In Problem 14.5 on page 583, you used horsepower and weight...

In Problem 14.5 on page 583, you used horsepower and weight to predict mileage (stored in Auto2010). Use the results from that problem. a. Construct a $95 \%$ confidence interval estimate of the population slope between mileage and horsepower. b. At the 0.05 level of significance, determine whether each independent variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables to include in this model.

In Problem 14.5 on page 583, you used horsepower and weight to predict mileage (stored in Auto2010). Use the results from that problem.
a. Construct a $95 \%$ confidence interval estimate of the population slope between mileage and horsepower.
b. At the 0.05 level of significance, determine whether each independent variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables to include in this model.

Key Concepts

Confidence Interval for Regression Slope

This concept involves estimating a range within which the true slope coefficient of a predictor variable lies with a specified level of confidence (typically 95%). The interval is calculated using the standard error of the estimated slope and the appropriate t-distribution critical value, providing insight into the precision and reliability of the slope estimate in the context of the overall regression model.

Hypothesis Testing in Linear Regression

In linear regression, hypothesis testing is used to assess whether each predictor variable has a statistically significant relationship with the response variable. This typically involves testing the null hypothesis that a regression coefficient is equal to zero versus the alternative that it is different from zero, allowing researchers to determine if a predictor contributes meaningful information to the model.

t-Test for Regression Coefficients

The t-test for a regression coefficient evaluates the significance of an individual predictor by comparing the estimated coefficient to its standard error. This test results in a t-statistic and a corresponding p-value, which are used to decide whether to reject the null hypothesis that the coefficient is zero, thereby indicating whether the predictor is significantly associated with the response variable.

Model Selection in Regression Analysis

Model selection involves determining which independent variables should be included in a regression model based on their statistical significance and contribution to explaining the variation in the dependent variable. By applying hypothesis tests and evaluating p-values, analysts can identify and retain significant predictors, refining the model to ensure it is both parsimonious and effective.

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