Question

The GRE is widely used to help predict the performance of applicants to graduate schools. The range of possible scores on a GRE is 200 to 900. The psychology department at a university finds that the students in their department have scores with a mean of 544 and standard deviation of 103. a. Find the probability that a student in the psychology department has a score less than 480. b. Find the probability that a student in the psychology department has a score between 480 and 730.

Answer 1

Answer:

a) 26.76% probability that a student in the psychology department has a score less than 480.

b) 69.73% probability that a student in the psychology department has a score between 480 and 730.

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 haev that:

$\mu = 544, \sigma = 103$

a. Find the probability that a student in the psychology department has a score less than 480.

This is the pvalue of Z when X = 480. So

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

$Z = \frac{480 - 544}{103}$

$Z = -0.62$

$Z = -0.62$ has a pvalue of 0.2676.

26.76% probability that a student in the psychology department has a score less than 480.

b. Find the probability that a student in the psychology department has a score between 480 and 730.

This probability is the pvalue of Z when X = 730 subtracted by the pvalue of Z when X = 480.

X = 730

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

$Z = \frac{730 - 544}{103}$

$Z = 1.81$

$Z = 1.81$ has a pvalue of 0.9649.

X = 480

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

$Z = \frac{480 - 544}{103}$

$Z = -0.62$

$Z = -0.62$ has a pvalue of 0.2676.

0.9649 - 0.2676 = 0.6973

69.73% probability that a student in the psychology department has a score between 480 and 730.