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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
a. Positive correlation. b. Negative correlation. c. Positive correlation. d. No or weak correlation. e. Positive correlation. f. Can't determine, likely positive correlation. g. Negative correlation. h. Both positive and negative correlation.

Step by step solution

01

Analyze the Relation between Variables

a. There is a positive correlation between daily temperature and cooling costs. The hotter the day, the more cooling or air condition will be required, resulting in higher energy consumption and thus costs. b. There is generally a negative correlation between interest rates and the number of loan applications. Higher interest rates make borrowing expensive, thus fewer applications are to be expected. c. We would expect a positive correlation in the incomes of husbands and wives when both have fulltime jobs. They are both working full time, which likely means they both have steady income. Also, they might be in similar socio-economic situations since they are married. d. There is likely no or weak correlation (close to 0) between height and IQ. These are relatively independent characteristics. e. There is a positive correlation between height and shoe size. Typically, taller people have larger shoe size. f. The correlation between the score on the math section and the verbal section of the SAT exam could vary significantly based on the individual. However, there could be a positive correlation if they are students who are generally good in academics. g. There is likely a negative correlation between time spent on homework and time spent watching television during the same day by elementary school children. The more time a child spends on homework, the less time they have to watch television, and vice versa. h. Up to a certain point, there is a positive correlation between the amount of fertilizer used per acre and crop yield. However, after that point, the correlation turns negative as more fertilizer can damage the yield. This is an example of a complex relationship that wouldn't fit neatly into positive or negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Correlation
Positive correlation in statistics is an association between two variables in which they both move in tandem鈥攊t鈥檚 a relationship where if one variable increases, the other one also increases. For example, the exercise mentions maximum daily temperature and cooling costs as positively correlated. This is because on hotter days, the need for cooling is higher, thus increasing related expenses.
Similarly, husbands and wives both with full-time jobs will typically earn more together than individually, indicating a positive correlation between their incomes. Identifying these relationships help in predicting outcomes and understanding trends in various fields, including economics and social sciences.
Negative Correlation
Negative correlation occurs when an increase in one variable leads to a decrease in another. In our exercise, the interest rate and the number of loan applications provide a perfect illustration. As interest rates climb, fewer people apply for loans due to the higher cost of borrowing.
Also, elementary school children's time spent on homework versus television watching is another example. More homework typically means less time available for watching TV. Being able to spot negative correlations is crucial for risk management and establishing preventative measures in policy and personal decisions.
Correlation Close to Zero
Sometimes, variables may exhibit a correlation close to zero, indicating no discernible association between their movements. Height and IQ are mentioned in the example; these traits are largely independent of each other, resulting in a very weak or nonexistent correlation.
Recognizing when correlations are negligible is as vital as identifying strong correlations, as it helps avoid misdirected efforts in research and strategy in fields like psychology and human development, where drawing connections between unrelated variables can lead to erroneous conclusions.
Correlation and Causation
A common misconception is that correlation implies causation. However, just because two variables change in consonance, it does not mean one directly causes the change in the other. While height and shoe size do correlate (taller people often need larger shoes), the exercise's suggestion of a correlation between SAT scores in math and verbal sections requires more caution. A positive correlation may exist, but it doesn't imply that proficiency in one causes proficiency in the other.
Understanding the distinction between correlation and causation is fundamental in research methodology and helps to prevent fallacies in logic when interpreting data.
Variables Relationship Analysis
The relationship analysis between variables can often be complex, which is eloquently displayed in the relationship between fertilizer use and crop yield. At first, as fertilizer use increases, crop yields tend to increase, showing a positive correlation. However, beyond a certain point, more fertilizer can lead to decreased yields鈥攁 negative correlation.
This nuance underscores the importance of careful variables relationship analysis, demonstrating that reliance on simple correlation coefficients without context can be misleading, particularly in scientific and agricultural studies where many factors interact in intricate ways.

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Most popular questions from this chapter

The paper "Effects of Age and Gender on Physical Performance" (Age [2007]: 77-85) describes a study of the relationship between age and 1 -hour swimming performance. Data on age and swim distance for over 10,000 men participating in a national long-distance 1 -hour swimming competition are summarized in the accompanying table. In Exercise 5.34 from Section 5.3 , a plot of the residuals from the least-squares line showed a curved pattern that suggested that a quadratic curve would do a better job of summarizing the relationship between \(x=\) representative age and \(y=\) average swim distance. Find the equation of the least-squares quadratic curve and use it to predict average swim distance at 40 years of age. \(\hat{y}=3843.027+10.619 x-0.360 x^{2}\) \(\hat{y}=3691.293\)

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 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.

The relationship between hospital patient-tonurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables, indicate whether you expect the slope of the least-squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of patient quality of care.

Some plant viruses are spread by insects and tend to spread from the edges of a field inward. The data on \(x=\) distance from the edge of the field (in meters) and \(y=\) proportion of plants with virus symptoms that appeared in the paper "Patterns of Spread of Two NonPersistently Aphid-Borne Viruses in Lupin Stands" \((A n-\) nals of Applied Biology [2005]: \(337-350\) ) was used to fit a least-squares regression line to describe the relationship between \(x\) and \(y^{\prime}=\ln \left(\frac{p}{1-p}\right) .\) Minitab output resulting from fitting the least-squares line is given below. The regression equation is \(\ln (p /(1-p))=-0.917-0.107\) Distance to Crop Edge \(\begin{array}{lrrrr}\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P. } \\ \text { Constant } & -0.9171 & 0.1249 & -7.34 & 0.000\end{array}\) \begin{tabular}{lrrrr} Constant & -0.9171 & 0.1249 & -7.34 & 0.000 \\ Distance to Crop Edge & -0.10716 & 0.01062 & -10.09 & 0.000 \\ \hline \end{tabular} \(\mathrm{S}=0.387646 \quad \mathrm{R}-\mathrm{Sq}=72.8 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{adj})=72.1 \%\) a. What is the logistic regression function relating \(x\) and the proportion of plants with virus symptoms? b. What would you predict for the proportion of plants with virus symptoms at a distance of 15 meters from the edge of the field? (Note: the \(x\) values in the data set ranged from 0 to \(20 .)\)

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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