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Ch. 9 - Correlation and Regression
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 9 - Correlation and RegressionProblema 9.3.29
Capitolo 9, Problema 9.3.29

"Constructing and Interpreting a Prediction Interval In Exercises 21-30, construct the indicated prediction interval and interpret the results.
29. New Vehicle Sales Construct a 95% prediction interval for new vehicle sales for General Motors in Exercise 19 when the number of new vehicles sold by Ford is 2028 thousand."

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Identify the regression model from Exercise 19, which relates new vehicle sales of General Motors (GM) to the number of new vehicles sold by Ford. This model typically has the form: \(\hat{y}\) = bx + a, where \(\hat{y}\) is the predicted GM sales, x is Ford sales, b is the slope, and a is the intercept.
Calculate the predicted GM sales (\(\hat{y}\)) by substituting x = 2028 (thousand vehicles sold by Ford) into the regression equation.
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 errors), the sample size, and the distance of x from the mean of the predictor variable.
Find the critical t-value for a 95% prediction interval using the appropriate degrees of freedom (usually n - 2, where n is the sample size). This value comes from the t-distribution table.
Construct the 95% prediction interval using the formula: \(\hat{y}\) \(\pm\) t_{\(\alpha\)/2, n-2} \(\times\) \(\text{standard error of prediction}\). Interpret this interval as the range in which we expect the actual GM sales to fall with 95% confidence when Ford sells 2028 thousand vehicles.

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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 mean responses, prediction intervals account for both the uncertainty in the estimated regression line and the variability of individual data points.
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Prediction Intervals

Simple Linear Regression

Simple linear regression models the relationship between two variables by fitting a straight line to observed data. It is used here to predict new vehicle sales for General Motors based on Ford's sales, assuming a linear association between the two.
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Intro to Least Squares Regression

Confidence Level and Interpretation

The confidence level (e.g., 95%) indicates the probability that the prediction interval contains the true future value. Interpreting the interval means understanding that there is a 95% chance the actual new vehicle sales for General Motors will fall within the calculated range when Ford's sales are 2028 thousand.
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Introduction to Confidence Intervals
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