I just need help with starting how to calculate the problem I am so lost.

A random sample of eight observations from the first population resulted in a standard deviation of 10. A random sample of six observations from the second population resulted in a standard deviation of 7. At the .02 significance level, is there a difference in the variation of the two populations?

To determine if there is a difference in the variation of the two populations, you can perform a hypothesis test. Specifically, you can conduct a two-sample F-test.

Here are the step-by-step instructions to calculate the F-test:

Step 1: State the hypothesis:
- Null hypothesis (H0): There is no significant difference in the variation of the two populations.
- Alternative hypothesis (Ha): There is a significant difference in the variation of the two populations.

Step 2: Define significance level:
- Given that the significance level is 0.02.

Step 3: Calculate the test statistic (F-value):
- The F-value can be calculated using the formula F = (S12 / S22), where S1 and S2 are the standard deviations of the first and second populations, respectively.
- In this case, S1 = 10 and S2 = 7.

F = (10^2 / 7^2)
= 100 / 49
≈ 2.04

Step 4: Determine the critical value:
- The critical value can be obtained from an F-table or using statistical software.
- Since the significance level is 0.02 and the sample sizes are 8 and 6, the degrees of freedom are 7 and 5, respectively.
- Using the F-table, the critical value for a two-tailed test at α = 0.02 with df (7,5) is approximately 5.59.

Step 5: Compare the test statistic with the critical value:
- If the test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Since 2.04 < 5.59, we fail to reject the null hypothesis.

Step 6: Make a conclusion:
- Based on the results of the hypothesis test, there is insufficient evidence to conclude that there is a significant difference in the variation of the two populations at the 0.02 significance level.

To compare the variation of two populations, we can use a statistical test called the F-test for equality of variances. In this case, we have two samples from two populations, and we are trying to determine if there is a significant difference in their variations.

To perform the F-test, we need to calculate the F-statistic and compare it to the critical value at the desired significance level. Here is how you can calculate the F-statistic and perform the test:

Step 1: State the hypotheses:
- Null hypothesis (H0): The variances of the two populations are equal.
- Alternative hypothesis (Ha): The variances of the two populations are not equal.

Step 2: Set the significance level (α):
The significance level, also known as alpha (α), is the probability of rejecting the null hypothesis when it is true. In this case, the significance level is given as 0.02.

Step 3: Calculate the F-statistic:
The F-statistic is calculated by dividing the larger sample variance by the smaller sample variance. In this case, the larger standard deviation is 10 (from the first population), and the smaller standard deviation is 7 (from the second population).

F-statistic = (larger standard deviation)^2 / (smaller standard deviation)^2

F-statistic = (10^2) / (7^2)

Step 4: Find the critical value:
The critical value is obtained from the F-distribution table, which depends on the significance level (α) and the degrees of freedom for the two samples.

The degrees of freedom for the first population is n1-1 (where n1 is the number of observations in the first sample), and the degrees of freedom for the second population is n2-1 (where n2 is the number of observations in the second sample). In this case, the first sample has 8 observations (n1=8) and the second sample has 6 observations (n2=6).

Step 5: Compare the F-statistic and critical value:
If the calculated F-statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference in the variation of the two populations.

If the calculated F-statistic is less than or equal to the critical value, we fail to reject the null hypothesis and conclude that there is no significant difference in the variation of the two populations.

Now, you can calculate the F-statistic and compare it to the critical value to determine if there is a difference in the variation of the two populations.