A school administrator wants to examine whether students' academic performance differs based on the type of instructional method used in their classes. A random sample of $18$ students is selected and divided evenly among the three teaching methods. After a semester, all students take the same standardized final exam. An ANOVA test is performed and results in a P-value of $1.403\bullet10^{-7}$. Interpret these results.

A regional sales director wants to determine whether different customer service training programs lead to different levels of employee performance across three branches. Each branch uses one of the following training programs: Program A. Program B, or Program C. After one month, the director measures the performance score (out of 100) for 5 randomly selected employees from each branch. Using $\alpha=0.05$, perform a one-way ANOVA to determine whether there is a statistically significant difference in mean performance among the three training programs.

A company wants to determine whether the average monthly sales differ among three different regions: North, South, and West. The company collects monthly sales data (in thousands of dollars) from four randomly selected stores in each region over the same month. Calculate the F-statistic given the Mean Square due to Treatments: MST = $226.6$ (variance between groups) and the Mean Square due to Error: MSE = $7.944$ (variance within groups).

Four different high schools in local towns took random samples of 100 students in three grades, $10^{\operatorname{th}}-12^{\operatorname{th}}$ and collected data on the weekly time spent studying to see if students in each of these grades study on average for the same amount of time per week. The four schools ran ANOVA tests on their samples, and the F-Statistics were $2.35$, $2.57$, $2.81$, and $3.93$. Which F-Statistic is most likely to indicate the average study times across grades are not all the same?

A school administrator wants to examine whether students' academic performance differs based on the type of instructional method used in their classes. A random sample of $18$ students is selected and divided evenly among the three teaching methods. After a semester, all students take the same standardized final exam. State the null and alternative hypotheses for a one-way ANOVA test.

A marketing manager wants to evaluate whether three different advertising platforms-TV, social media, and print media-lead to different average sales performance across regional stores. She runs a 4-week advertising campaign, assigning one platform to a group of 5 stores each (15 stores total). After the campaign, she collects the average weekly sales (in $1,000s) for each store during the campaign period. She wants to determine whether there is a statistically significant difference in mean sales among the three advertising platforms. State the null & alternative hypotheses for a one-way ANOVA test.

A marketing manager wants to evaluate whether three different advertising platforms-TV, social media, and print media-lead to different average sales performance across regional stores. She runs a 4-wook advertising campaign, assigning one platform to a group of 5 stores each (15 stores total). After the campaign, she collects the weekly soles (in $1,000s) for each store during the campaign period. She wants to determine whether there is a statistically significant difference in mean sales among the three advertising platforms. In an ANOVA test a P-value of 0.03 is obtained. What can be concluded about mean weekly sales for different advertising platforms?

A regional sales director wants to determine whether different customer service training programs lead to different levels of employee performance across three branches. Each branch uses one of the following training programs: Program A. Program B, or Program C. After one month, the director measures the performance score (out of 100) for 5 randomly selected employees from each branch. Using $\alpha=0.05$, perform a one-way ANOVA to determine whether there is a statistically significant difference in mean performance among the three training programs.