" height="13.5" align-x="86.25" align-y="2580" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">pCI</ml:id><ml:apply><ml:id xml:space="preserve">confidence</ml:id><ml:sequence><ml:id xml:space="preserve">vx</ml:id><ml:id xml:space="preserve">vy</ml:id><ml:id xml:space="preserve">lsf</ml:id><ml:id xml:space="preserve">params</ml:id><ml:apply><ml:minus /><ml:real>1</ml:real><ml:id xml:space="preserve">α</ml:id></ml:apply></ml:sequence></ml:apply></ml:define></math><rendering item-idref="49" /></region><region region-id="2217" left="264" top="2562.75" width="98.25" height="29.25" align-x="283.5" align-y="2580" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">pCI</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="2"><ml:real>10.656620958435955</ml:real><ml:real>1.2933622129380302</ml:real><ml:real>2.9979515668685264</ml:real><ml:real>0</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="50" /></region><region region-id="2223" left="36" top="2600.25" width="320.25" height="48" align-x="43.5" align-y="2610" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="true" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The first column gives the confidence interval on the parameters intercept and slope, and the entry in the second column is the value of the t statistic. Divide the first column by t to get back the standard errors in both parameters:</p></text></region><region region-id="2225" left="66" top="2655" width="81" height="33" align-x="99.75" align-y="2676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:div /><ml:apply><ml:matcol /><ml:id xml:space="preserve">pCI</ml:id><ml:real>0</ml:real></ml:apply><ml:id xml:space="preserve">t</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:matrix rows="2" cols="1"><ml:real>3.5546341295857551</ml:real><ml:real>0.43141531278605544</ml:real></ml:matrix></result></ml:eval></math><rendering item-idref="51" /></region><region region-id="2013" left="36" top="2696.25" width="308.25" height="48" align-x="36" align-y="2706" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If we wish to calculate a confidence interval for the regression itself, that is, what is the uncertainty on any prediction of a y value made by the regression formula, we would find</p></text></region><region region-id="2160" left="66" top="2754.75" width="240.75" height="52.5" align-x="111" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">confy</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:div /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">x0</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="52" /></region><region region-id="2093" left="36" top="2822.25" width="296.25" height="24" align-x="36" align-y="2832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Then you can calculate the confidence interval for any predicted y value:</p></text></region><region region-id="2161" left="66" top="2859" width="104.25" height="12.75" align-x="123" align-y="2868" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">confy</ml:id><ml:apply><ml:mult style="default" /><ml:real>20</ml:real><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>18.548215072697438</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="53" /></region><region region-id="2095" left="36" top="2882.25" width="288" height="12" align-x="36" align-y="2892" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can do the same thing with the matrix computations:</p></text></region><region region-id="2157" left="66" top="2903.25" width="162.75" height="30.75" align-x="96.75" align-y="2922" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">CI</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:matrix rows="1" cols="2"><ml:real>1</ml:real><ml:id xml:space="preserve">x0</ml:id></ml:matrix><ml:id xml:space="preserve">H</ml:id></ml:apply><ml:matrix rows="2" cols="1"><ml:real>1</ml:real><ml:id xml:space="preserve">x0</ml:id></ml:matrix></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="54" /></region><region region-id="2164" left="246" top="2913" width="78" height="12.75" align-x="276.75" align-y="2922" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">CI</ml:id><ml:real>20</ml:real></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>18.548215072697435</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="55" /></region><region region-id="2102" left="36" top="2948.25" width="288.75" height="24" align-x="36" align-y="2958" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can view the entire confidence interval for the regression region:</p></text></region><region region-id="2158" left="66" top="2985" width="140.25" height="12.75" align-x="112.5" align-y="2994" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">Upper</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">confy</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="56" /></region><region region-id="2104" left="216" top="2985" width="147" height="12.75" align-x="262.5" align-y="2994" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">Lower</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:minus /><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply></ml:parens><ml:apply><ml:id xml:space="preserve">confy</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="57" /></region><region region-id="2159" left="66" top="3003" width="87.75" height="12.75" align-x="80.25" align-y="3012" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">dr</ml:id><ml:range><ml:sequence><ml:apply><ml:mult style="default" /><ml:real>0</ml:real><ml:id xml:space="preserve">mi</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:real>1</ml:real><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:sequence><ml:apply><ml:mult style="default" /><ml:real>20</ml:real><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:range></ml:define></math><rendering item-idref="58" /></region><region region-id="2228" left="60" top="3018" width="263.25" height="198" align-x="60" align-y="3018" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="59" /><rendering item-idref="60" /></region><region region-id="2229" left="0" top="3210" width="6000" height="6" align-x="0" align-y="3210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="2048" left="36" top="3230.25" width="279.75" height="60" align-x="36" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The confidence region for predicted values using the regression formula has a waist near the center of the measured values. This is because these formulas are mean-based, so values predicted closer to the mean of the data are more accurate.</p></text></region><region region-id="2107" left="36" top="3308.25" width="296.25" height="24" align-x="36" align-y="3318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that the confidence limits on the measured values are somewhat different than those for predicted values.</p></text></region><region region-id="2120" left="66" top="3342.75" width="258.75" height="52.5" align-x="112.5" align-y="3366" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">confm</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:div /><ml:real>1</ml:real><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply><ml:apply><ml:div /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">x0</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">x</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="61" /></region><region region-id="2125" left="66" top="3405" width="92.25" height="12.75" align-x="120" align-y="3414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">confm</ml:id><ml:apply><ml:mult style="default" /><ml:real>8</ml:real><ml:id xml:space="preserve">mi</ml:id></ml:apply></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>20.499779741711869</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="62" /></region><region region-id="2126" left="36" top="3428.25" width="177.75" height="12" align-x="36" align-y="3438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Likewise, for the matrix calculations</p></text></region><region region-id="2167" left="66" top="3449.25" width="186" height="30.75" align-x="103.5" align-y="3468" show-border="false" show-highlight="false" is-protected="false" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:function><ml:id xml:space="preserve">CIm</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">t</ml:id><ml:apply><ml:id xml:space="preserve">stderr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply></ml:apply><ml:apply><ml:sqrt /><ml:apply><ml:plus /><ml:real>1</ml:real><ml:apply><ml:mult style="default" /><ml:apply><ml:mult style="default" /><ml:matrix rows="1" cols="2"><ml:real>1</ml:real><ml:id xml:space="preserve">x0</ml:id></ml:matrix><ml:id xml:space="preserve">H</ml:id></ml:apply><ml:matrix rows="2" cols="1"><ml:real>1</ml:real><ml:id xml:space="preserve">x0</ml:id></ml:matrix></ml:apply></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="63" /></region><region region-id="2122" left="66" top="3495" width="71.25" height="12.75" align-x="99" align-y="3504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:id xml:space="preserve">CIm</ml:id><ml:real>8</ml:real></ml:apply><ml:unitOverride><ml:id xml:space="preserve">min</ml:id></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>20.499779741711865</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="64" /></region><region region-id="2127" left="36" top="3518.25" width="303" height="24" align-x="36" align-y="3528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="false"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">These limits apply only at the measured data points, and are probably best viewed as error bars around each value.</p></text></region><region region-id="2138" left="66" top="3555" width="141.75" height="12.75" align-x="112.5" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">Upper</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">confm</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="65" /></region><region region-id="2135" left="228" top="3555" width="148.5" height="12.75" align-x="274.5" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define warning="WarnRedefinedUDFunction"><ml:function><ml:id xml:space="preserve">Lower</ml:id><ml:boundVars><ml:id xml:space="preserve">x0</ml:id></ml:boundVars></ml:function><ml:apply><ml:minus /><ml:parens><ml:apply><ml:plus /><ml:id xml:space="preserve">b</ml:id><ml:apply><ml:mult style="default" /><ml:id xml:space="preserve">m</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply></ml:parens><ml:apply><ml:id xml:space="preserve">confm</ml:id><ml:id xml:space="preserve">x0</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="66" /></region><region region-id="2133" left="36" top="3582" width="338.25" height="234" align-x="36" align-y="3582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="67" /><rendering item-idref="68" /></region><region region-id="2137" left="36" top="3824.25" width="301.5" height="36" align-x="36" align-y="3834" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This can be used as a form of outlier detection, where measured values outside the confidence intervals can indicate an outlier.</p></text></region><region region-id="2230" left="0" top="3864" width="6000" height="6" align-x="0" align-y="3864" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1224" left="36" top="3877.5" width="113.25" height="12.75" align-x="36" align-y="3888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Relationship to statistics</p></text></region><region region-id="1226" left="36" top="3902.25" width="193.5" height="12" align-x="36" align-y="3912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">An alternate formula for <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">slope(x, y)</c></inlineAttr></b></f> is</p></text></region><region region-id="1235" left="66" top="3926.25" width="78.75" height="27.75" align-x="105" align-y="3942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:apply><ml:id xml:space="preserve">stdev</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">stdev</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply></ml:apply></math><rendering item-idref="69" /></region><region region-id="1249" left="36" top="3968.25" width="300" height="36" align-x="36" align-y="3978" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">look at the definitions for <b><u><link href="./dispersion.xmcd" title="Links to section" popup="false">standard deviation</link></u></b> and the <b><u><link href="./association.xmcd" title="Links to section" popup="false">correlation coefficient</link></u></b> to see what this means in terms of your data.</p></text></region><region region-id="1247" left="36" top="4016.25" width="319.5" height="12" align-x="36" align-y="4026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">An alternative formula for <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">intercept(x, y)</c></inlineAttr></b></f> is</p></text></region><region region-id="1248" left="66" top="4041" width="123.75" height="12.75" align-x="105" align-y="4050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:id xml:space="preserve">slope</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:apply></math><rendering item-idref="70" /></region><region region-id="1184" left="36" top="4070.25" width="305.25" height="24" align-x="36" align-y="4080" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This means that the least-squares line always passes through the point <f family="Verdana" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">(mean(x), mean(y))</c></inlineAttr></f>, as illustrated below.</p></text></region><region region-id="1188" left="54" top="4122" width="173.25" height="186" align-x="54" align-y="4122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><plot disable-calc="false" item-idref="71" /><rendering item-idref="72" /></region><region region-id="1264" left="36" top="4321.5" width="62.25" height="12.75" align-x="36" align-y="4332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="goodfit"><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Quality of Fit</p></text><indexes><phrase>quality of fit</phrase></indexes></region><region region-id="1639" left="36" top="4340.25" width="283.5" height="24" align-x="36" align-y="4350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">A linear regression model can be broken down into components of variation, one due to residual error,</p></text></region><region region-id="1295" left="66" top="4377" width="88.5" height="30" align-x="87.75" align-y="4392" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">SSE</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">y</ml:id><ml:apply><ml:id xml:space="preserve">ti</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="73" /></region><region region-id="1296" left="180" top="4379.25" width="102.75" height="15" align-x="180" align-y="4392" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">also known as <f family="Symbol" charset="2" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">c</c></inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"><sup>2</sup></c></inlineAttr></f><f family="Verdana" charset="0" size="10"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000">,</c></inlineAttr></f></p></text></region><region region-id="1360" left="66" top="4407" width="77.25" height="18.75" align-x="87" align-y="4422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">SSE</ml:id><ml:unitOverride><ml:apply><ml:pow /><ml:id xml:space="preserve">min</ml:id><ml:real>2</ml:real></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>292.1243805748266</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" power-numerator="2" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="74" /></region><region region-id="1297" left="36" top="4442.25" width="252" height="12" align-x="36" align-y="4452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">and another due to the regression equation itself. </c></p></text></region><region region-id="1302" left="66" top="4461" width="118.5" height="30" align-x="89.25" align-y="4476" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">SSR</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">ti</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="75" /></region><region region-id="1358" left="222" top="4461" width="94.5" height="18.75" align-x="244.5" align-y="4476" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">SSR</ml:id><ml:unitOverride><ml:apply><ml:pow /><ml:id xml:space="preserve">min</ml:id><ml:real>2</ml:real></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>1612.0978416473954</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" power-numerator="2" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="76" /></region><region region-id="1303" left="36" top="4496.25" width="273" height="24" align-x="36" align-y="4506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">The sum of these components equals what we call the </c><i><c val="#000">total sum of squares</c></i><inlineAttr font-style="normal"><c val="#000">:</c></inlineAttr></p></text><indexes><phrase>total sum of squares</phrase></indexes></region><region region-id="2232" left="0" top="4524" width="6000" height="6" align-x="0" align-y="4524" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1305" left="66" top="4533" width="106.5" height="30" align-x="88.5" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">SST</ml:id><ml:apply><ml:vectorSum /><ml:apply><ml:pow /><ml:parens><ml:apply><ml:minus /><ml:id xml:space="preserve">y</ml:id><ml:apply><ml:id xml:space="preserve">mean</ml:id><ml:id xml:space="preserve">y</ml:id></ml:apply></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="77" /></region><region region-id="1357" left="222" top="4533" width="93.75" height="18.75" align-x="243.75" align-y="4548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">SST</ml:id><ml:unitOverride><ml:apply><ml:pow /><ml:id xml:space="preserve">min</ml:id><ml:real>2</ml:real></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>1904.2222222222224</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" power-numerator="2" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="78" /></region><region region-id="1362" left="66" top="4563" width="122.25" height="18.75" align-x="116.25" align-y="4578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:plus /><ml:id xml:space="preserve">SSE</ml:id><ml:id xml:space="preserve">SSR</ml:id></ml:apply><ml:unitOverride><ml:apply><ml:pow /><ml:id xml:space="preserve">min</ml:id><ml:real>2</ml:real></ml:apply></ml:unitOverride><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>1904.2222222222219</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="minute" power-numerator="2" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="79" /></region><region region-id="1366" left="36" top="4598.25" width="300" height="36" align-x="36" align-y="4608" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The proportion of variation in y that can be explained by the linear regression rather than by using the mean of y alone is given by </p></text></region><region region-id="1367" left="66" top="4647" width="105" height="18.75" align-x="138" align-y="4662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:pow /><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:id xml:space="preserve">dist</ml:id><ml:id xml:space="preserve">Time</ml:id></ml:sequence></ml:apply><ml:real>2</ml:real></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.84659123438129069</ml:real></result></ml:eval></math><rendering item-idref="80" /></region><region region-id="1368" left="36" top="4682.25" width="288.75" height="36" align-x="36" align-y="4692" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">which is the percentage of the variation in Time explained by the regression model. The rest of the variation is explained by standard deviations in the data.</p></text></region><region region-id="1309" left="36" top="4730.25" width="48" height="12" align-x="36" align-y="4740" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are</p></text></region><region region-id="1307" left="66" top="4755" width="104.25" height="16.5" align-x="110.25" align-y="4764" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="RESID">DF</ml:id><ml:apply><ml:minus /><ml:apply><ml:id xml:space="preserve">length</ml:id><ml:id xml:space="preserve">x</ml:id></ml:apply><ml:real>2</ml:real></ml:apply></ml:define></math><rendering item-idref="81" /></region><region region-id="1311" left="36" top="4778.25" width="304.5" height="36" align-x="36" align-y="4788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">degrees of residual freedom associated with the sum of squares for residual error, because there are two parameters.</p></text></region><region region-id="1319" left="36" top="4832.25" width="328.5" height="12" align-x="36" align-y="4842" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The degrees of freedom for the sum of squares due to regression,</p></text></region><region region-id="1320" left="66" top="4857" width="47.25" height="16.5" align-x="101.25" align-y="4866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve" subscript="REG">DF</ml:id><ml:real>2</ml:real></ml:define></math><rendering item-idref="82" /></region><region region-id="1321" left="36" top="4880.25" width="298.5" height="48" align-x="36" align-y="4890" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">are the same as the number of independent variables in the model. The degrees of freedom, like the corresponding sum of squares, are additive, so that the degrees of freedom for the total sum of squares equals </p></text></region><region region-id="1325" left="66" top="4941" width="141.75" height="16.5" align-x="120.75" align-y="4950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">DF_TOTAL</ml:id><ml:apply><ml:plus /><ml:id xml:space="preserve" subscript="RESID">DF</ml:id><ml:id xml:space="preserve" subscript="REG">DF</ml:id></ml:apply></ml:define></math><rendering item-idref="83" /></region><region region-id="1326" left="66" top="4965" width="71.25" height="12.75" align-x="120" align-y="4974" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">DF_TOTAL</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>9</ml:real></result></ml:eval></math><rendering item-idref="84" /></region><region region-id="1643" left="36" top="4988.25" width="283.5" height="36" align-x="36" align-y="4998" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Dividing a sum of squares by its degrees of freedom provides us with a variance estimate. Mean square error is</p></text></region><region region-id="1335" left="66" top="5036.25" width="74.25" height="31.5" align-x="91.5" align-y="5052" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">MSE</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">SSE</ml:id><ml:id xml:space="preserve" subscript="RESID">DF</ml:id></ml:apply></ml:define></math><rendering item-idref="85" /></region><region region-id="1337" left="36" top="5078.25" width="282" height="39" align-x="36" align-y="5088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><c val="#000">and can be used as an estimate of the population error variance, </c><f family="Symbol" charset="2"><c val="#000">s</c></f><c val="#000"><sup>2</sup></c><c val="#000">, that is, the variance unexplained by the regression model. Similarly, the mean square regression, </c></p></text></region><region region-id="1370" left="0" top="5124" width="6000" height="6" align-x="0" align-y="5124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1371" left="66" top="5138.25" width="66.75" height="31.5" align-x="93" align-y="5154" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">MSR</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">SSR</ml:id><ml:id xml:space="preserve" subscript="REG">DF</ml:id></ml:apply></ml:define></math><rendering item-idref="86" /></region><region region-id="1339" left="36" top="5180.25" width="260.25" height="24" align-x="36" align-y="5190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">supplies us with an estimate for the variance that is explained by the regression.</p></text></region><region region-id="1621" left="36" top="5221.5" width="66.75" height="12.75" align-x="36" align-y="5232" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The F Statistic</p></text><indexes><phrase>F test</phrase></indexes></region><region region-id="1590" left="36" top="5240.25" width="300" height="60" align-x="36" align-y="5250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The ratio of explained variance MSR to unexplained variance MSE gives us another measure of the regression relationship. The test will tell us whether it's reasonable to model our data with a linear regression. Under the null hypothesis of</p></text></region><region region-id="1372" left="66" top="5306.25" width="231.75" height="27" align-x="66" align-y="5316" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">H</c></inlineAttr></b></f><f family="Times New Roman" charset="0" size="12"><b><inlineAttr line-through="false"><c val="#000"><sub>0</sub></c></inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">:</c></inlineAttr></b></f><c val="#000"> no linear regression relationship (that is, the slope is equal to zero)</c></p></text></region><region region-id="1641" left="36" top="5336.25" width="84.75" height="12" align-x="36" align-y="5346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">the test statistic</p></text></region><region region-id="1471" left="66" top="5354.25" width="45.75" height="27.75" align-x="80.25" align-y="5370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">Ft</ml:id><ml:apply><ml:div /><ml:id xml:space="preserve">MSR</ml:id><ml:id xml:space="preserve">MSE</ml:id></ml:apply></ml:define></math><rendering item-idref="87" /></region><region region-id="1654" left="162" top="5361" width="51" height="12.75" align-x="175.5" align-y="5370" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">Ft</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>19.31486319171028</ml:real></result></ml:eval></math><rendering item-idref="88" /></region><region region-id="1473" left="36" top="5390.25" width="126" height="12" align-x="36" align-y="5400" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">has an F distribution with</p></text></region><region region-id="1481" left="66" top="5415" width="52.5" height="16.5" align-x="81.75" align-y="5424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">n1</ml:id><ml:id xml:space="preserve" subscript="REG">DF</ml:id></ml:define></math><rendering item-idref="89" /></region><region region-id="1482" left="162" top="5414.25" width="18" height="12" align-x="162" align-y="5424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">and</p></text></region><region region-id="1615" left="222" top="5415" width="61.5" height="16.5" align-x="237.75" align-y="5424" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">n2</ml:id><ml:id 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xmlns="http://schemas.mathsoft.com/math20"><ml:real>4.73741412777588</ml:real></result></ml:eval></math><rendering item-idref="93" /></region><region region-id="1646" left="36" top="5510.25" width="174.75" height="12" align-x="36" align-y="5520" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The p-value of the test is given by</p></text></region><region region-id="1647" left="66" top="5529" width="89.25" height="12.75" align-x="77.25" align-y="5538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:define><ml:id xml:space="preserve">p</ml:id><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:id 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style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The small value of p and <region region-id="1656" left="161.25" top="5553" width="51.75" height="16.5" align-x="195.75" align-y="5562" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:greaterThan /><ml:id xml:space="preserve">Ft</ml:id><ml:id xml:space="preserve" subscript="a">F</ml:id></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>1</ml:real></result></ml:eval></math><rendering item-idref="96" /></region>means we can reject the null hypothesis, and we have <region region-id="1655" left="184.5" top="5570.25" width="427.5" height="13.5" align-x="212.25" align-y="5580" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" 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/></region><region region-id="1556" left="114" top="5667" width="90" height="18.75" align-x="135.75" align-y="5682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">SST</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><unitedValue><ml:real>6855200.0000000009</ml:real><unitMonomial xmlns="http://schemas.mathsoft.com/units10"><unitReference unit="second" power-numerator="2" /></unitMonomial></unitedValue></result></ml:eval></math><rendering item-idref="106" /></region><region region-id="1557" left="306" top="5667" width="75.75" height="18.75" align-x="316.5" align-y="5682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:id xml:space="preserve">p</ml:id><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.0014140853211306315</ml:real></result></ml:eval></math><rendering item-idref="107" /></region><region region-id="1629" left="36" top="5701.5" width="137.25" height="12.75" align-x="36" align-y="5712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Coefficient of Determination:</p></text></region><region region-id="1651" left="210" top="5697" width="79.5" height="18.75" align-x="256.5" align-y="5712" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:eval><ml:apply><ml:pow /><ml:apply><ml:id xml:space="preserve">corr</ml:id><ml:sequence><ml:id xml:space="preserve">x</ml:id><ml:id xml:space="preserve">y</ml:id></ml:sequence></ml:apply><ml:real>2</ml:real></ml:apply><result xmlns="http://schemas.mathsoft.com/math20"><ml:real>0.84659123438129069</ml:real></result></ml:eval></math><rendering item-idref="108" /></region><region region-id="1570" left="0" top="5724" width="6000" height="6" align-x="0" align-y="5724" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><pageBreak /></region><region region-id="1571" left="36" top="5732.25" width="195.75" height="12" align-x="36" align-y="5742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">At a glance, we can see the following. </p></text></region><region region-id="1572" left="36" top="5756.25" width="9.75" height="12" align-x="36" align-y="5766" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">1.</p></text></region><region region-id="1573" left="54" top="5756.25" width="267" height="36" align-x="54" align-y="5766" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The amount of variability explained by the linear regression (<i><inlineAttr line-through="false">MSR</inlineAttr></i>) is larger than the amount due to residual error (<i><inlineAttr line-through="false">MSE</inlineAttr></i>) occurring naturally in the sample.</p></text></region><region region-id="1574" left="36" top="5810.25" width="9.75" height="12" align-x="36" align-y="5820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">2.</p></text></region><region region-id="1575" left="54" top="5810.25" width="280.5" height="36" align-x="54" align-y="5820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">This difference is large enough to conclude that there is significant evidence (the p-value is close to 0) that a linear relationship exists between time and distance.</p></text></region><region region-id="1576" left="36" top="5864.25" width="9.75" height="12" align-x="36" align-y="5874" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">3.</p></text></region><region region-id="1633" left="54" top="5864.25" width="285" height="24" align-x="54" align-y="5874" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The coefficient of determination 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