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Ch. 9 - Correlation and Regression
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 9 - Correlation and Regression문제 9.R.20
9장, 문제 9.R.20

"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."

검증된 단계별 안내
1
Identify the regression equation from Exercise 10, which relates the average time adults ages 25 to 34 spend watching television (independent variable, x) to the average time adults ages 35 to 44 spend watching television (dependent variable, y). This equation typically has the form: \(\hat{y}\) = b_0 + b_1 x.
Calculate the predicted average time for adults ages 35 to 44 by substituting x = 2.25 hours into the regression equation to find \(\hat{y}\).
Determine the standard error of the prediction, which accounts for both the variability of the estimate of the mean response and the variability of individual observations around the regression line. This involves using the formula for the prediction interval standard error: SE_{pred} = s \(\sqrt{1 + \frac{1}{n}\) + \(\frac{(x - \bar{x}\))^2}{\(\sum\) (x_i - \(\bar{x}\))^2}}, where s is the standard error of the estimate, n is the sample size, \(\bar{x}\) is the mean of the x-values, and x is 2.25.
Find the critical t-value for a 90% prediction interval with degrees of freedom equal to n - 2. This value comes from the t-distribution table and reflects the desired confidence level.
Construct the 90% prediction interval using the formula: \(\hat{y}\) \(\pm\) t_{\(\alpha\)/2, n-2} \(\times\) SE_{pred}. This interval estimates the range in which an individual adult aged 35 to 44 is expected to spend watching television, given the average time for adults aged 25 to 34 is 2.25 hours.

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주요 개념

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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 means, prediction intervals account for both the variability in the sample mean and the individual data points, making them wider and more appropriate for predicting individual outcomes.
추천 영상:
가이드 코스
09:00
Prediction Intervals

Confidence Level

The confidence level, such as 90%, represents the probability that the interval constructed from sample data contains the true value of the parameter or future observation. A 90% prediction interval means we are 90% confident that the new observation will lie within the calculated range.
추천 영상:
06:33
Introduction to Confidence Intervals

Regression and Prediction Using Independent Variables

When predicting a response variable based on an independent variable, regression analysis models the relationship between them. Using the regression equation, we can estimate the expected value and construct prediction intervals for new observations at given predictor values, accounting for uncertainty in the estimate.
추천 영상:
가이드 코스
04:57
Using Regression Lines to Predict Values
관련 실천
교과서 질문

"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.

17. The table shows the times (in seconds) to accelerate from 0 to 60 miles per hour and the top speeds (in miles per hour) for eight electric cars. The regression equation is y =- 14.399x + 196.996. (Source: Car and Driver)

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교과서 질문

"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"

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교과서 질문

"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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교과서 질문

"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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교과서 질문

"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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교과서 질문

"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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