Instructions for Exercises 12.21 and 12.22: (a) Perform a...

Instructions for Exercises 12.21 and 12.22: (a) Perform a regression using MegaStat or Excel. (b) State the null and alternative hypotheses for a two-tailed test for a zero slope. (c) Report the $p$-value and the 95 percent confidence interval for the slope shown in the regression results. (d) Is the slope significantly different from zero? Explain your conclusion. $$ \begin{array}{ccc} \hline & {\begin{array}{l} \text { College Student Weekly } \\ \text { Earnings in Dollars }(n=5) \\ \text { WeekPay } \end{array}} \\ \hline {\text { Hours Worked (X) }} & \text { Weekly Pay }(Y) \\ \hline & 10 & 93 \\ & 15 & 171 \\ & 20 & 204 \\ & 20 & 156 \\ & 35 & 261 \\ \hline \end{array} $$

Instructions for Exercises 12.21 and 12.22: (a) Perform a regression using MegaStat or Excel. (b) State the null and alternative hypotheses for a two-tailed test for a zero slope. (c) Report the $p$-value and the 95 percent confidence interval for the slope shown in the regression results. (d) Is the slope significantly different from zero? Explain your conclusion.
$$
\begin{array}{ccc}
\hline & {\begin{array}{l}
\text { College Student Weekly } \\
\text { Earnings in Dollars }(n=5) \\
\text { WeekPay }
\end{array}} \\
\hline {\text { Hours Worked (X) }} & \text { Weekly Pay }(Y) \\
\hline & 10 & 93 \\
& 15 & 171 \\
& 20 & 204 \\
& 20 & 156 \\
& 35 & 261 \\
\hline
\end{array}
$$

Key Concepts

p-value

The p-value is a measure used in hypothesis testing to determine the significance of the results. It quantifies the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. A small p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, suggesting that the observed effect is statistically significant.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a dependent variable and a single independent variable. It involves estimating the best-fitting straight line through the data points, which is characterized by an intercept and a slope. The slope indicates the change in the dependent variable for a one-unit change in the independent variable, while the intercept represents the expected value of the dependent variable when the independent variable is zero.

Hypothesis Testing for the Slope

Hypothesis testing for the slope in regression analysis involves assessing whether the independent variable has a statistically significant relationship with the dependent variable. The null hypothesis (H0) typically states that the slope is equal to zero, meaning there is no linear association, while the alternative hypothesis (H1) posits that the slope is not equal to zero, indicating a significant linear relationship. A two-tailed test is used when deviations in both directions (positive or negative) are considered.

Confidence Interval for the Slope

A confidence interval for the slope provides a range of values within which the true slope is likely to fall with a specified level of confidence, usually 95%. This interval not only shows the estimate of the slope but also reflects the precision of the estimate. If the confidence interval does not contain zero, it offers evidence that the relationship between the independent and dependent variables is statistically significant.

Statistical Significance and Conclusion

Determining whether the slope is significantly different from zero involves examining the p-value and the confidence interval obtained from the regression analysis. If the p-value is below the chosen significance level (e.g., 0.05) and the confidence interval does not include zero, one concludes that the slope is statistically significant. This means there is sufficient evidence to declare that a linear relationship exists between the independent and dependent variables.

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