Select the TWO options that are TRUE statements about two-sample t-tests.
Select one or more:
The same formula is used for the ESE (estimated standard error) in the two-sample t-test with a common variance and the two-sample t-test with unequal variances.
A researcher has collected two samples of data and the sample variances are 1.45 and 2.11. It would not be appropriate to use the two-sample t-test with a common variance.
A researcher has collected two samples of data and the sample variances are 2.16² and 4.82². It would not be appropriate to use the two-sample t-test with a common variance.
The two-sample t-test with unequal variances has, as its null hypothesis, that the means of the two populations involved are the same.
It is not possible to find a confidence interval for the difference between two means based on the two sample t-test with unequal variances.
To determine which statements are true about two-sample t-tests, let's go through each option:
1. The same formula is used for the ESE (estimated standard error) in the two-sample t-test with a common variance and the two-sample t-test with unequal variances:
To verify this statement, we need to understand the formulas used for each type of t-test. In the case of the two-sample t-test with a common variance (also known as pooled t-test), the formula for calculating the estimated standard error is the same for both samples. In the case of the two-sample t-test with unequal variances (also known as Welch's t-test), different formulas are used to calculate the estimated standard error for each sample. Therefore, this statement is FALSE.
2. A researcher has collected two samples of data, and the sample variances are 1.45 and 2.11. It would not be appropriate to use the two-sample t-test with a common variance:
To determine the appropriateness of using a two-sample t-test with a common variance, we need to consider the assumption of equal variances between the two populations. If the sample variances are significantly different, using the t-test with a common variance may not be appropriate. In this case, the statement suggests that the sample variances are 1.45 and 2.11, which are not significantly different. This implies that it would be appropriate to use the two-sample t-test with a common variance. Therefore, this statement is FALSE.
3. A researcher has collected two samples of data, and the sample variances are 2.16² and 4.82². It would not be appropriate to use the two-sample t-test with a common variance:
Similar to the previous statement, we need to consider the assumption of equal variances between the two populations. If the sample variances are significantly different, using the t-test with a common variance may not be appropriate. In this case, the statement suggests that the sample variances are 2.16² and 4.82², which are significantly different. This implies that it would not be appropriate to use the two-sample t-test with a common variance but rather the two-sample t-test with unequal variances. Therefore, this statement is TRUE.
4. The two-sample t-test with unequal variances has, as its null hypothesis, that the means of the two populations involved are the same:
The null hypothesis in a two-sample t-test with unequal variances is that the means of the two populations are equal. Therefore, this statement is TRUE.
5. It is not possible to find a confidence interval for the difference between two means based on the two-sample t-test with unequal variances:
A confidence interval estimates the range of plausible values for a parameter, such as the difference between two means. While the calculation of a confidence interval may be more complex in a two-sample t-test with unequal variances compared to a t-test with a common variance, it is still possible to find a confidence interval using appropriate statistical methods. Therefore, this statement is FALSE.
To summarize, the two TRUE statements about two-sample t-tests are:
- A researcher has collected two samples of data, and the sample variances are 2.16² and 4.82². It would not be appropriate to use the two-sample t-test with a common variance.
- The two-sample t-test with unequal variances has, as its null hypothesis, that the means of the two populations involved are the same.