College Attendance Rates. The following data show the...

College Attendance Rates. The following data show the percentage of 17 - to 24 -year-olds who are attending college in several metropolitan statistical areas in four geographic regions of the United States (U.S. Census Bureau website, https $J / / \mathrm{www}$ .census.gov/data.html). $$ \begin{array}{cccc} \text { Northeast } & \text { Midwest } & \text { South } & \text { West } \\ 28.6 & 36.7 & 59.9 & 16.4 \\ 39.9 & 33.4 & 37.2 & 33.5 \\ 31.9 & 22.8 & 28.0 & 22.3 \\ 46.3 & 43.8 & 41.1 & 12.4 \\ 32.5 & 32.1 & 33.9 & 43.7 \\ 14.9 & 58.3 & 18.8 & 26.8 \\ 36.8 & 31.1 & 30.3 & 57.3 \\ 36.3 & 64.0 & 67.4 & 14.3 \\ 37.7 & 27.6 & 32.6 & 37.0 \\ 58.4 & 55.5 & 30.0 & 28.1 \\ 60.6 & 78.8 & 39.1 & 17.5 \\ & 42.2 & 29.7 & 32.3 \\ & 74.7 & 29.8 & 52.4 \\ & 36.5 & 23.7 & 51.5 \\ & 28.7 & 34.0 & 25.4 \\ & 60.4 & 24.5 & 29.6 \\ & 58.2 & 54.2 & 27.6 \\ & 21.0 & 31.0 & 31.5 \\ & 28.8 & 41.9 & 22.8 \\ & 25.5 & 70.2 & 34.6 \\ & 73.9 & 22.7 & 33.0 \end{array} $$ $$ \begin{array}{cccc} \text { Northeast } & \text { Midwest } & \text { South } & \text { West } \\ & 36.8 & 30.7 & 37.0 \\ 28.4 & 30.8 & 33.8 \\ 27.2 & 21.6 & 28.7 \\ 31.8 & 31.5 & 21.8 \\ & 56.8 & 38.2 & \\ 28.3 & 40.2 & \\ 33.3 & 35.4 & \\ 39.4 & 21.6 & \\ & 39.2 & 35.5 & \\ & & 26.1 & \\ & & 32.7 & \end{array} $$ Use $\alpha=.05$ to test whether the mean percentage of 17 - to 24 -year-olds who are attending college is the same for the four geographic regions.

College Attendance Rates. The following data show the percentage of 17 - to 24 -year-olds who are attending college in several metropolitan statistical areas in four geographic regions of the United States (U.S. Census Bureau website, https $J / / \mathrm{www}$ .census.gov/data.html).
$$
\begin{array}{cccc}
\text { Northeast } & \text { Midwest } & \text { South } & \text { West } \\
28.6 & 36.7 & 59.9 & 16.4 \\
39.9 & 33.4 & 37.2 & 33.5 \\
31.9 & 22.8 & 28.0 & 22.3 \\
46.3 & 43.8 & 41.1 & 12.4 \\
32.5 & 32.1 & 33.9 & 43.7 \\
14.9 & 58.3 & 18.8 & 26.8 \\
36.8 & 31.1 & 30.3 & 57.3 \\
36.3 & 64.0 & 67.4 & 14.3 \\
37.7 & 27.6 & 32.6 & 37.0 \\
58.4 & 55.5 & 30.0 & 28.1 \\
60.6 & 78.8 & 39.1 & 17.5 \\
& 42.2 & 29.7 & 32.3 \\
& 74.7 & 29.8 & 52.4 \\
& 36.5 & 23.7 & 51.5 \\
& 28.7 & 34.0 & 25.4 \\
& 60.4 & 24.5 & 29.6 \\
& 58.2 & 54.2 & 27.6 \\
& 21.0 & 31.0 & 31.5 \\
& 28.8 & 41.9 & 22.8 \\
& 25.5 & 70.2 & 34.6 \\
& 73.9 & 22.7 & 33.0
\end{array}
$$
$$
\begin{array}{cccc}
\text { Northeast } & \text { Midwest } & \text { South } & \text { West } \\
& 36.8 & 30.7 & 37.0 \\
28.4 & 30.8 & 33.8 \\
27.2 & 21.6 & 28.7 \\
31.8 & 31.5 & 21.8 \\
& 56.8 & 38.2 & \\
28.3 & 40.2 & \\
33.3 & 35.4 & \\
39.4 & 21.6 & \\
& 39.2 & 35.5 & \\
& & 26.1 & \\
& & 32.7 &
\end{array}
$$
Use $\alpha=.05$ to test whether the mean percentage of 17 - to 24 -year-olds who are attending college is the same for the four geographic regions.

Key Concepts

p-value

The p-value indicates the probability of observing the data, or something more extreme, if the null hypothesis were true. If the p-value is below the significance level (0.05 in this case), it suggests that the observed differences among group means are statistically significant.

F-Statistic

The F-statistic in an ANOVA compares the between-group variability to the within-group variability. A larger F value suggests that the group means are more spread out than what would be expected by random variation, thereby providing evidence against the null hypothesis.

Degrees of Freedom

Degrees of freedom (df) in ANOVA are associated with the number of groups and the total number of observations. They are used to determine the appropriate critical value for the F-statistic from the F-distribution.

Null and Alternative Hypotheses

The null hypothesis (H?) typically states that there is no difference in the means across all groups, meaning the mean college attendance rate is the same in each geographic region. The alternative hypothesis (H?) claims that at least one region's mean differs from the others.

One-Way ANOVA

A one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of three or more independent groups. In this context, it tests if the average college attendance rates among the different geographic regions are equal.

Significance Level

The significance level, denoted by alpha (?), is the threshold for deciding whether an observed effect is statistically significant. Here, ? = 0.05 means there is a 5% risk of concluding that a difference exists when there is none.

Assumptions of ANOVA

ANOVA relies on a few key assumptions: independence of observations, normally distributed populations for each group, and homogeneity of variances (equal variances) across groups. Verifying these assumptions is crucial to ensure the validity of the ANOVA results.

Transcript

Please give Ace some feedback