Question

A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following GPAs.

College A College B
3.7 3.8 2.8
3.2 3.2 4.0
3.0 3.0 3.6
2.5 3.9 2.6
2.7 3.8 4.0
3.6 2.5 3.6
2.8 3.9
3.4

Construct a 95% confidence interval for ₁−₂μ₁−μ₂, the difference between the mean GPA of college A students and the mean GPA of college B students.

A. -0.91 to 0.25
B. -0.81 to 0.15
C. -072 to 0.07
D. -0.65 to -0.01

Answer 1

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 *
`\sqrt`(s²p * (1/n₁ + 1/n₂))

ME = 2.131 *
`\sqrt`(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.