Question

Based on data from a​ college, scores on a certain test are normally distributed with a mean of 1530 and a standard deviation of 322.

Find the percentage of scores greater than 2317. (Round to two decimal places as needed.)
Find the percentage of scores less than 1190. % (Round to two decimal places as needed.)
Find the percentage of scores between 1351 and 1673.

Answer 1

0.73% of the scores are greater than 2317.

14.46% of the scores are less than 1190.

38.23% of the scores are between 1351 and 1673.

Explanation:

Problems of normally distributed samples can be 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 = (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 = 1530, \sigma = 322`

Find the percentage of scores greater than 2317.

This is 1 subtracted by the pvalue of Z when X = 2317. So:

`Z = (X - \mu)/(\sigma)`

`Z = (2317 - 1530)/(322)`

`Z = 2.44`

`Z = 2.44` has a pvalue of 0.9927.

So 1-0.9927 = 0.0073 = 0.73% of the scores are greater than 2317.

Find the percentage of scores less than 1190.

This is the pvalue of Z when X = 1190. So:

`Z = (X - \mu)/(\sigma)`

`Z = (1190 - 1530)/(322)`

`Z = -1.06`

`Z = -1.06` has a pvalue of 0.1446.

So 14.46% of the scores are less than 1190.

Find the percentage of scores between 1351 and 1673.

This is the pvalue of Z when X = 1673 subtracted by the pvalue of Z when X = 1351. So

X = 1673

`Z = (X - \mu)/(\sigma)`

`Z = (1673- 1530)/(322)`

`Z = 0.44`

`Z = 0.44` has a pvalue of 0.67

X = 1351

`Z = (X - \mu)/(\sigma)`

`Z = (1351- 1530)/(322)`

`Z = -0.56`

`Z = -0.56` has a pvalue of 0.2877

So 0.67-0.2877 = 0.3823 = 38.23% of the scores are between 1351 and 1673.