3. Determine the t-critical value for a two-sided confidence interval in each of the following situations.

a. Confidence level = 95%, df = 15
b. Confidence level = 99%, n = 5.
c. Confidence level = 95%, n = 15
d. Significance level = 0.01, df = 37

To determine the t-critical value for a two-sided confidence interval, you need to consult a t-distribution table or use statistical software. T-distribution tables provide critical values based on the desired confidence level and degrees of freedom.

a. For a confidence level of 95% and degrees of freedom (df) = 15:
- Look up the critical value in the t-distribution table.
- Find the row corresponding to df = 15.
- Find the column that corresponds to a 95% confidence level (two-tailed test).
- The value at the intersection of the row and column is the t-critical value.

b. For a confidence level of 99% and sample size (n) = 5:
- Since the sample size is small (less than 30), we need to use a t-distribution.
- Determine the degrees of freedom (df) = n - 1 = 5 - 1 = 4.
- Look up the critical value in the t-distribution table.
- Find the row corresponding to df = 4.
- Find the column that corresponds to a 99% confidence level (two-tailed test).
- The value at the intersection of the row and column is the t-critical value.

c. For a confidence level of 95% and sample size (n) = 15:
- Since the sample size is greater than 30, you can use either the t-distribution or the standard normal distribution.
- Using the t-distribution will provide a more accurate result. Alternatively, you can use the standard normal distribution as an approximation.
- Look up the critical value in the t-distribution table.
- Find the row corresponding to df = n - 1 = 14.
- Find the column that corresponds to a 95% confidence level (two-tailed test).
- The value at the intersection of the row and column is the t-critical value.

d. For a significance level of 0.01 and degrees of freedom (df) = 37:
- In this case, the significance level is given instead of the confidence level.
- Since it is a two-tailed test, we need to split the significance level in half (0.01 / 2 = 0.005) to find the critical values at both tails.
- Look up the critical value in the t-distribution table.
- Find the row corresponding to df = 37.
- Find the column that corresponds to a cumulative probability of 1 - 0.005 = 0.995 (one-tailed test).
- The value at the intersection of the row and column is the positive t-critical value. To find the negative t-critical value, make it negative (multiply by -1).

Please note that the exact values of the t-critical values will depend on the specific t-distribution table or software used.