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, we need to understand the concepts and criteria for these tests.

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.
This statement is TRUE. The formula for the estimated standard error (ESE) is the same in both types of two-sample t-tests. The only difference is in how the degrees of freedom are calculated.

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.
This statement is not clear. To determine whether it would be appropriate to use the two-sample t-test with a common variance, we would need additional information, specifically the sample sizes and the significance level of the test. With only the information on sample variances, it is not possible to definitively state if it would be appropriate or not.

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.
This statement is TRUE. When the sample variances of the two groups being compared are significantly different, it is not appropriate to use the two-sample t-test with a common variance assumption. In this case, the sample variances are different (2.16² and 4.82²), indicating unequal variances.

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.
This statement is TRUE. In the two-sample t-test with unequal variances, the null hypothesis is that the means of the two populations being compared are equal. The alternative hypothesis is that the means are not equal.

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.
This statement is FALSE. It is possible to calculate a confidence interval for the difference between two means using the two-sample t-test with unequal variances. The confidence interval provides an estimate of the range within which the true difference between the means is likely to fall.

Based on the explanations above:
- The first statement is TRUE.
- The third statement is TRUE.
- The fourth statement is TRUE.
- The fifth statement is FALSE.

Therefore, the correct answer would be the first and third options.