High School

A study was conducted on students from a particular high school over the last 8 years. The following information was found regarding standardized tests used for college admittance:

- Scores on the SAT test are normally distributed with a mean of 1070 and a standard deviation of 204.
- Scores on the ACT test are normally distributed with a mean of 19.1 and a standard deviation of 5.2.

It is assumed that the two tests measure the same aptitude but use different scales.

(A) If a student gets an SAT score that is the 51st percentile, find the actual SAT score. Round your answer to a whole number.

SAT score =

Answer :

The answer explains how to calculate z-scores for SAT scores, interpret them, and compare scores between SAT and ACT tests.

Z-score Calculation:

a. For an SAT score of 720, the z-score is calculated as (720-520)/115 = 1.74. This indicates that a score of 720 is 1.74 standard deviations above the mean.

b. A math SAT score 1.5 standard deviations above the mean of 520 is 520 + 1.5(115) = 692.5.

c. To determine who did better, we calculate the z-scores for the SAT score of 700 and the ACT score of 30. The z-score for the SAT score of 700 is (700-514)/117 ≈ 1.59, while the z-score for the ACT score of 30 is (30-21)/5.3 ≈ 1.70. Comparing these values, the person with the ACT score of 30 performed slightly better.

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