In a study of differences in levels of community demand for...

In a study of differences in levels of community demand for firefighters, the following sample regression was obtained, based on data from 39 towns in Maryland: $$ \begin{aligned} y= & -\underset{(0.00010)}{0.00232}-\underset{(0.000018)}{0.00024 x_1}-\underset{(0.00012)}{0.00002 x_2}+\underset{(0.00005)}{0.00034 x_3} \\ & +\underset{(0.77951)}{0.48122 x_4}+\underset{(0.01172)}{0.04950 x_5}-\underset{(0.06306)}{0.00010 x_6}+\underset{\left(0.00645 x_7\right.}{0.0 .3572} \\ R^2= & 0.357 \end{aligned} $$ (0.00005) (0.00306) where $y=$ number of full-time firefighters per capita $x_1=$ maximum base salary of firefighters, in thousands of dollars $x_2=$ percentage of population $x_3=$ estimated per capita income, in thousands of dollars $x_4=$ population density $x_5=$ amount of intergovernmental grants per capita, in thousands of dollars $x_6=$ number of miles from the regional city $x_7=$ percentage of the population that is male and between 12 and 21 years of age The numbers in parentheses under the coefficients are the estimated coefficient standard errors. a. Find and interpret a $99 \%$ confidence interval for $\beta_5$. b. Test, against a two-sided alternative, the null hypothesis that $\beta_4$ is 0 , and interpret your result. c. Test, against a two-sided alternative, the null hypothesis that $\beta_7$ is 0 , and interpret your result.

In a study of differences in levels of community demand for firefighters, the following sample regression was obtained, based on data from 39 towns in Maryland:
$$
\begin{aligned}
y= & -\underset{(0.00010)}{0.00232}-\underset{(0.000018)}{0.00024 x_1}-\underset{(0.00012)}{0.00002 x_2}+\underset{(0.00005)}{0.00034 x_3} \\
& +\underset{(0.77951)}{0.48122 x_4}+\underset{(0.01172)}{0.04950 x_5}-\underset{(0.06306)}{0.00010 x_6}+\underset{\left(0.00645 x_7\right.}{0.0 .3572} \\
R^2= & 0.357
\end{aligned}
$$
(0.00005)
(0.00306)
where
$y=$ number of full-time firefighters per capita
$x_1=$ maximum base salary of firefighters, in thousands of dollars
$x_2=$ percentage of population
$x_3=$ estimated per capita income, in thousands of dollars
$x_4=$ population density
$x_5=$ amount of intergovernmental grants per capita, in thousands of dollars
$x_6=$ number of miles from the regional city
$x_7=$ percentage of the population that is male and between 12 and 21 years of age

The numbers in parentheses under the coefficients are the estimated coefficient standard errors.
a. Find and interpret a $99 \%$ confidence interval for $\beta_5$.
b. Test, against a two-sided alternative, the null hypothesis that $\beta_4$ is 0 , and interpret your result.
c. Test, against a two-sided alternative, the null hypothesis that $\beta_7$ is 0 , and interpret your result.

Key Concepts

Confidence Intervals

A confidence interval for a regression coefficient provides a range of values within which the true coefficient is expected to lie with a specified level of confidence (e.g., 99%). It combines the estimated coefficient with its standard error and a multiplier from the t-distribution, offering insight into the reliability of the estimate.

Hypothesis Testing in Regression

Hypothesis testing in regression involves determining whether a particular coefficient is significantly different from a hypothesized value, typically zero. This is done by computing a t-statistic (the coefficient divided by its standard error) and comparing it to a critical value from the t-distribution to decide whether to reject the null hypothesis.

t-Distribution

The t-distribution is used in regression analysis when the sample size is small or when the population standard deviation is unknown. It provides the basis for deriving confidence intervals and conducting hypothesis tests by accounting for extra variability from estimating the population standard error.

Standard Errors

The standard error of a regression coefficient measures the average amount that the estimated coefficient will vary from the actual population coefficient. It is essential for assessing the precision of the estimated coefficient and is used to construct confidence intervals and perform hypothesis tests.

Multiple Linear Regression

Multiple linear regression is a statistical method used to model the relationship between one dependent variable and two or more independent variables. It estimates coefficients for each predictor that demonstrate how a one-unit change in the predictor is associated with a change in the response variable while holding other factors constant.

Regression Coefficient Interpretation

Each coefficient in a regression model represents the expected change in the dependent variable for a one-unit change in an independent variable, with all other variables held constant. This interpretation is fundamental when discussing the impact of predictors on the outcome in a multivariable framework.

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