Eating raw fish (and enjoying it) is an acquired taste for many people who have never tried sushi. Ten people who had never before eaten sushi were asked to rate how much they liked it on a 100-point scale (100=fantastic, 0=yuck) on each of four occasions one week apart. The following are the TOTALS of the 10 subjects ratings on each occasion:

Following is the subjects' total ratings of sushi enjoyment

Time 1 / Time 2 / Time 3 / Time 4
550 / 570 / 600 / 800

When these data were analyzed, it was found that 25% of the total variance in the subjects' ratings was attributable to the effect of time (or occasion of eating sushi) while 37.5% of the total variance was attributable to indivdiual differences among the subjects themselves.

a) Is the total variance of the subjects ratings different from 400, the population variance for these ratings, alpha = .05?

ANSWER:
Chi-squared-obtained = 39.8, do not reject Ho

This looks to be some sort of repeated measures test. You may want to look into something like Friedman's nonparametric test using the chi-square distribution.

To determine if the total variance of the subjects' ratings is different from 400, we can use the Friedman's nonparametric test. This test is appropriate for analyzing data where the same subjects are measured on multiple occasions.

Here's how you can perform the test and obtain the chi-squared-obtained value:

Step 1: Calculate the average ratings for each time period.
- Time 1 average = 550 / 10 = 55
- Time 2 average = 570 / 10 = 57
- Time 3 average = 600 / 10 = 60
- Time 4 average = 800 / 10 = 80

Step 2: Calculate the ranks for the ratings.
- Assign each average rating a rank from 1 to 10, with 1 being the lowest and 10 being the highest. If there are ties, assign an average rank to each tied value. For example, if two averages are tied for the highest, both are assigned a rank of 10. If there are three tied for the lowest, all three are assigned a rank of 1.

Step 3: Calculate the sum of squared ranks for each time.
- Time 1 sum of squared ranks = 1^2 + 2^2 + ... + 10^2 = 385
- Time 2 sum of squared ranks = 1^2 + 2^2 + ... + 10^2 = 385
- Time 3 sum of squared ranks = 1^2 + 2^2 + ... + 10^2 = 385
- Time 4 sum of squared ranks = 1^2 + 2^2 + ... + 10^2 = 385

Step 4: Calculate the chi-squared-obtained value.
- The chi-squared-obtained value can be calculated using the following formula:
chi-squared-obtained = (12 * n) / [(k * (k + 1))] * [(sum of squared ranks) - (k * (k + 1)^2) / 4]
where n is the number of subjects, k is the number of time periods, and the sum of squared ranks is the sum calculated in Step 3.

In this case, n = 10 and k = 4.

Using the formula, we can calculate the chi-squared-obtained value:
chi-squared-obtained = (12 * 10) / [(4 * (4 + 1))] * [(385) - (4 * (4 + 1)^2) / 4]
= (120) / (20) * (385 - (4 * 5)^2 / 4)
= (120) / (20) * (385 - 4 * 25)
= (120) / (20) * (385 - 100)
= (120) / (20) * (285)
= (120) / (20) * (285)
= 720

Step 5: Determine the critical chi-squared value.
- The critical chi-squared value can be obtained from a chi-squared distribution table. The degrees of freedom for this test are (k-1).

In this case, k = 4, so the degrees of freedom are 3.

Step 6: Compare the chi-squared-obtained value with the critical chi-squared value.
- If the chi-squared-obtained value is less than the critical chi-squared value, we do not reject the null hypothesis (Ho) and conclude that the total variance of the subjects' ratings is not different from 400.

From our calculations, we obtained a chi-squared-obtained value of 720. To determine if this value is significant, you would need to compare it to the critical chi-squared value with 3 degrees of freedom. If the chi-squared-obtained value is less than the critical chi-squared value, you would not reject the null hypothesis and conclude that the total variance of the subjects' ratings is not different from 400.

Please note that the critical chi-squared value depends on the desired significance level (alpha), which you stated as alpha = .05.