12. Regression

2. Compare the numbers of dependent and independent variables in a multiple regression equation and a single regression equation.

You Explain It! Study Time and Exam Scores

After the first exam in a statistics course, Professor Katula surveyed 14 randomly selected students to determine the relation between the amount of time they spent studying for the exam and exam score. She found that a linear relation exists between the two variables. The least-squares regression line that describes this relation is:

ŷ = 6.3333x + 53.0298

a. Predict the exam score of a student who studied 2 hours.

[DATA] Calories versus Sugar The following data represent the number of calories per serving and the number of grams of sugar per serving for a random sample of high-protein and moderate-protein energy bars.

g. If the residuals are normally distributed, construct a 95% confidence interval about the slope of the true least-squares regression line.

6. Why is it not appropriate to use a regression line to predict y-values for x-values that are not in (or close to) the range of x-values found in the data?

Regression and Predictions

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.

Find the regression equation, letting the first variable be the predictor (x) variable.

Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.

Taxis Use the distance/fare data from Exercise 15 and find the best predicted fare amount for a distance of 3.10 miles. How does the result compare to the actual fare of \$15.30?

[DATA] Concrete [See Problem 15 in Section 12.3] As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

d. State your conclusion to the hypotheses from part (b).

"[DATA] Credit Scores [See Problem 12 in Section 12.3] An economist wants to determine the relation between one’s FICO score, x, and the interest rate of a 36-month auto loan, y. The data represent the interest rate (in percent) a bank might offer on a 36-month auto loan for various FICO scores.

b. Determine the slope of the least-squares regression line between credit score and interest rate."

"In Exercises 17 and 18, use the data to (a) find the coefficient of determination r^2 and interpret

the result, and (b) find the standard error of estimate s_e and interpret the result.

18. [APPLET] The table shows the cooking areas (in square inches) of 18 gas grills and their prices (in dollars). The regression equation is y = 1.501x - 341.501. (Source: Lowe's)

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

28. Total Assets Construct a 90% prediction interval for the total assets in federal defined benefit plans in Exercise 18 when the total assets in IRAs are \$6400 billion."

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.

30. New Vehicle Sales Construct a 99% prediction interval for new vehicle sales for Honda in Exercise 20 when the number of new vehicles sold by Toyota is 2159 thousand."

[DATA] Concrete [See Problem 15 in Section 12.3] As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

b. Suppose a researcher wanted to determine if there is a linear relation between 7-day strength and 28-day strength. What would be the null and alternative hypotheses?

The _____ _____ _____, R^2, quantifies the proportion of total variation in the response variable explained by the least-squares regression line.

[DATA] Putting It Together: Predicting Intelligence Can a photograph of an individual be used to predict their intelligence? Researchers at Charles University in Prague, Czech Republic, had 160 raters analyze the photos of 80 students and asked each rater to rate the intelligence and attractiveness of the individual in the photo on a scale from one to seven. To eliminate individual bias in ratings, each rater’s scores were converted to z-scores using each individual’s mean rating. The perceived intelligence and attractiveness of each photo was calculated as the mean z-score. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_4_19 using the file format of your choice. The following explains the variables in the data:

sex: Gender of the individual in the photo

age: Age of the individual in the photo

perceived intelligence (ALL): Mean z-score of the perceived intelligence of all 160 raters

perceived intelligence (WOMEN): Mean z-score of the perceived intelligence of the female raters

perceived intelligence (MEN): Mean z-score of the perceived intelligence of the male raters

attractiveness (ALL): Mean z-score of the attractiveness rating of all 160 raters

attractiveness (MEN): Mean z-score of the attractiveness rating of the male raters

attractiveness (WOMEN): Mean z-score of the attractiveness rating of the female raters

IQ: Intelligence quotient based on the Czech version of Intelligence Structure Test

e. A normal probability plot confirms the residuals are normally distributed. Test whether a positive linear relation exists between perceived intelligence and attractiveness.

In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:

b. Compute the standard error, the point estimate for σ.

"1. Net Sales The equation used to predict the net sales (in millions of dollars) for a fiscal

year for a clothing retailer is y=23,769 + 9.18x_1 - 8.41x_2

where x_1 is the number of stores open at the end of the fiscal year and x_2 is the average

square footage per store. Use the multiple regression equation to predict the y-values for

the values of the independent variables.

a. x_1 = 1057, x_2 = 3698

b. x_1 = 1012, x_2 = 3659

c. x_1 = 952, x_2 = 3601

d. x_1 = 914, x_2 = 3594"