Exercise 8.25 presents regression output from a model for...

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. $$\begin{array}{rrrrr} \hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \hline \text { (Intercept) } & -29.901 & 7.789 & -3.839 & 0.001 \\ \text { poverty\% } & 2.559 & 0.390 & 6.562 & 0.000 \\ \hline\end{array}$$ $s=5.512 \quad R^{2}=70.52 \% \quad R_{a d j}^{2}=68.89 \%$ (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.

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.
$$\begin{array}{rrrrr}
\hline & \text { Estimate } & \text { Std. Error } & \text { t value } & \operatorname{Pr}(>|\mathrm{t}|) \\
\hline \text { (Intercept) } & -29.901 & 7.789 & -3.839 & 0.001 \\
\text { poverty\% } & 2.559 & 0.390 & 6.562 & 0.000 \\
\hline\end{array}$$
$s=5.512 \quad R^{2}=70.52 \% \quad R_{a d j}^{2}=68.89 \%$
(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

Interpreting Hypothesis Testing and Confidence Intervals

Both hypothesis testing and confidence interval estimation are methods to infer the significance of a predictor's effect. Consistency between the two is expected: if the hypothesis test rejects the null hypothesis and the confidence interval for the slope does not include zero, both methods indicate that the predictor has a statistically significant impact on the response variable. This dual approach helps in validating the findings and provides both a test statistic and a range estimate for the effect size.

Confidence Interval for the Slope

A confidence interval for the slope coefficient in regression provides a range of plausible values for the true effect of the predictor on the response variable, based on the data and the chosen level of confidence (commonly 95%). This interval is constructed using the estimated coefficient, its standard error, and the critical value from the t-distribution. If zero is not contained within the interval, it supports the conclusion that the predictor is significantly associated with the response.

t-Statistic and p-Value

The t-statistic, calculated by dividing the estimated regression coefficient by its standard error, measures the strength of evidence against the null hypothesis that the coefficient is zero. The corresponding p-value indicates the probability of observing such a t-statistic or one more extreme if the null hypothesis were true. A small p-value, typically below a predetermined significance level (e.g., 0.05), leads to the rejection of the null hypothesis, implying that the predictor is statistically significant.

Hypothesis Testing for Predictor Significance

In simple linear regression, hypothesis testing is employed to assess whether a predictor variable is significantly associated with the response variable. This typically involves testing the null hypothesis that the slope (the coefficient quantifying the change in the outcome per unit change in the predictor) is zero, against an alternative hypothesis that it is not zero. A rejection of the null hypothesis suggests that the predictor provides meaningful information about the response.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a single predictor (independent variable) and an outcome (dependent variable) by fitting a linear equation to the observed data. The regression line represents the best fit to the data, allowing predictions of the outcome variable based on values of the predictor, and provides parameters such as the slope and intercept that quantify the relationship between the variables.

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