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Ch. 9 - Correlation and Regression
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 9 - Correlation and RegressionProblem 9.R.24
Chapter 9, Problem 9.R.24

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.
24. Construct a 99% prediction interval for the price of a gas grill in Exercise 18 with a usable cooking area of 900 square inches."

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Identify the regression equation from Exercise 18, which relates the price of a gas grill to its usable cooking area. This equation typically has the form: \(\hat{y}\) = b_0 + b_1 x, where \(\hat{y}\) is the predicted price and x is the cooking area.
Calculate the predicted price \(\hat{y}\) for a grill with 900 square inches of cooking area by substituting x = 900 into the regression equation.
Find the standard error of the prediction, which accounts for both the variability in the estimate of the mean response and the variability of individual observations. This involves the residual standard error from the regression and the formula for prediction intervals: SE_{pred} = s \(\sqrt{1 + \frac{1}{n}\) + \(\frac{(x_0 - \bar{x}\))^2}{\(\sum\) (x_i - \(\bar{x}\))^2}}, where s is the standard error of the estimate, n is the sample size, x_0 is 900, and \(\bar{x}\) is the mean of the cooking areas.
Determine the critical t-value for a 99% prediction interval with n - 2 degrees of freedom from the t-distribution table.
Construct the prediction interval using the formula: \(\hat{y}\) \(\pm\) t^* \(\times\) SE_{pred}. This interval estimates the range in which the price of a gas grill with 900 square inches of cooking area is likely to fall with 99% confidence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, with a specified level of confidence. Unlike confidence intervals for the mean, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual observations.
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Confidence Level (99%)

The confidence level represents the probability that the interval contains the true value. A 99% confidence level means we are 99% confident that the prediction interval includes the actual price of a gas grill with 900 square inches of cooking area.
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Regression and Usable Cooking Area as Predictor

Regression analysis models the relationship between a dependent variable (price) and an independent variable (usable cooking area). Using the regression equation from Exercise 18, the cooking area of 900 square inches is plugged in to predict the price and construct the prediction interval.
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Related Practice
Textbook Question

"9. Stock Price The equation used to predict the stock price (in dollars) at the end of the year for a restaurant chain is y=- 86+7.46x_1 - 1.61x_2

where x_1 is the total revenue (in billions of dollars) and x_2 is the shareholders' equity (in

billions of dollars). Use the multiple regression equation to predict the y-values for the

values of the independent variables.

a. x_1 = 27.6, x_2 = 15.3

b. x_1 = 24.1, x_2 = 14.6

c. x_1 = 23.5, x_2 = 13.4

d. x_1 = 22.8, x_2 =15.3"

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Textbook Question

"[APPLET] For Exercises 1–8, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for secondary and elementary school teachers, excluding special and vocational education teachers, in the United States for 11 years. (Source: U.S. Bureau of Labor Statistics)

8. Construct a 95% prediction interval for the average annual salary of elementary school teachers when the average annual salary of secondary school teachers is \$63,500. Interpret the results."

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Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

20. Construct a 90% prediction interval for the average time adults ages 35 to 44 spend per day watching television in Exercise 10 when the average time adults ages 25 to 34 spend per day watching television is 2.25 hours."

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Textbook Question

"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)

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Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

19. Construct a 90% prediction interval for the amount of milk produced in Exercise 9 when there are an average of 9275 thousand milk cows."

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Textbook Question

"In Exercises 19-24, construct the indicated prediction interval and interpret the results.

23. Construct a 99% prediction interval for the top speed of an electric car in Exercise 17 that takes 5.9 seconds to accelerate from 0 to 60 miles per hour."

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