Question

(a)
A guidance counselor at a school wants to compare the effectiveness of two different SAT prep courses in improving students’ math scores.
​He ​collects data for 400
students who took the SAT for the first time, completed ​one of two prep courses, and then took the SAT for the second time.
​
​Half of ​the students completed prep course A
, and the other half completed prep ​course B
.
​
​The data of the change in test scores is summarized in the side-by-side box plots shown. (Note: A
A(-) value represents a decrease in ​student score and a (+) value represents an increase in student score.) kk-342874-1446703199991

Answer 1

I don’t have the boxplots image/data you referenced, so I can’t give a specific conclusion yet. Please either upload the boxplots or give the key numbers (medians, quartiles, whisker endpoints, any outliers, or summary statistics for each course).

While you get that to me, here is how to compare the two courses and what to report (so you can interpret the plots yourself or I can do it once you supply the image/data):

1. Visual comparison from the boxplots
- Center: Compare the medians. The course with the higher median change produced the larger typical gain.
- Spread: Compare IQRs (box heights) and full ranges (whiskers). A smaller IQR means more consistent improvement; a larger IQR means more variability.
- Position relative to zero: Check where the box and median lie relative to 0 (no change). If most of a course’s box is above 0, that course tended to improve scores; if below 0, it tended to decrease them.
- Skewness and tails: Long whiskers or many outliers on one side indicate more extreme gains or losses.
- Outliers: Note any extreme gains or losses and whether one course has more/bigger outliers.

2. Quantitative comparison / hypothesis test
- Null hypothesis: mean (or median) change is the same for A and B.
- Because you have change scores for independent students in each course (200 each), a two-sample t-test on the change scores is appropriate if approximate normality or large-sample conditions hold. If the distributions are strongly nonnormal, use the Mann–Whitney (Wilcoxon rank-sum) test.
- Report the estimated difference in means (A − B) with a 95% confidence interval, the test statistic, and the p-value.
- With n = 200 per group, the tests have high power to detect modest differences.

3. Alternative / complementary analyses
- Report proportion of students who improved (change > 0) in each course and compare with a two-proportion (chi-square) test.
- Report effect size (Cohen’s d for mean difference) to describe practical significance.
- If you want to account for covariates (e.g., baseline score), run an ANCOVA or regression with baseline score as a covariate and course indicator as predictor.

4. How to word conclusions
- Example (if A median > B median and p < 0.05): “Students in course A had a statistically significantly larger average increase in SAT math scores than students in course B (mean difference ≈ X points, 95% CI [L, U], p = <0.05). The effect size was Y (small/medium/large).”
- Example (if no significant difference): “There is no evidence of a meaningful difference in score change between courses A and B (mean difference ≈ X points, 95% CI [L, U], p = P).”

If you upload the boxplots or paste the medians/IQRs or the raw summary numbers, I’ll give a specific conclusion and can run the appropriate test and report numbers and interpretation.