Question

Exercises 9.5 to 9.8 show some computer output for fitting simple linear models. State the value of the sample slope for each model and give the null and alternative hypotheses for testing if the slope in the population is different from zero. Identify the p-value and use it (and a \(5 \%\) significance level) to make a clear conclusion about the effectiveness of the model. $$ \begin{array}{lrrrr} \text { The regression equation is } \mathrm{Y}=89.4 & -8.20 \mathrm{X} & \\ \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathrm{T} & \mathrm{P} \\ \text { Constant } & 89.406 & 4.535 & 19.71 & 0.000 \\ \mathrm{X} & -8.1952 & 0.9563 & -8.57 & 0.000 \end{array} $$

Short answer

The sample slope is \(-8.1952\). The null hypothesis (\(H0: \Beta = 0\)) and the alternative hypothesis (\(H1: \Beta ≠ 0\)). The p-value is \(0.000\). Since the p-value is less than \(0.05\), we reject the null hypothesis and conclude that the model is effective.

Step-by-step solution

Identify the Sample Slope

The sample slope is the coefficient of the predictor variable \(X\) in the regression equation. Looking at the regression results, the estimated sample slope coefficient is \(-8.1952\). This is the average change in \(Y\) for each unit change in \(X\), provided that all other conditions remain constant.

State the Null and Alternative Hypotheses

The null hypothesis states that there is no relationship between the predictor variable (in this case \(X\)) and the outcome variable \(Y\). Mathematically, this means the slope of the regression line in the population equals zero. Thus, \(H0: \Beta = 0\). The alternative hypothesis is that there is a relationship between \(X\) and \(Y\), which means the slope of the regression line in the population is not zero. Hence, \(H1: \Beta ≠ 0\).

Identify the p-value

The p-value is a probability that measures the evidence against the null hypothesis. In the regression results given, the p-value is shown as 0.000.

Decision and Conclusion

Given a \(5\%\) significance level, a p-value less than \(0.05\) would lead to a rejection of the null hypothesis in favor of the alternative. Thus, since the p-value found (0.000) is less than \(0.05\), we conclude that at the \(5 \%\)-level, there is significant evidence to suggest that the slope of the regression line in the population is different from zero. Therefore, the model is effective.