Instructions for Exercises 12.27-12.29: (a) Use Excel's Data...

Instructions for Exercises 12.27-12.29: (a) Use Excel's Data Analysis > Regression (or MegaStat or MINITAB) to obtain regression estimates. (b) Interpret the 95 percent confidence interval for the slope. Does it contain zero? (c) Interpret the $t$ test for the slope and its $p$-value. (d) Interpret the $F$ statistic. (e) Verify that the $p$-value for $F$ is the same as for the slope's $t$ statistic, and show that $t^2=F$. (f) Describe the fit of the regression. $$ \begin{array}{cccc} \hline & \text { Moviegoer Snack Spending }(\boldsymbol{n}=\mathbf{1 0}) & \\ & \text { Movies } & \\ \text { Age }(\boldsymbol{X}) & \text { Spent }(Y) & \text { Age }(\boldsymbol{X}) & \text { Spent }(Y) \\ \hline 30 & 2.85 & 33 & 6.75 \\ 50 & 6.50 & 36 & 3.60 \\ 34 & 1.50 & 26 & 6.10 \\ 12 & 6.35 & 18 & 8.35 \\ 37 & 6.20 & 46 & 4.35 \\ \hline \end{array} $$

Instructions for Exercises 12.27-12.29: (a) Use Excel's Data Analysis > Regression (or MegaStat or MINITAB) to obtain regression estimates. (b) Interpret the 95 percent confidence interval for the slope. Does it contain zero? (c) Interpret the $t$ test for the slope and its $p$-value. (d) Interpret the $F$ statistic. (e) Verify that the $p$-value for $F$ is the same as for the slope's $t$ statistic, and show that $t^2=F$. (f) Describe the fit of the regression.
$$
\begin{array}{cccc}
\hline & \text { Moviegoer Snack Spending }(\boldsymbol{n}=\mathbf{1 0}) & \\
& \text { Movies } & \\
\text { Age }(\boldsymbol{X}) & \text { Spent }(Y) & \text { Age }(\boldsymbol{X}) & \text { Spent }(Y) \\
\hline 30 & 2.85 & 33 & 6.75 \\
50 & 6.50 & 36 & 3.60 \\
34 & 1.50 & 26 & 6.10 \\
12 & 6.35 & 18 & 8.35 \\
37 & 6.20 & 46 & 4.35 \\
\hline
\end{array}
$$

Key Concepts

F Statistic

The F statistic in regression analysis evaluates the overall significance of the model by comparing the model with predictors to a model with no predictors (the intercept-only model). It tests whether at least one predictor variable has a relationship with the dependent variable. In simple linear regression, the F statistic is mathematically equivalent to the square of the t statistic for the slope, linking the overall model test to the individual coefficient test.

Model Fit

Model fit refers to how well a regression model captures the relationship between the independent and dependent variables. It is often quantified by metrics like the coefficient of determination (R-squared), which indicates the proportion of variability in the dependent variable that is explained by the model, and by analyzing residuals to assess any patterns that suggest a poor fit.

p-value

The p-value is a measure that indicates the probability of observing the data, or something more extreme, assuming the null hypothesis is true. In the context of regression, p-values help determine the statistical significance of the estimated coefficients and overall model fit, guiding decisions on whether to reject the null hypothesis.

t Test for Slope

The t test for the slope assesses whether the estimated slope in a regression model is statistically different from zero. This helps determine if there is a meaningful linear relationship between the independent and dependent variables. The test produces a t statistic and an associated p-value, where a small p-value indicates strong evidence against the null hypothesis of no effect.

Confidence Interval for Slope

A confidence interval for the slope is a range of values that likely contains the true slope parameter of the regression line with a specified level of confidence, such as 95%. It provides insight into the precision of the slope estimate; if the interval contains zero, it suggests that there might not be a statistically significant association between the variables.

Linear Regression Analysis

Linear regression analysis is a statistical method used to model and analyze the relationship between a dependent variable and one independent variable. It involves estimating the coefficients (such as the intercept and slope) that define the linear relationship, thereby enabling predictions and inferences about the association between the variables.

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