Question

A Regional College uses the SAT to admit students to the school. The university notices that a lot of students apply even though they are not eligible. Given SAT scores are normally distributed, have a population mean of 500 and standard deviation of 100, what is the probability that a group of 12 randomly selected applicants would have a mean SAT score that is greater than 525 but below the current admission standard of 584?

Answer 1

Answer:

0.191 is the probability that a group of 12 randomly selected applicants would have a mean SAT score that is greater than 525 but below the current admission standard of 584.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 500

Standard Deviation, σ = 100

n = 12

We are given that the distribution of SAT score is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

P(greater than 525 but 584)

Standard error due to sampling =

$\displaystyle\frac{\sigma}{\sqrt{n}} = \frac{100}{\sqrt{12}}$

$P(525 < x < 584) = P(\displaystyle\frac{525 - 500}{\frac{100}{\sqrt{12}}} < z < \displaystyle\frac{584-500}{\frac{100}{\sqrt{12}}}) = P(0.866 < z < 2.909)\\\\= P(z \leq 2.909) - P(z < 0.866)\\= 0.998 - 0.807 = 0.191 = 19.1\%$

$P(525 < x < 584) = 19.1\%$

0.191 is the probability that a group of 12 randomly selected applicants would have a mean SAT score that is greater than 525 but below the current admission standard of 584.