When you open peclass you will notice that the mile

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When you open PEClass you will notice that the mile run times have been recorded in “time format.” We want to convert this to numerical values. For example, in Minitab choose Data > Change Data Type > Data/Time to Numeric. Specify C3 as the column to be changed and C4 as the storage location for the converted values. The data are now numeric, but in terms of a 24-hour day. To convert these back to minutes, type MTB> let c5=c4*24 or use Calc > Calculator, letting miletime = 24*c4. Now you should have the number of minutes (including the fraction of minute) for each student. Because we aren’t considering either of these as a response variable, it does not matter which variable we denote as the y -variable and which as the x -variable. If we plot mile time vs. push-ups, we see there is a strong negative association. Students who do more push-ups also tend to run the mile in faster times. However, there is some evidence that the relationship is not linear. There is also an unusual observation, a student who did a larger number of push-ups but was one of the slowest runners. Carrying out the regression and examining residual plots confirms these observations.
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Chance/Rossman, 2015 ISCAM III Example 5.3 412 The residual vs. explanatory variable graph also reveals some differences in the amount of variation in the residuals at different values of the explanatory variable (indicating a violation of the constant variance condition). It appears that transforming these data might be helpful. Because both distributions appear skewed to the right (if you looked at histograms of each variable individually), we could considering taking the log of each variable. The log-log scatterplot does appear more well behaved. (We have used log base ten but natural logs would also work.) We still have some outliers but appear to now have a linear relationship. If we also examine the normality of the residuals: This condition also seems to be reasonably met for the transformed data. There is slight evidence of skewness to the right in the residuals but coupled with the large sample size we will not be concerned with this minor deviation. The correlation coefficient for the transformed variables is ˗0 .624, indicating a moderately strong negative linear relationship between log(time) and log(push-ups). The least-squares regression equation is computed by Minitab to be ^ log mile = 1.23 ± 0.213 log( push-ups ). The intercept coefficient here would indicate the predicted log-time for a student who only completes 1 push up (so log( push-ups ) = 0) to be 1.23. This corresponds to a time of 10 1.23 or about 17 minutes. The slope coefficient predicts the average multiplicative change in the log-mile times for each unit increase in log( push-ups ). A unit increase in “log( push-ups )” corresponds to the push -ups increasing by a factor of 10. So for each 10 fold increase in the number of push-ups (e.g., 1 push up to 10 push-ups), the mile time decreases on average by a factor of 10 ± .213 = 0.61. [Note: our prediction for 10 push-ups is 1.017, corresponding to 10 1.107 | 10.4 minutes, which is 0.61(17).]
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