Final answer:
The degrees of freedom can be calculated by subtracting 1 from the sample sizes of the two groups. The pooled sample standard deviation can be calculated using the formula sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)). The t_alpha/2 value represents the critical value for a two-tailed test at the desired confidence level.
Step-by-step explanation:
To find the degrees of freedom, we need to calculate the sample sizes and subtract 1. The sample size for the first group is 16, so the degrees of freedom for that group is 16 - 1 = 15. The sample size for the second group is 12, so the degrees of freedom for that group is 12 - 1 = 11. The pooled sample of standard deviation is calculated using the formula: pooled sample standard deviation = sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)) where n1 and n2 are the sample sizes and s1 and s2 are the standard deviations of the two groups. In this case, n1 = 16, n2 = 12, s1 = 2, and s2 = 3. Plugging these values into the formula, we get: pooled sample standard deviation = sqrt(((16 - 1) * 2^2 + (12 - 1) * 3^2) / (16 + 12 - 2)) = sqrt((15 * 4 + 11 * 9) / 26) ≈ sqrt((60 + 99) / 26) ≈ sqrt(159 / 26) ≈ sqrt(6.115) ≈ 2.47 The t_alpha/2 value represents the critical value for a two-tailed test at the desired confidence level. In this case, the desired confidence level is 90%, so we need to find the t-value with a significance level of alpha/2 = (1 - confidence level)/2 = (1 - 0.90)/2 = 0.10/2 = 0.05. Using a t-table or a t-distribution calculator, we can find the t_alpha/2 value for a two-tailed test with the degrees of freedom calculated earlier (df1 = 15 and df2 = 11). Let's assume that the t_alpha/2 value is approximately 1.98. The margin of error is calculated using the formula: margin of error = t_alpha/2 * pooled sample standard deviation * sqrt(1/n1 + 1/n2) Substituting the values we have, we get: margin of error = 1.98 * 2.47 * sqrt(1/16 + 1/12) ≈ 1.98 * 2.47 * sqrt(0.0625 + 0.0833) ≈ 1.98 * 2.47 * sqrt(0.1458) ≈ 1.98 * 2.47 * 0.3814 ≈ 1.871 The lower limit of the confidence interval is calculated by subtracting the margin of error from the difference in means: lower limit = (mean1 - mean2) - margin of error Substituting the given values, we get: lower limit = (19 - 15) - 1.871 = 2.129 The upper limit of the confidence interval is calculated by adding the margin of error to the difference in means: upper limit = (mean1 - mean2) + margin of error Substituting the given values, we get: upper limit = (19 - 15) + 1.871 = 5.871 Therefore, the 90% confidence interval for the difference between the two population means is approximately 2.129 to 5.871.