The 95% confidence interval for the difference between the mean GPA of college A students and the mean GPA of college B students is -0.65 to -0.01 (Option D).
How can you construct the 95% confidence interval?
College A:
Sample mean (μ₁) = (3.7 + 3.8 + 2.8 + 3.2 + 3.2 + 3.0 + 2.5 + 2.7) / 8 = 3.1875
Sample standard deviation (s₁) = √[(3.7 - 3.1875)² + (3.8 - 3.1875)² + ... + (2.7 - 3.1875)²] / (8 - 1) = 0.4063
College B:
Sample mean (μ₂) = (3.9 + 3.8 + 4.0 + 2.5 + 3.9 + 2.6 + 3.8 + 4.0 + 3.6 + 2.5 + 3.6 + 3.9) / 13 = 3.4846
Sample standard deviation (s₂) = √[(3.9 - 3.4846)² + (3.8 - 3.4846)² + ... + (3.9 - 3.4846)²] / (13 - 1) = 0.4494
Since we are assuming independent samples from two populations with unknown variances, we need to calculate the pooled variance (s²p) to estimate the population variance:
s²p = [(n₁ - 1)s₁² + (n₂ - 1)s₂²] / (n₁ + n₂ - 2)
s²p = [(8 - 1) * 0.4063² + (13 - 1) * 0.4494²] / (8 + 13 - 2)
s²p = 0.1883
The margin of error (ME) is calculated based on the critical value from the t-distribution for a 95% confidence interval with degrees of freedom (df) equal to n₁ + n₂ - 2:
df = n₁ + n₂ - 2 = 8 + 13 - 2 = 19
Critical value (t_α/2) from the t-distribution table for df = 19 and α = 0.05 (one-tailed) is 2.131.
ME = t_α/2 *
(s²p * (1/n₁ + 1/n₂))
ME = 2.131 *
(0.1883 * (1/8 + 1/13))
ME = 0.3531
Finally, the 95% confidence interval for the difference between the population means (μ₁ - μ₂) is:
(μ₁ - μ₂) ± ME
(3.1875 - 3.4846) ± 0.3531
-0.2971 ± 0.3531
(-0.6502, -0.0140)
Therefore, the 95% confidence interval for the difference between the mean GPA of college A students and the mean GPA of college B students is -0.65 to -0.01 (Option D). This means that we are 95% confident that the true difference between the mean GPAs lies within this range.