Question

Independent random samples are taken at a university to compare the average GPA of seniors to the average GPA of sophomores. Given a 95% confidence interval for the difference between the true average GPAs (seniors - sophomores) of (0.53, 1.61), what can you conclude?

A. We are 95% confident that the average GPA of seniors is greaterthanequalto the average GPA of sophomores.
B. We do not have enough information to make a conclusion.
C. We are 95% confident that the difference between the two sample GPAs falls within the interval.
D. There is no significant difference between the true average GPA for seniors and sophomores.
E. We are 95% confident that the average GPA of seniors is less than the average GPA of sophomores.

Answer 1

Correct options is (A).

Explanation:

A (1 - α)% confidence interval can be constructed to determine whether there is any difference between the average GPA of seniors and the average GPA of sophomores.

The hypothesis can be defined as:

H₀: There is no difference between the average GPA of seniors and the average GPA of sophomores, i.e. μ₁ - μ₂ = 0.

Hₐ: There is a difference between the average GPA of seniors and the average GPA of sophomores, i.e. μ₁ - μ₂ ≠ 0.

The (1 - α)% confidence interval for the difference between two means is:

`CI=(\bar x_(1)-\bar x_(1))\pm z_(\alpha /2)* SE`

Decision rule:

If the confidence interval contains the mull value, i.e. 0, then the null hypothesis will be failed to be rejected and vice-versa.

The 95% confidence interval for the difference between the true average GPAs (seniors - sophomores) is (0.53, 1.61).

The 95% confidence interval dos not contains 0. Thus, the null hypothesis will be rejected.

Conclusion:

There is a significant difference between the average GPA of seniors and the average GPA of sophomores.

The 95% confidence interval for the difference between the true average GPAs (0.53, 1.61), implies that there is a 95% confidence that the average GPA of seniors is greater than the average GPA of sophomores.

Thus, the correct options is (A).