The following model was fitted to data on 50 states: where ŷ= annual salary of the attorney

The following model was fitted to data on 50 states: where ...

The following model was fitted to data on 50 states: where $$ \begin{aligned} & \hat{y}=\text { annual salary of the attorney general of the } \\ & \text { state } \\ & x_1=\text { average annual salary of lawyers, in thou- } \\ & \text { sands of dollars } \\ & x_2=\text { number of bills enacted in previous legisla- } \\ & \text { tive session } \\ & x_3=\text { number of due process reviews by state } \\ & \text { courts that resulted in overturn of legislation } \\ & \text { in previous } 40 \text { years } \\ & x_4=\text { length of term of the attorney general of the } \\ & \text { state } \\ & x_5=\text { dummy variable taking value } 1 \text { if justices of } \\ & \text { the state supreme court can be removed from } \\ & \end{aligned} $$ office by the governor, judicial review board, or majority vote of the supreme court and 0 otherwise $x_6=$ dummy variable taking value 1 if supreme court justices are elected on partisan ballots and 0 otherwise The numbers in parentheses under the coefficients are the estimated coefficient standard errors. a. Interpret the estimated coefficient on the dummy variable $x_5$. b. Interpret the estimated coefficient on the dummy variable $x_6$. c. Test, at the $5 \%$ level, the null hypothesis that the true coefficient on the dummy variable $x_5$ is 0 against the alternative that it is positive. d. Test, at the $5 \%$ level, the null hypothesis that the true coefficient on the dummy variable $x_6$ is 0 against the alternative that it is negative. e. Find and interpret a $95 \%$ confidence level for the parameter $\beta_1$.

The following model was fitted to data on 50 states:
where
$$
\begin{aligned}
& \hat{y}=\text { annual salary of the attorney general of the } \\
& \text { state } \\
& x_1=\text { average annual salary of lawyers, in thou- } \\
& \text { sands of dollars } \\
& x_2=\text { number of bills enacted in previous legisla- } \\
& \text { tive session } \\
& x_3=\text { number of due process reviews by state } \\
& \text { courts that resulted in overturn of legislation } \\
& \text { in previous } 40 \text { years } \\
& x_4=\text { length of term of the attorney general of the } \\
& \text { state } \\
& x_5=\text { dummy variable taking value } 1 \text { if justices of } \\
& \text { the state supreme court can be removed from } \\
&
\end{aligned}
$$
office by the governor, judicial review board, or majority vote of the supreme court and 0 otherwise
$x_6=$ dummy variable taking value 1 if supreme court justices are elected on partisan ballots and 0 otherwise
The numbers in parentheses under the coefficients are the estimated coefficient standard errors.
a. Interpret the estimated coefficient on the dummy variable $x_5$.
b. Interpret the estimated coefficient on the dummy variable $x_6$.
c. Test, at the $5 \%$ level, the null hypothesis that the true coefficient on the dummy variable $x_5$ is 0 against the alternative that it is positive.
d. Test, at the $5 \%$ level, the null hypothesis that the true coefficient on the dummy variable $x_6$ is 0 against the alternative that it is negative.
e. Find and interpret a $95 \%$ confidence level for the parameter $\beta_1$.

Key Concepts

Dummy Variables

Dummy variables are binary indicators used in regression analysis to represent categorical information, usually taking on the values 0 or 1. These variables allow the model to capture the qualitative differences between groups by indicating the presence or absence of a certain attribute, thereby enabling the assessment of how categorical distinctions affect the dependent variable.

Interpretation of Dummy Variable Coefficients

The coefficient on a dummy variable in a regression model measures the difference in the predicted value of the dependent variable when the dummy variable takes the value one as compared to the value zero, holding all other variables constant. This interpretation helps in understanding the impact of the categorical attribute on the outcome relative to a reference group.

Hypothesis Testing of Regression Coefficients

Hypothesis testing in the context of regression is used to determine whether a coefficient significantly differs from a hypothesized value, usually zero. This involves setting up a null hypothesis (e.g., the true coefficient is zero) against an alternative hypothesis (e.g., the coefficient is either positive or negative) and using statistical tests to decide whether there is sufficient evidence to reject the null hypothesis at a given significance level.

t-Statistic and Significance Levels

The t-statistic is calculated by dividing the estimated regression coefficient by its standard error. It serves as a measure of how many standard deviations the estimated coefficient is away from the hypothesized value under the null hypothesis. The significance level, often set at 5%, defines the threshold for determining whether the observed t-statistic is extreme enough to reject the null hypothesis, thereby indicating that the corresponding coefficient is statistically significant.

Confidence Intervals for Regression Coefficients

A confidence interval for a regression coefficient provides a range of values within which the true parameter value is expected to fall with a specified level of confidence, commonly 95%. This interval is constructed around the estimated coefficient using its standard error and an appropriate critical value from the t-distribution. Interpreting a confidence interval involves understanding that if the interval does not include a specific value (such as zero), it suggests that the parameter is significantly different from that value at the chosen level of confidence.

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