12. Regression

If the linear correlation between two variables is negative, what can be said about the slope of the regression line?

"Old Vehicles In Exercises 31–34, use the figure shown at the left.

Regression Line Find and draw the regression line."

"Predicting y-Values In Exercises 3-6, use the multiple regression equation to predict the y-values for the values of the independent variables.

4. Sorghum Yield The equation used to predict the annual sorghum yield (in bushels per

acre) is y = 80.1-20.2x_1 +21.2x_2

where x_1 is the number of acres planted (in millions) and x_2 is the number of acres harvested (in millions). (Adapted from United States Department of Agriculture)

a. x_1 = 5.5, x_2 = 3.9

b. x_1 = 8.3, x_2 = 7.3

c. x_1 = 6.5, x_2 = 5.7

d. x_1 = 9.4, x_2= 7.8"

[DATA] Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following table lists the temperature (in degrees Fahrenheit, °F) and the number of chirps per second for the striped ground cricket.

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

"The U.S. Food and Drug Administration (FDA) requires nutrition labeling for most foods. Un FDA regulations, manufacturers are required to list the amounts of certain nutrients in their foods, such as calories, sugar, fat, and carbohydrates. This nutritional information is displayed in the ""Nutrition Facts"" panel on the food's package.

The table shows the nutritional content below for one cup of each of 21 different breakfast

cereals.

C = calories

S = sugar in grams

F = fat in grams

R = carbohydrates in grams

7. Use the equations from Exercise 6 to predict the calories in 1 cup of cereal that has 7 grams of sugar, 0.5 gram of fat, and 31 grams of carbohydrates."

Bull Markets A bull market is defined as a market condition in which the price of a security rises for an extended period of time. A bull market in the stock market is often defined as a condition in which a market rises by 20% or more without a 20% decline. The data to the right represent the number of months and percentage change in the S&P 500 (a group of 500 stocks) during the 25 bull markets dating back to 1929 (the year of the famous market crash).

d. Find the least-squares regression line treating length of the bull market as the explanatory variable.

[DATA] Concrete 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:

e. Assuming the residuals are normally distributed, construct a 95% confidence interval for the slope of the true least-squares regression line.

[DATA] Concrete 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:

c. Determine sb_1.

[DATA] Age versus HDL Cholesterol A doctor wanted to determine whether there is a relation between a male’s age and his HDL (so-called good) cholesterol. He randomly selected 17 of his patients and determined their HDL levels. He obtained the following data.

d. Assuming the residuals are normally distributed, construct a 95% confidence interval for the slope of the true least-squares regression line.

"[APPLET] For Exercises 2–9, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years. (Source: U.S. Bureau of Labor Statistics)

6. Use the regression equation that you found in Exercise 5 to predict the average annual salary of postsecondary library science teachers when the average annual salary of librarians is \$61,000."

[NOW WORK] Using the sample data from Problem 5 in Section 12.3,

a. Predict the mean value of y if x = 7.

"In Exercises 27 and 28, use the multiple regression equation to predict the y-values for the values of the independent variables.

27. An equation that can be used to predict fuel economy (in miles per gallon) for automobiles is

y=41.3- 0.004x_1 - 0.0049x_2

where x_1 is the engine displacement (in cubic inches) and x_2 is the vehicle weight (in

pounds).

a. x_1 = 305, x_2 = 3750

b. x_1 = 225, x_2 = 3100

c. x_1 = 105, x_2 = 2200

d. x_1 = 185, x_2 = 3000"

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

d. A student who studied 5 hours for the exam scored 81 on the exam. Is this student’s exam score above or below average among all students who studied 5 hours?

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.