A consumer agency wanted to investigate if four insurance companies differed with regard to the premiums they charge for auto insurance. The agency randomly selected a few auto drivers who were insured by each of these four companies and had similar driving records,autos, and insurance policies. The following table gives the premiums paid per month by these drivers insured with these four insurance companies.


Company A - 75 83 68 52

Company B - 59 75 100

Company C - 65 70 97 90 73

Company D - 76 60 52 58

Using the 5% significance level, test the null hypothesis that the mean auto insurance premium paid per month by all drivers insured by each of these four companies is the same.

To test the null hypothesis that the mean auto insurance premium paid per month by all drivers insured by each of these four companies is the same, we can use a one-way analysis of variance (ANOVA) test.

The ANOVA test compares the variation among multiple groups (in this case, the insurance companies) with the variation within each group (the premiums paid by the drivers insured by each company). If the variation among the groups is significantly larger than the variation within each group, it suggests that there are differences in the means of the groups.

Here are the steps to perform the ANOVA test using the provided data:

Step 1: Calculate the means (average) of the premiums paid by drivers insured by each company:
- Company A mean: (75 + 83 + 68 + 52) / 4 = 69.5
- Company B mean: (59 + 75 + 100) / 3 = 78
- Company C mean: (65 + 70 + 97 + 90 + 73) / 5 = 79
- Company D mean: (76 + 60 + 52 + 58) / 4 = 61.5

Step 2: Calculate the overall mean of all premiums paid:
- Overall mean = (69.5 + 78 + 79 + 61.5) / 4 = 72.25

Step 3: Calculate the sum of squared differences within each group (SSwithin):
- SSwithin = (75 - 69.5)^2 + (83 - 69.5)^2 + (68 - 69.5)^2 + (52 - 69.5)^2 + (59 - 78)^2 + (75 - 78)^2 + (100 - 78)^2 + (65 - 79)^2 + (70 - 79)^2 + (97 - 79)^2 + (90 - 79)^2 + (73 - 79)^2 + (76 - 61.5)^2 + (60 - 61.5)^2 + (52 - 61.5)^2 + (58 - 61.5)^2

Step 4: Calculate the sum of squared differences between each group mean and the overall mean (SSbetween):
- SSbetween = 4 * [ (69.5 - 72.25)^2 + (78 - 72.25)^2 + (79 - 72.25)^2 + (61.5 - 72.25)^2 ]

Step 5: Calculate the degrees of freedom for within-group and between-group variations:
- dfwithin = total number of premiums - number of groups = 16 - 4 = 12
- dfbetween = number of groups - 1 = 4 - 1 = 3

Step 6: Calculate the mean squares (MS) for within-group and between-group variations:
- MSwithin = SSwithin / dfwithin
- MSbetween = SSbetween / dfbetween

Step 7: Calculate the F-statistic:
- F = MSbetween / MSwithin

Step 8: Determine the critical F-value at a 5% significance level with the degrees of freedom (dfbetween, dfwithin) and compare it with the calculated F-value. If the calculated F-value is greater than the critical F-value, we reject the null hypothesis.

Step 9: Interpret the results. If we reject the null hypothesis, it implies that there are significant differences in the mean auto insurance premiums among the insurance companies.

Note: The calculation of SSwithin, SSbetween, MSwithin, and MSbetween can be done manually or using statistical software such as Excel, SPSS, or R.

Performing these steps will help you test the null hypothesis that the mean auto insurance premium paid per month by all drivers insured by each of these four companies is the same.