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

Larson 8th Edition

Ch. 9 - Correlation and Regression

문제 9.R.19

9장, 문제 9.R.19

"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."

검증된 단계별 안내

Identify the regression model from Exercise 9, which relates the number of milk cows to the amount of milk produced. This model typically has the form:

\(\hat{y}\) = b + m x

, where

\(\hat{y}\)

is the predicted milk production and

x

is the number of milk cows (in thousands).

Calculate the predicted value of milk production,

\(\hat{y}\)

, by substituting

x = 9275

into the regression equation.

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. The formula for the standard error of prediction is:

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 9275, and

\(\bar{x}\)

is the mean of the

x

values.

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 corresponds to the desired confidence level.

Construct the 90% prediction interval using the formula:

\(\pm\) t_{\(\alpha\)/2, n-2} \(\times\) SE_{pred}

. This interval estimates the range in which a single new observation of milk production is expected to fall when there are 9275 thousand milk cows.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Prediction Interval

A prediction interval estimates the range within which a single future 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.

Confidence Level

The confidence level, such as 90%, represents the probability that the prediction interval contains the true future value. It reflects the degree of certainty we have in the interval estimate, with higher levels indicating wider intervals to capture more uncertainty.

Regression and Use of Predictor Variables

Prediction intervals often rely on regression models that relate a predictor variable (e.g., number of milk cows) to a response variable (e.g., milk production). Understanding how to use the regression equation and the variability around the predicted value is essential to construct accurate prediction intervals.

"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."