Question

The distribution of SAT scores is normal with a mean of µ = 500 and a standard deviation of σ = 100. a. What SAT score (i.e., X score) separates the top 15% of the distribution from the rest?b. What SAT score (i.e., X score) separates the top 10% of the distribution from the rest?c. What SAT score (i.e., X score) separates the top 2% of the distribution from the rest?

Answer 1

Answer:

a) 604

b) 628

c) 705.5

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 500, \sigma = 100$

a. What SAT score (i.e., X score) separates the top 15% of the distribution from the rest?

This is the value of X when Z has a pvalue of 0.85. So it is X when Z = 1.04.

$Z = \frac{X - \mu}{\sigma}$

$1.04 = \frac{X - 500}{100}$

$X - 500 = 100*1.04$

$X = 604$

b. What SAT score (i.e., X score) separates the top 10% of the distribution from the rest?

This is the value of X when Z has a pvalue of 0.90. So it is X when Z = 1.28.

$Z = \frac{X - \mu}{\sigma}$

$1.28 = \frac{X - 500}{100}$

$X - 500 = 100*1.28$

$X = 628$

c. What SAT score (i.e., X score) separates the top 2% of the distribution from the rest?

This is the value of X when Z has a pvalue of 0.98.So it is X when Z = 2.055.

$Z = \frac{X - \mu}{\sigma}$

$2.055 = \frac{X - 500}{100}$

$X - 500 = 100*2.055$

$X = 705.5$