invest_3ed.pdf

# How do you think this will change the behavior of the

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How do you think this will change the behavior of the distribution of sample slopes and the distribution of the sample intercepts? Change to scatterplot: Prediction: (q) Press the Draw Samples button. Was your conjecture in (p) correct?

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Chance/Rossman, 2015 ISCAM III Investigation 5.10 386 (r) Change the value of x std back to 16.89 ( Create Population ) and change the sample size from 247 to 125 . Conjecture what will happen to the sampling distribution of the sample slopes. (s) Press Draw Samples . Was your prediction correct? (t) Summarize how each of these quantities affect the sampling distribution of the sample slope: x The variability about the regression line, V (sigma) x The variability in the explanatory variable, SD(X) x The sample size, n . You should have made the following observations (when the technical conditions are met): o The distribution of sample slopes is approximately normal. o The mean of the distributions of sample slopes equals the population slope E 1 . o The variability in the sample slopes increases if we increase V . o The variability in the sample slopes increases if we decrease s x . o The variability in the sample slopes decreases if we increase n . (u) Explain why each of the last 3 observations make intuitive sense. [You may want to recall the pictures from the Applet Exploration.]
Chance/Rossman, 2015 ISCAM III Investigation 5.10 387 (v) Are the last three observations consistent with the following formula for the standard deviation of the sample slope, SD ( b 1 )? Explain. 2 1 ) 1 ( 1 ) ( x s n b SD ± V Back to the question at hand: Is it plausible that the observed sample slope ( b 1 = 0.140) came from a population with no association between the variables and so a population slope E 1 = 0? (w) Return to (or recreate) the dotplot of the sample slopes for the first simulation. Where does 0.140 fall in this null distribution? Is it plausible that the population slope is really zero and we obtained a sample slope as big as 0.140 just by chance? How often did such a sample slope occur in your 1000 samples? In all the samples obtained by the class? (x) Change the population slope to .10 and press Create Population . Draw 1000 samples and examine the new hypothesized population and the new sampling distribution of the sample slopes. (You may want to press Rescale .) Where is the center? Roughly how often did you get a sample slope as big as 0.140 or bigger? Does it seem plausible that the students’ regression line came from a population with E 1 = 0.10? Study Conclusions These sample data provide extremely strong evidence of a relationship between age and finishing time for the population of amateur runners similar to those in this sample. The above simulation shows that if there were no relationship in the population ( E 1 = 0), then we would pretty much never see a sample slope as large as 0.140 just by chance (random sampling). We should be a bit cautious in generalizing these results to a larger sample as they were not a random sample. We might want to limit ourselves to 5K racers in similar types of towns.

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