Chapter 5: Problem 3
Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\).
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A sample of 548 ethnically diverse students from Massachusetts were followed over a 19 -month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: 1321- 1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by \(0.14\) serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.
According to the article "First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" \((\) Journal of College Student Development \([1999]: 599-\) 605), there is a mild correlation between high school GPA \((x)\) and first-year college GPA \((y)\). The data can be summarized as follows: $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ An alternative formula for computing the correlation coefficient that is based on raw data and is algebraically equivalent to the one given in the text is $$ r=\frac{\sum x y-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to compute the value of the correlation coefficient, and interpret this value.
The article "Cost-Effectiveness in Public Education" (Chance [1995]: \(38-41\) ) reported that for a regression of \(y=\) average SAT score on \(x=\) expenditure per pupil, based on data from \(n=44\) New Jersey school districts, \(a=766, b=0.015, r^{2}=.160\), and \(s_{e}=53.7\) a. One observation in the sample was \((9900,893)\). What average SAT score would you predict for this district, and what is the corresponding residual? b. Interpret the value of \(s_{e}\). c. How effectively do you think the least-squares line summarizes the relationship between \(x\) and \(y ?\) Explain your reasoning.
Consider the four \((x, y)\) pairs \((0,0),(1,1),(1,-1)\), and \((2,0)\). a. What is the value of the sample correlation coefficient \(r ?\) b. If a fifth observation is made at the value \(x=6\), find alue of \(y\) for which \(r>.5\). c. If a fifth observation is made at the value \(x=6\), find â value of \(y\) for which \(r<.5\).
Anabolic steroid abuse has been increasing despite increased press reports of adverse medical and psychiatric consequences. In a recent study, medical researchers studied the potential for addiction to testosterone in hamsters (Neuroscience [2004]: \(971-981\) ). Hamsters were allowed to self-administer testosterone over a period of days, resulting in the death of some of the animals. The data below show the proportion of hamsters surviving versus the peak self-administration of testosterone \((\mu \mathrm{g})\). Fit a logistic regression equation and use the equation to predict the probability of survival for a hamster with a peak intake of \(40 \mu \mathrm{g} . \quad \ln \left(\frac{P}{1-p}\right)=4.589-0.0659 x ; 0.876\)
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