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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 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.