Question

In a survey of 1000 high school students in Oregon, the average SAT score for 500 students who chose to go out of state for college (group 1) was 1225 and the average SAT score for 500 students who chose to stay in state for college (group 2) was 1130. The population standard deviation for students who choose to go out of state is 95, and the population standard deviation for students who choose to stay in state is 103. Find a 95% confidence interval and decide if the SAT scores between the two groups are significantly different.

Options:
a. μ1 - μ2 < 51.35
b. μ1 - μ2 > 51.35
c. μ1 - μ2 ≠ 51.35
d. μ1 - μ2 = 51.35

Answer 1

To find a 95% confidence interval and determine if the SAT scores between the two groups are significantly different, we can use the formula for the confidence interval. Plugging in the given values and calculating, the confidence interval is (82.73, 107.27). Since the confidence interval does not include 51.35, we can conclude that the SAT scores between the two groups are significantly different. The correct option is c. μ1 - μ2 ≠ 51.35.

Step-by-step explanation:

To find a 95% confidence interval and determine if the SAT scores between the two groups are significantly different, we can use the formula for the confidence interval:

CI = (X1 - X2) ± Z * √((σ1^2/n1) + (σ2^2/n2))

Where:

• X1 and X2 are the sample means
• σ1 and σ2 are the population standard deviations
• n1 and n2 are the sample sizes
• Z is the critical value

Plugging in the given values:

X1 = 1225, X2 = 1130, σ1 = 95, σ2 = 103, n1 = n2 = 500

The critical value for a 95% confidence interval is approximately 1.96.

Calculating the confidence interval:

CI = (1225 - 1130) ± 1.96 * √((95^2/500) + (103^2/500))

CI = 95 ± 1.96 * √(9025/500 + 10609/500)

CI = 95 ± 1.96 * √(18.05 + 21.218)

CI = 95 ± 1.96 * √(39.268)

CI = 95 ± 1.96 * 6.266

CI = 95 ± 12.27

CI = (82.73, 107.27)

Since the confidence interval does not include 51.35, we can conclude that the SAT scores between the two groups are significantly different. Therefore, the correct option is c. μ1 - μ2 ≠ 51.35.