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Chapter 5: Problem 3
Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\).
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Each individual in a sample was asked to indicate on a quantitative scale how willing he or she was to spend money on the environment and also how strongly he or she believed in God ("Religion and Attitudes Toward the Environment," Journal for the Scientific Study of Religion [1993]: \(19-28\) ). The resulting value of the sample correlation coefficient was \(r=-.085 .\) Would you agree with the stated conclusion that stronger support for environmental spending is associated with a weaker degree of belief in God? Explain your reasoning.
The following data on \(x=\) score on a measure of test anxiety and \(y=\) exam score for a sample of \(n=9\) students are consistent with summary quantities given in the paper "Effects of Humor on Test Anxiety and Performance" (Psychological Reports [1999]: 1203-1212): $$ \begin{array}{rrrrrrrrrr} x & 23 & 14 & 14 & 0 & 17 & 20 & 20 & 15 & 21 \\ y & 43 & 59 & 48 & 77 & 50 & 52 & 46 & 51 & 51 \end{array} $$ Higher values for \(x\) indicate higher levels of anxiety. a. Construct a scatterplot, and comment on the features of the plot. b. Does there appear to be a linear relationship between the two variables? How would you characterize the relationship? c. Compute the value of the correlation coefficient. Is the value of \(r\) consistent with your answer to Part (b)? d. Is it reasonable to conclude that test anxiety caused poor exam performance? Explain.
A sample of automobiles traversing a certain stretch of highway is selected. Each one travels at roughly a constant rate of speed, although speed does vary from auto to auto. Let \(x=\) speed and \(y=\) time needed to traverse this segment of highway. Would the sample correlation coefficient be closest to \(.9, .3,-.3\), or \(-.9 ?\) Explain.
The following data on \(x=\) soil depth (in centimeters) and \(y=\) percentage of montmorillonite in the soil were taken from a scatterplot in the paper "Ancient Maya Drained Field Agriculture: Its Possible Application Today in the New River Floodplain, Belize, C.A." (Agricultural Ecosystems and Environment \([1984]: 67-84)\) : $$ \begin{array}{lllllllr} x & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\ y & 58 & 34 & 32 & 30 & 28 & 27 & 22 \end{array} $$ a. Draw a scatterplot of \(y\) versus \(x\). b. The equation of the least-squares line is \(\hat{y}=64.50-\) \(0.45 x\). Draw this line on your scatterplot. Do there appear to be any large residuals? c. Compute the residuals, and construct a residual plot. Are there any unusual features in the plot?
An article on the cost of housing in California that appeared in the San Luis Obispo Tribune (March 30,2001 ) included the following statement: "In Northern California, people from the San Francisco Bay area pushed into the Central Valley, benefiting from home prices that dropped on average \(\$ 4000\) for every mile traveled east of the Bay area." If this statement is correct, what is the slope of the least-squares regression line, \(\hat{y}=a+b x\), where \(y=\) house price (in dollars) and \(x=\) distance east of the Bay (in miles)? Explain.
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