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Chapter 4: Problem 2

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. Interest rate and number of loan applications b. Height and \(\mathrm{IQ}\) c. Height and shoe size d. Minimum daily temperature and cooling cost

Short Answer

Expert verified
a. Negative correlation. b. Correlation close to 0. c. Positive correlation. d. Positive correlation.

Step by step solution

01

Analyze the relationship between interest rates and number of loan applications

Higher interest rates may deter individuals from applying for loans due to the increased cost of borrowing. As such, one could expect a negative correlation between these two variables.
02

Assess the relationship between height and IQ

Height and IQ are, generally speaking, not directly related. The height of an individual does not determine their intelligence, therefore one could expect a correlation close to 0.
03

Analyze the relationship between height and shoe size

Typically, taller individuals tend to have larger shoe sizes due to proportional body growth. Therefore, a positive correlation can be expected between these two variables
04

Evaluate the relationship between minimum daily temperature and cooling cost

As the minimum daily temperature increases, the need for cooling (via air conditioning for example) also increases, leading to higher cooling costs. As such, one would expect a positive correlation between these two variables.

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Most popular questions from this chapter

It may seem odd, but biologists can tell how old a lobster is by measuring the concentration of pigment in the lobster's eye. The authors of the paper "Neurolipofuscin Is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster, in Florida" (Biological Bulletin [2007]: 55-66) wondered if it was sufficient to measure the pigment in just one eye, which would be the case if there is a strong relationship between the concentration in the right eye and the concentration in the left eye. Pigment concentration (as a percentage of tissue sample) was measured in both eyes for 39 lobsters, resulting in the following summary quantities (based on data from a graph in the paper): $$ \begin{array}{cll} n=39 & \sum_{x}=88.8 & \sum y=86.1 \\ \sum x y=281.1 & \sum x^{2}=288.0 & \sum y^{2}=286.6 \end{array} $$ An alternative formula for calculating the correlation coefficient that doesn't involve calculating the z-scores 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 calculate the value of the correlation coefficient, and interpret this value.

Studies have shown that people who suffer sudden cardiac arrest have a better chance of survival if a defibrillator shock is administered very soon after cardiac arrest. How is survival rate related to the length of time between cardiac arrest and the defibrillator shock being delivered? This question is addressed in the paper "Improving Survival from Sudden Cardiac Arrest: The Role of Home Defibrillators" (www.heartstarthome.com). The accompanying data give \(y=\) survival rate (percent) and \(x=\) mean call-to-shock time (minutes) for a cardiac rehabilitation center (in which cardiac arrests occurred while victims were hospitalized and so the call-to-shock time tended to be short) and for four other communities of different sizes. Mean call-to-shock time, \(x\) 1 12 30 \begin{tabular}{l} \hline \end{tabular} Survival rate, \(y\) 90 45 5 a. Find the equation of the least squares line. b. Interpret the slope of the least squares line in the context of this study. c. Does it make sense to interpret the intercept of the least squares regression line? If so, give an interpretation. If not, explain why it is not appropriate for this data set. d. Use the least squares line to predict survival rate for a community with a mean call-to-shock time of 10 minutes.

The article "Examined Life: What Stanley 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 first-year 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 least squares regression 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.

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. Weight of a car and gas mileage b. Size and selling price of a house c. Height and weight d. Height and number of siblings

Briefly explain why it is important to consider the value of \(r^{2}\) in addition to the value of \(s\) when evaluating the usefulness of the least squares regression line.
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