Question

"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."

Accepted Answer

Identify the regression equation from Exercise 18, which relates total assets in federal defined benefit plans (dependent variable) to total assets in IRAs (independent variable). This equation typically has the form: \hat{y} = $$b_0$$ + $$b_1 x$$, where \hat{y} is the predicted total assets in federal defined benefit plans and x is the total assets in IRAs. Calculate the predicted value of total assets in federal defined benefit plans by substituting x = 6400 billion into the regression equation: \hat{y} = $$b_0$$ + $$b_1$$ 	imes 6400. Determine the standard error of the prediction, which accounts for both the variability of the regression line and the individual prediction. This involves the residual standard error (or standard deviation of the errors), the sample size, and the distance of 6400 from the mean of the x values. 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 residual standard error, n is the sample size, $$x_0 = 6400$$, 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: \hat{y} \pm t^* 	imes SE_{pred}. Interpret this interval as the range in which we expect the total assets in federal defined benefit plans to fall with 90% confidence when the total assets in IRAs are 6400 billion.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Prediction Interval

Linear Regression and Prediction

Confidence Level and Interpretation