Consider Example 9.1.1. Using the results calculated in the...

Consider Example 9.1.1. Using the results calculated in the example and the parameters calculated by the MINITAB command REGRESS and shown in Figure 9.1.2, do the following: (a) Find the $95 \%$ confidence interval for $\beta_1$. (b) Test the null hypothesis $H_0: \beta_1=0.07$ against $H_1: \beta_1 \neq 0.07$ at the $5 \%$ level of significance.

Consider Example 9.1.1. Using the results calculated in the example and the parameters calculated by the MINITAB command REGRESS and shown in Figure 9.1.2, do the following:
(a) Find the $95 \%$ confidence interval for $\beta_1$.
(b) Test the null hypothesis $H_0: \beta_1=0.07$ against $H_1: \beta_1 \neq 0.07$ at the $5 \%$ level of significance.

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. In the context of the problem, the estimation of the slope coefficient (??) represents the expected change in the response variable for a one-unit increase in the predictor variable.

Confidence Interval Estimation

Confidence interval estimation is used to quantify the uncertainty associated with sample estimates of a population parameter. For the slope ?? in regression analysis, a 95% confidence interval provides a range within which the true slope is expected to fall with 95% probability, taking into account the variability inherent in the data.

Hypothesis Testing for Regression Coefficients

Hypothesis testing for regression coefficients involves comparing the estimated coefficient to a hypothesized value to determine if there is statistically significant evidence to reject the null hypothesis. In the provided question, the test assesses whether the true slope is significantly different from 0.07 using a two-tailed approach at a 5% significance level.

t-Distribution

The t-distribution is used in situations where the sample size is small and the population variance is unknown. In linear regression analysis, the t-distribution is used for constructing confidence intervals and performing hypothesis tests on the regression coefficients, including the slope ??, due to the reliance on the estimated standard error of the coefficient.

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