Question

The College Boards, which are administered each year to many thousands of high school students, are scored so as to yield a mean of 500 and a standard deviation of 100. These scores are close to being normally distributed. What percentage of the scores can be expected to satisfy each condition? a. Greater than 600 b. Greater than 700 c. Less than 450 d. Between 450 and 600

Answer 1

a) 15.87% of the scores are expected to be greater than 600.

b) 2.28% of the scores are expected to be greater than 700.

c) 30.85% of the scores are expected to be less than 450.

d) 53.28% of the scores are expected to be between 450 and 600.

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 = (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. Greater than 600

This is 1 subtracted by the pvalue of Z when X = 600. So

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

`Z = (600 - 500)/(100)`

`Z = 1`

`Z = 1` has a pvalue of 0.8413.

1 - 0.8413 = 0.1587

15.87% of the scores are expected to be greater than 600.

b. Greater than 700

This is 1 subtracted by the pvalue of Z when X = 700. So

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

`Z = (700 - 500)/(100)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772

1 - 0.9772 = 0.0228

2.28% of the scores are expected to be greater than 700.

c. Less than 450

Pvalue of Z when X = 450. So

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

`Z = (450 - 500)/(100)`

`Z = -0.5`

`Z = -0.5` has a pvalue of 0.3085.

30.85% of the scores are expected to be less than 450.

d. Between 450 and 600

pvalue of Z when X = 600 subtracted by the pvalue of Z when X = 450. So

X = 600

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

`Z = (600 - 500)/(100)`

`Z = 1`

`Z = 1` has a pvalue of 0.8413.

X = 450

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

`Z = (450 - 500)/(100)`

`Z = -0.5`

`Z = -0.5` has a pvalue of 0.3085.

0.8413 - 0.3085 = 0.5328

53.28% of the scores are expected to be between 450 and 600.