Answer: To test for unequal variances, we use Welch's t-test, which is a modification of the Student's t-test that adjusts for unequal variances. The null hypothesis for Welch's t-test is that the population means are equal, and the alternative hypothesis is that they are not.
The critical values for a two-tailed test at alpha level 0.10 with degrees of freedom df = 23.99 can be found using a t-distribution table or a calculator and are ±1.717.
For the first sample, we have:
Sample 1: s1 = 10.2, n1 = 22
Sample 2: s2 = 6.4, n2 = 16
Test: Two-tailed
The degrees of freedom can be calculated as follows:
df = ((s1^2 / n1) + (s2^2 / n2))^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
= ((10.2^2 / 22) + (6.4^2 / 16))^2 / ((10.2^2 / 22)^2 / 21 + (6.4^2 / 16)^2 / 15)
= 23.81
The calculated t-value is:
t = (x1 - x2) / sqrt(s1^2 / n1 + s2^2 / n2)
= (0 - 0) / sqrt(10.2^2 / 22 + 6.4^2 / 16) = 0
Since the calculated t-value is within the critical region (-1.717, 1.717), we fail to reject the null hypothesis. We can conclude that there is insufficient evidence to suggest that the population means are different.
For the second sample, we have:
Sample 1: s1 = 0.89, n1 = 25
Sample 2: s2 = 0.67, n2 = 18
Test: Right-tailed
The degrees of freedom can be calculated as follows:
df = ((s1^2 / n1) + (s2^2 / n2))^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
= ((0.89^2 / 25) + (0.67^2 / 18))^2 / ((0.89^2 / 25)^2 / 24 + (0.67^2 / 18)^2 / 17)
= 34.13
The critical value for a right-tailed test at alpha level 0.10 with degrees of freedom df = 34.13 can be found using a t-distribution table or a calculator and is 1.311.
The calculated t-value is:
t = (x1 - x2) / sqrt(s1^2 / n1 + s2^2 / n2)
= (0.89 - 0.67) / sqrt(0.89^2 / 25 + 0.67^2 / 18)
= 2.42
Since the calculated t-value (2.42) is greater than the critical value (1.311), we reject the null hypothesis. We can conclude that there is sufficient evidence to suggest that the population mean of Sample 1 is greater than the population mean of Sample 2.
For the third sample, we have:
Sample 1: s1 = 124, n1 = 12
Sample 2: s2 = 260, n2 = 10
Test: Left-tailed
The degrees of freedom can be calculated as follows:
df = ((s1^2 / n1) + (s2^2 / n2))^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
= ((124^2 / 12) + (260^2 / 10))^2 / ((124^2 / 12)^2 / 11 + (260^2 / 10)^2 / 9)
= 14.23
The critical value for a left-tailed test at alpha level 0.10 with degrees of freedom df = 14.23 can be found using a t-distribution table or a calculator and is -1.345.
The calculated t-value is:
t = (x1 - x2) / sqrt(s1^2 / n1 + s2^2 / n2)
= (0 - 0) / sqrt(124^2 / 12 + 260^2 / 10) = 0
Since the calculated t-value is not less than the critical value (-1.345), we fail to reject the null hypothesis. We can conclude that there is insufficient evidence to suggest that the population mean of Sample 1 is less than the population mean of Sample 2.
The decision rule for all three tests is:
If the calculated t-value is within the critical region, fail to reject the null hypothesis.
If the calculated t-value is greater than the critical value for a right-tailed test, reject the null hypothesis in favor of the alternative hypothesis that the population mean of Sample 1 is greater than the population mean of Sample 2.
If the calculated t-value is less than the critical value for a left-tailed test, reject the null hypothesis in favor of the alternative hypothesis that the population mean of Sample 1 is less than the population mean of Sample 2.
Explanation: