Answer :
Final answer:
The standard deviation of SAT scores is crucial when constructing a confidence interval for the mean SAT score, as it is needed to determine the reliability and precision of the interval. Comparisons across different test scales like the SAT and ACT require normalization.
Explanation:
When constructing a confidence interval for the mean mathematics SAT score, it is crucial to determine the standard deviation of the SAT scores. Option (a) is the correct answer because when calculating a confidence interval for a mean, the standard deviation is an essential component in establishing the reliability and precision of the interval.
The formula to calculate a z-score for an SAT score is z = (X - µ) / σ, where X is the SAT score, µ is the mean score, and σ is the standard deviation. For an SAT score of 720, the z-score would be calculated as follows:
- Subtract the mean score (µ = 520) from the SAT score (X = 720): 720 - 520 = 200.
- Divide the difference by the standard deviation (σ = 115): 200 / 115 ≈ 1.7391.
The z-score is approximately 1.7391, meaning that a score of 720 is roughly 1.7391 standard deviations above the mean.
When determining a score that is 1.5 standard deviations above the mean, simply multiply the standard deviation by 1.5 and add it to the mean: (1.5 x 115) + 520 = 692.5. This score can be regarded as significantly above average, since it is more than one standard deviation away from the mean.
In comparing SAT and ACT scores, one must consider the differences in distributions and scales. Normalize the scores by converting to z-scores or by comparing percentiles to determine who did better with respect to the test each person took.
