The combined SAT scores for the students at a local high school are normally distributed with a mean of 1507 and a standard deviation of 308. The local college requires a minimum score of 645 for admission. What percentage of students from this school earn scores that satisfy the admission requirement?

Question

19-09-2023
Mathematics

High School

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The combined SAT scores for the students at a local high school are normally distributed with a mean of 1507 and a standard deviation of 308. The local college requires a minimum score of 645 for admission. What percentage of students from this school earn scores that satisfy the admission requirement?

Answer :

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KurtDavies KurtDavies · Answer 1 · 2024-03-08T20:59:08-05:00

Final answer:

Nearly 100% of students from the high school will have SAT scores above the local college admission requirement of 645, as a score of 645 is far below the mean when considering the normal distribution of combined SAT scores at the high school.

Explanation:

To find the percentage of students from the high school whose SAT scores satisfy the local college admission requirement of 645, we will use the given normal distribution parameters: a mean (μ) of 1507 and a standard deviation (σ) of 308. First, we calculate the z-score for the minimum score of 645 using the following formula:

z = (X - μ) / σ

Where X is the score in question. So:

z = (645 - 1507) / 308 ≈ -2.80

A z-score of -2.80 tells us that a score of 645 is 2.80 standard deviations below the mean. Using a standard normal distribution table or a calculator, we can find the percentage of scores that fall above this z-score, since SAT scores higher than 645 would be required for college admission.

This percentage corresponds to the area to the right of z = -2.80 on the normal distribution curve. Practically, this encompasses almost all the area under the curve, since a z-score that low is in the far left tail of the distribution. Therefore, nearly 100% of the students at the high school will have SAT scores above 645, meeting the college admission requirement.