Murders and poverty, Part II. D Exercise 8.25 presents...

Murders and poverty, Part II. D Exercise 8.25 presents regression output from a model for predicting annual murders per million from percentage living in poverty based on a random sample of 20 metropolitan areas. The model output is also provided below. \begin{tabular}{rrrrr} \hline & Estimate & Std. Error & t value & $\operatorname{Pr}(>|\mathrm{t}|)$ \\ \hline (Intercept) & -29.901 & 7.789 & -3.839 & 0.001 \\ poverty\% & 2.559 & 0.390 & 6.562 & 0.000 \\ \hline \end{tabular} $s=5.512 \quad R^{2}=70.52 \% \quad R_{a d j}^{2}=68.89 \%$ (a) What are the hypotheses for evaluating whether poverty percentage is a significant predictor of murder rate? (b) State the conclusion of the hypothesis test from part (a) in context of the data. (c) Calculate a $95 \%$ confidence interval for the slope of poverty percentage, and interpret it in context of the data. (d) Do your results from the hypothesis test and the confidence interval agree? Explain.

Murders and poverty, Part II. D Exercise 8.25 presents regression output from a model for predicting annual murders per million from percentage living in poverty based on a random sample of 20 metropolitan areas. The model output is also provided below.
\begin{tabular}{rrrrr}
\hline & Estimate & Std. Error & t value & $\operatorname{Pr}(>|\mathrm{t}|)$ \\
\hline (Intercept) & -29.901 & 7.789 & -3.839 & 0.001 \\
poverty\% & 2.559 & 0.390 & 6.562 & 0.000 \\
\hline
\end{tabular} $s=5.512 \quad R^{2}=70.52 \% \quad R_{a d j}^{2}=68.89 \%$
(a) What are the hypotheses for evaluating whether poverty percentage is a significant predictor of murder rate?
(b) State the conclusion of the hypothesis test from part (a) in context of the data.
(c) Calculate a $95 \%$ confidence interval for the slope of poverty percentage, and interpret it in context of the data.
(d) Do your results from the hypothesis test and the confidence interval agree? Explain.

Key Concepts

Agreement Between Hypothesis Testing and Confidence Intervals

This concept emphasizes that the outcomes from hypothesis testing and confidence interval estimation should be consistent. If a hypothesis test leads to rejection of the null hypothesis (i.e., the coefficient is significantly different from zero), the corresponding confidence interval should not encompass zero. This consistency reinforces the reliability of the inference regarding the predictor's significance in the regression model.

Hypothesis Testing for Regression Coefficients

This concept involves setting up statistical hypotheses to determine whether a predictor variable has a significant effect on the response variable. Typically, the null hypothesis states that the regression slope is equal to zero (indicating no effect), while the alternative hypothesis states that the slope is not equal to zero (indicating some effect). This process is fundamental in evaluating the contribution of predictors in a linear regression model.

t-Statistic and p-Value in Regression Analysis

The t-statistic is used to assess how many standard errors the estimated regression coefficient is away from zero. Correspondingly, the p-value indicates the probability of observing a t-statistic as extreme as, or more extreme than, the value computed under the null hypothesis. A small p-value (commonly less than 0.05) implies that such an extreme result is unlikely under the null hypothesis, thereby supporting the claim that the predictor contributes significantly to the model.

Confidence Interval for a Regression Slope

A confidence interval for a regression coefficient provides a range where the true slope is likely to be found with a specified level of confidence (usually 95%). If this interval does not include zero, it infers that the effect of the predictor on the response variable is statistically significant. This interval offers both an estimate of the parameter's value and a measure of its precision.

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