The combined SAT scores for the students at a local high school are normally distributed with a mean of 1527 and a standard deviation of 291. The local college includes a minimum score of 1207 in its admission requirements. What percentage of students from this school earn scores that satisfy the admission requirement?

Find \( P(X > 1207) \).

The percentage of students from the high school that satisfy the local college's minimum SAT score admission requirement is approximately 86.43%. This was obtained by first calculating the Z-score and using it to find the probability in the normal distribution.

Explanation:

The subject of this question is Mathematics, specifically statistics. This problem is asking us to find the percentage of students from a high school with certain SAT scores that satisfy the minimum admission requirement of a local college. To solve this, first, we need to calculate a Z-score. In statistics, a Z-score is a measure of how many standard deviations an element is from the mean.

The formula for a Z-score is:

Z = (X - μ) / σ

where:

Z is the Z-score
X is the value we are comparing to the mean
μ is the mean
σ is the standard deviation.

So, we plug in our values:

Z = (1207 - 1527) / 291 = -1.1

We then use the Z-score to find the probability. Looking at the Z-table for a Z-score of -1.1 gives us a probability of 0.1357. However, since we want P(X > 1207), we need to subtract this value from 1 since the total probability under the normal curve is 1.

p = 1 - 0.1357 = 0.8643 = 86.43%

So, about 86.43% of students from the high school satisfy the local college's admission requirement based on their SAT scores.

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JaymaMays JaymaMays · Answer 2 · 2024-11-13T00:37:08-05:00

Approximately 86.43% of students from the local high school will score above the minimum required SAT score of 1207, satisfying the college's admission requirement calculated using the normal distribution properties and z-scores.

We are asked to determine the percentage of students from a local high school who earn SAT scores that satisfy a local college's admission requirement. The SAT scores are normally distributed with a mean of 1527 and a standard deviation of 291. To calculate the percentage of students scoring above the minimum score requirement of 1207, we will use the normal distribution properties and z-scores.

First, we compute the z-score for 1207:

Z = (X - μ)/ σ

Z = (1207 - 1527) / 291 ≈ -1.1

Since the z-score is negative, it indicates that the score of 1207 is below the mean. Next, using the z-score and a standard normal distribution table or a calculator, we can find the probability of a student scoring higher than 1207, which we denote as P(X > 1207).

From the standard normal distribution, we find that P(Z > -1.1) is approximately 0.8643, which means that about 86.43% of students will score above 1207 on the SAT and satisfy the college's admission requirement.