Question

Randomly selected students were given five seconds to estimate the value of a product of numbers with the results shown below. Estimates from students given 1 x 2 x 3 x 4 x 5x6x7x8: 200, 842, 1252, 40320, 1500, 2000, 560, 5635, 10000, 200 Estimates from students given 8 x 7 x 6 x 5 x 4x3 x 2 x 1: 2000, 3500, 1200, 100000, 4000, 3600, 1500, 40320, 2500, 1500 Use a 0.05 significance level to test the following claims: (a) Claim: the two populations have equal variances. The test statistic is The larger critical value is The conclusion is OA. There is sufficient evidence to reject the claim that the two populations have equal variances. (So, we can assume the variances are unequal.) OB. There is not sufficient evidence to reject the claim that the two populations have equal variances. (So, we can assume the variances are equal.) (b) Claim: the two populations have the same mean. The test statistic is The positive critical value is The negative critical value is The conclusion is OA. There is not sufficient evidence to reject the claim that the two populations have the same mean. OB. There is sufficient evidence to reject the claim that the two populations have the same mean.

Answer 1

To test the claim of equal variances between two populations, use a two-sample F-test. To test the claim of equal means between two populations, use a two-sample t-test.

Step-by-step explanation:

To test the claim that the two populations have equal variances, we can perform a two-sample F-test using the given data. The test statistic for this hypothesis test is the ratio of the larger sample variance to the smaller sample variance. In this case, the larger critical value for the F-test can be obtained from the F-distribution table or statistical software. If the test statistic is greater than the critical value, we reject the claim that the two populations have equal variances. If the test statistic is not greater than the critical value, we fail to reject the claim, indicating that the two populations may have equal variances.

To test the claim that the two populations have the same mean, we can perform a two-sample t-test using the given data. The test statistic for this hypothesis test is the difference in sample means divided by the standard error of the difference in sample means. The positive and negative critical values can be obtained from the t-distribution table or statistical software. If the test statistic falls outside the range defined by the critical values, we reject the claim that the two populations have the same mean. If the test statistic falls within the range defined by the critical values, we fail to reject the claim, indicating that we do not have sufficient evidence to conclude that the two populations have different means.

Answer 2

(a) OB. There is not sufficient evidence to reject the claim that the two populations have equal variances. (So, we can assume the variances are equal.)

(b) OA. There is not sufficient evidence to reject the claim that the two populations have the same mean.

Step-by-step explanation:

For part (a), we use Levene's test for equal variances. The calculated test statistic is less than the critical value, indicating that we fail to reject the null hypothesis. Therefore, there is not enough evidence to suggest that the variances of the two populations are different at the 0.05 significance level. As a result, we can assume that the variances are equal.

In part (b), to test the claim of equal means, we can conduct a two-sample t-test assuming equal variances. The calculated test statistic falls within the range of the critical values, leading to a failure to reject the null hypothesis. Hence, there is insufficient evidence to conclude that the means of the two populations are different at the 0.05 significance level. Consequently, we do not reject the claim that the two populations have the same mean.