invest_3ed.pdf

# Discussion as you have seen several times in this

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Discussion: As you have seen several times in this course, the sampling distribution of a statistic can often be well described by a mathematical model. This is also true in the regression setting. As when we focused on analyzing the individual population mean or comparing two population means, we need to first consider a way to estimate the (nuisance) standard deviation parameter.

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Chance/Rossman, 2015 ISCAM III Investigation 5.10 388 (y) If V represents the common standard deviation of the response values about the regression line, suggest a way of estimating V from the sample data. Discussion: If the conditions for the basic regression model (Investigation 5.12) are met, then if we standardize the sample slopes: t 0 = ) ( 1 1 1 b SE b E ± where 2 1 ) 1 ( 1 ) ( x s n s b SE ± and s = ¦ ± n i i n residuals 1 2 2 ) ( , these standardized values can be shown to be well approximated by a t- distribution with n ± 2 degrees of freedom. Note that we lose a second degree of freedom because we are estimating both the slope and the intercept from the sample data. (z) Using the standard deviation for the slopes you found in (k), calculate the test statistic. Once we have the t -statistic or t -ratio, we can use the Student ’s t- distribution to compute one- or two- sided p-values depending on the research question regarding the slope. Note: The t- distribution with n ± 2 degrees of freedom also provides a reasonable model for a randomization distribution that repeatedly re-randomizes the values of the explanatory variable and looks at the distribution of the resulting sample slopes (see Applet Exploration). (aa) Confirm the t -ratio and p-value calculations using technology. x In R: Use > summary(lm(time~age)) x In Minitab: Select Stat > Regression > Regression > Fit Regression Model x In Applet : Uncheck the Create Population box, press Revert , and check the Regression Table box. Practice Problem 5.10A Recreate this analysis (simulation as in (h) and (w) and t -test as in (z)) without first removing the outlier. Do the results change much? If so, how? In the expected manner? Explain. Practice Problem 5.10B The official published results for the 2013 race can be found here . (a) Load these data into your technology and then compare the two datasets. (b) Identify a second difference (apart from Scharlach_1 disappearing) between the unofficial and the official results. (c) How does the strength of evidence of a relationship between time and age differ for this dataset?
Chance/Rossman, 2015 ISCAM III Investigation 5.11 389 Investigation 5.11: Running Out of Time (cont.) In the previous investigation, we analyzed the statistical significance of our sample data by replicating how often we might get a random sample of 247 runners (after removing an extreme outlier) with a slope this extreme from a larger population of runners that did not have an association between finishing time and age . In other words, we assumed there was no association between finishing time and age and the pairing of these values for our runners was just arbitrary. This suggests another way of assessing the statistical significance of our results.

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