Answer :
To compare the Math SAT scores of girls at an elite math and science academy with the average SAT scores for high school girls nationally, the appropriate statistical test to use is the two-sample t-test.
Here’s a step-by-step explanation:
Who: Marcus and Joiner are comparing two groups: 1) Girls from an elite math and science academy, and 2) High school girls nationwide.
What: They want to compare Math SAT scores between these two groups.
Why: The goal is to determine if there is a statistically significant difference between the Math SAT scores of the academy's students and the national average.
When and Where: The test can be performed once the scores from both groups are collected.
How:
Assumptions: Before using the two-sample t-test, check the assumptions:
a. The distribution of scores in each group should be approximately normal.
b. The variances of the scores in the two groups should be approximately equal (this can be tested using a variance test).
c. The samples should be independent of each other.Hypotheses:
- Null hypothesis (H₀): There is no difference in mean Math SAT scores between the academy and national average. ([tex]\mu_1 = \mu_2[/tex])
- Alternative hypothesis (H₁): There is a difference in mean Math SAT scores. ([tex]\mu_1 \neq \mu_2[/tex])
Test: Use the two-sample t-test formula:
[tex]t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{ \left(\frac{s_1^2}{n_1}\right) + \left(\frac{s_2^2}{n_2}\right)}}[/tex]
where [tex]\bar{x}_1[/tex] and [tex]\bar{x}_2[/tex] are the sample means, [tex]s_1^2[/tex] and [tex]s_2^2[/tex] are the sample variances, and [tex]n_1[/tex] and [tex]n_2[/tex] are the sample sizes.Decision: Calculate the t-value and compare it against the critical t-value from the t-distribution table for the desired level of significance (e.g., [tex]\alpha = 0.05[/tex]). If the calculated t-value is greater than the critical value, reject the null hypothesis.
In summary, the two-sample t-test is an effective statistical method for determining if there are significant differences between the SAT scores of girls from the academy and those of all high school girls nationwide. This approach provides insights into whether the educational programs might be impacting student performance.
