Late-Night Talk Show Viewership. Jimmy Kimmel Live! on ABC,...

Late-Night Talk Show Viewership. Jimmy Kimmel Live! on ABC, The Tonight Show Starring Jimmy Fallon on NBC, and The Late Show with Stephen Colbert on CBS are three popular late-night talk shows. The following table shows the number of viewers in millions for a 10 -week period during the spring for each of these shows (TV by the Numbers website, https://tvbythenumbers.zap2itcom/).$$ \begin{array}{|c|c|c|c|} \hline \text { Week } & \begin{array}{l} \text { Jimmy Kimmel } \\ \text { Live (ABC) } \end{array} & \begin{array}{l} \text { The Tonight Show } \\ \text { Starring Jimmy } \\ \text { Fallon (NBC) } \end{array} & \begin{array}{l} \text { The Late Show with } \\ \text { Stephen Colbert (CBS) } \end{array} \\ \hline \text { June 13-June } 17 & 2.67 & 3.24 & 2.27 \\ \hline \text { June 6-June } 10 & 2.58 & 3.32 & 2.05 \\ \hline \text { May 3D-June } 3 & 2.64 & 2.66 & 2.08 \\ \hline \text { May 23-May } 27 & 2.47 & 3.30 & 2.07 \\ \hline \text { May 16-May } 20 & 1.97 & 3.10 & 2.31 \\ \hline \text { May 9-May } 16 & 2.21 & 3.31 & 2.45 \\ \hline \text { May 2-May } 6 & 2.12 & 3.20 & 2.57 \\ \hline \text { April 25-April } 29 & 2.24 & 3.15 & 2.45 \\ \hline \text { April 18-April } 22 & 2.10 & 2.77 & 2.56 \\ \hline \text { April 11-April } 15 & 2.21 & 3.24 & 2.16 \\ \hline \end{array} $$ At the .05 level of significance, test for a difference in the mean number of viewers per week for the three late-night talk shows.

Late-Night Talk Show Viewership. Jimmy Kimmel Live! on ABC, The Tonight Show Starring Jimmy Fallon on NBC, and The Late Show with Stephen Colbert on CBS are three popular late-night talk shows. The following table shows the number of viewers in millions for a 10 -week period during the spring for each of these shows (TV by the Numbers website, https://tvbythenumbers.zap2itcom/).$$
\begin{array}{|c|c|c|c|}
\hline \text { Week } & \begin{array}{l}
\text { Jimmy Kimmel } \\
\text { Live (ABC) }
\end{array} & \begin{array}{l}
\text { The Tonight Show } \\
\text { Starring Jimmy } \\
\text { Fallon (NBC) }
\end{array} & \begin{array}{l}
\text { The Late Show with } \\
\text { Stephen Colbert (CBS) }
\end{array} \\
\hline \text { June 13-June } 17 & 2.67 & 3.24 & 2.27 \\
\hline \text { June 6-June } 10 & 2.58 & 3.32 & 2.05 \\
\hline \text { May 3D-June } 3 & 2.64 & 2.66 & 2.08 \\
\hline \text { May 23-May } 27 & 2.47 & 3.30 & 2.07 \\
\hline \text { May 16-May } 20 & 1.97 & 3.10 & 2.31 \\
\hline \text { May 9-May } 16 & 2.21 & 3.31 & 2.45 \\
\hline \text { May 2-May } 6 & 2.12 & 3.20 & 2.57 \\
\hline \text { April 25-April } 29 & 2.24 & 3.15 & 2.45 \\
\hline \text { April 18-April } 22 & 2.10 & 2.77 & 2.56 \\
\hline \text { April 11-April } 15 & 2.21 & 3.24 & 2.16 \\
\hline
\end{array}
$$
At the .05 level of significance, test for a difference in the mean number of viewers per week for the three late-night talk shows.

Key Concepts

Assumptions of ANOVA

ANOVA relies on several key assumptions: independence of observations, normally distributed residuals, and homogeneity of variances (equal variances) across the groups. These assumptions ensure the validity and accuracy of the test results when comparing group means.

One-Way Analysis of Variance (ANOVA)

One-way ANOVA is a statistical method used to test whether there are any statistically significant differences between the means of three or more independent (unrelated) groups. It helps determine if at least one group mean is different from the others by analyzing total variability into components attributable to various sources.

Null and Alternative Hypotheses

In the context of comparing means across groups, the null hypothesis states that all group means are equal, while the alternative hypothesis states that at least one group mean is different. This formulation is crucial in testing whether any observed differences in sample means are due to random chance or reflect true differences in the populations.

F-Statistic

The F-statistic is a ratio of the variance between group means to the variance within the groups. It quantifies whether the differences observed among the group means are larger than would be expected by chance. A larger F-statistic indicates a higher likelihood that the group means are significantly different.

Significance Level

The significance level, often denoted by ? (alpha), is the threshold probability for rejecting the null hypothesis. A commonly used value is 0.05, meaning there is a 5% risk of committing a Type I error—rejecting the null hypothesis when it is actually true.

Between-Group and Within-Group Variability

Between-group variability measures the differences between the group means and the overall mean, reflecting the effect of the treatment or group membership. Within-group variability, on the other hand, measures the variation observed within each group, due to random error or individual differences. The comparison of these two variabilities is central in the ANOVA framework.

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