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Chapter 5: Problem 1

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Maximum daily temperature and cooling costs b. Interest rate and number of loan applications c. Incomes of husbands and wives when both have fulltime jobs d. Height and IQ e. Height and shoe size f. Score on the math section of the SAT exam and score on the verbal section of the same test g. Time spent on homework and time spent watching television during the same day by elementary school children h. Amount of fertilizer used per acre and crop yield (Hint: As the amount of fertilizer is increased, yield tends to increase for a while but then tends to start decreasing.)

Short Answer

Expert verified
The expected correlations are as follows: a. Positive, b. Negative, c. Positive, d. Close to 0, e. Positive, f. Generally Positive, g. Negative, h. Initially Positive then Negative.

Step by step solution

01

Maximum daily temperature and cooling costs

This pair is likely to have a positive correlation since the cooler a building needs to be to counteract the heat, the higher the associated costs.
02

Interest rate and number of loan applications

This pair is likely to have a negative correlation. When interest rates rise, loans become more expensive which can discourage people from applying for them.
03

Incomes of husbands and wives when both have fulltime jobs

This pair is likely to have a positive correlation because couples often have similar levels of education and work in similar sectors, which can lead to similar incomes.
04

Height and IQ

This pair is expected to have a correlation close to zero, as there is not generally a link between a person's height and their intellectual capabilities.
05

Height and shoe size

This pair is likely to have a positive correlation as people with larger body sizes usually have larger shoe sizes.
06

Score on the math section of the SAT exam and score on the verbal section

Depending on the student, this pair could have a positive, negative, or no correlation. However, students who work hard and are academically gifted are likely to score high in both sections, so this pair generally is expected to have a positive correlation.
07

Time spent on homework and time spent watching television during the same day by elementary school children

This pair is likely to have a negative correlation. The more time a child spends on homework, the less free time they have for activities like television watching.
08

Amount of fertilizer used per acre and crop yield

This pair is likely to display first a positive correlation and then a negative one. As the hint indicates, initially increasing fertilizer tends to increase yield, but after a certain point, adding more fertilizer starts to decrease yield.

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Most popular questions from this chapter

The article "Examined Life: What Stanley \(\mathrm{H}\). Kaplan Taught Us About the SAT" (The New yorker [December 17, 2001]: 86-92) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict firstyear college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at \(15.4\) percent, and SAT I was last at \(13.3\) percent." a. If the data from this study were used to fit a leastsquares line with \(y=\) first-year college GPA and \(x=\) high school GPA, what would be the value of \(r^{2}\) b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would be very accurate? Explain why or why not.

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.

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.

Determining the age of an animal can sometimes be a difficult task. One method of estimating the age of harp seals is based on the width of the pulp canal in the seal's canine teeth. To investigate the relationship between age and the width of the pulp canal, researchers measured age and canal width in seals of known age. The following data on \(x=\) age (in years) and \(y=\) canal length (in millimeters) are a portion of a larger data set that appeared in the paper "Validation of Age Estimation in the Harp Seal Using Dentinal Annuli" (Canadian journal of Fisheries and Aquatic Science [1983]: \(1430-1441):\) \(0.75\) \(\begin{array}{rrrrrr}x & 0.25 & 0.25 & 0.50 & 0.50 & 0.50 \\ y & 700 & 675 & 525 & 500 & 400\end{array}\) \(\begin{array}{rr}0.75 & 1.00 \\ 300 & 300\end{array}\) 350 \(\begin{array}{ll}1.25 & 1.25\end{array}\) \(\begin{array}{rrrrrr}x & 1.00 & 1.00 & 1.00 & 1.00 & 1 \\ y & 250 & 230 & 150 & 100 & 2\end{array}\) 1 \(\begin{array}{rr}1.50 & 1.50 \\ 100 & 125\end{array}\) \(y \quad 250\) 230 150 \(\begin{array}{ccccc}100 & 200 & 100 & 100 & 125\end{array}\) \(\begin{array}{cc}4.00 & 5.00\end{array}\) \(\begin{array}{rrrrrrr}x & 2.00 & 2.00 & 2.50 & 2.75 & 3.00 & 4.00 \\ y & 60 & 140 & 60 & 50 & 10 & 10\end{array}\) \(10 \quad 10\) \(\begin{array}{lll}5.00 & 6.00 & 6.00\end{array}\) \(\begin{array}{rrrr}x & 5.00 & 5.00 & 5 \\ y & 15 & 10-0\end{array}\) \(\begin{array}{llll}10 & 10 & 15 & 10\end{array}\) Construct a scatterplot for this data set. Would you describe the relationship between age and canal length as linear? If not, suggest a transformation that might straighten the plot.

The accompanying data were read from graphs that appeared in the article "Bush Timber Proposal Runs Counter to the Record" (San Luis Obispo Tribune, September 22,2002 ). The variables shown are the number of acres burned in forest fires in the western United States and timber sales. \begin{tabular}{lcc} & Number of Acres Burned (thousands) & Timber Sales (billions of board feet) \\\ \hline 1945 & 200 & \(2.0\) \\ 1950 & 250 & \(3.7\) \\ 1955 & 260 & \(4.4\) \\ 1960 & 380 & \(6.8\) \\ 1965 & 80 & \(9.7\) \\ 1970 & 450 & \(11.0\) \\ 1975 & 180 & \(11.0\) \\ 1980 & 240 & \(10.2\) \\ 1985 & 440 & \(10.0\) \\ 1990 & 400 & \(11.0\) \\ 1995 & 180 & \(3.8\) \\ \hline \end{tabular} a. Is there a correlation between timber sales and acres burned in forest fires? Compute and interpret the value of the correlation coefficient. b. The article concludes that "heavier logging led to large forest fires." Do you think this conclusion is justified based on the given data? Explain.
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