John wants to advocate for making study rooms available to students. In an effort to learn whether the time of day or the day of the week would be most important to consider in deciding when to make such rooms available, he polls a randomly selected group of students to find out what time of day and what days of the week students at his school tend to study. Morning

Question

John wants to advocate for making study rooms available to students. In an effort to learn whether the time of day or the day of the week would be most important to consider in deciding when to make such rooms available, he polls a randomly selected group of students to find out what time of day and what days of the week students at his school tend to study. Morning

5
12
18
16
9
Afternoon
14
24
26
19
5
Evening
12
14
17
13
6
Run an appropriate test to help John decide whether it makes more difference what time of day the rooms are available or what day of the week. Use alpha = 0.05.

Answer 1

To determine whether it makes more difference what time of day the study rooms are available or what day of the week, John can conduct a statistical test. In this case, a two-way analysis of variance (ANOVA) would be appropriate.

Here's a step-by-step explanation of how John can perform the ANOVA test:

Step 1: State the null and alternative hypothesis:
The null hypothesis (H0) is that the means of study times are equal for all times of day and all days of the week. The alternative hypothesis (Ha) is that at least one mean is different.

Step 2: Set the significance level:
In this case, John has already specified α (the significance level) as 0.05.

Step 3: Compute the sum of squares within groups (SSW):
To calculate SSW, he needs to find the sum of squared deviations for each group, which is the squared difference of each observation from the group mean.

Step 4: Compute the sum of squares between groups (SSB):
To calculate SSB, he needs to find the squared difference between each group mean and the overall mean, multiplied by the number of observations in each group.

Step 5: Calculate the degrees of freedom:
John should determine the degrees of freedom for the within-groups (dfW) and between-groups (dfB) sum of squares.

dfW = (number of times of day - 1) * (number of days of the week - 1)
dfB = (number of times of day - 1) + (number of days of the week - 1)

Step 6: Calculate the mean square (MS):
To calculate the mean square, divide the corresponding sum of squares (SSW / dfW, SSB / dfB).

Step 7: Compute the F-statistic:
John can compute the F-statistic by dividing the mean square between groups (MSB) by the mean square within groups (MSW).

Step 8: Determine the critical value:
Using the significance level α and the degrees of freedom for the between-groups and within-groups, John can find the critical value from the F-distribution table.

Step 9: Compare the F-statistic and critical value:
If the F-statistic is greater than the critical value, John can reject the null hypothesis and conclude that there is a significant difference between the means. Conversely, if the F-statistic is less than the critical value, he fails to reject the null hypothesis.

By following these steps, John can perform the appropriate ANOVA test to determine whether it makes more difference what time of day the study rooms are available or what day of the week.