Question

"Scores on an English test are normally distributed with a mean of 37.4 and a standard deviation of 7.9. Find the score that separates the top 59% from the bottom 41%"

Answer 1

Score that separates the top 59% from the bottom 41% is 35.6

Explanation:

We are given that Scores on an English test are normally distributed with a mean of 37.4 and a standard deviation of 7.9, i.e.;
`\mu` = 37.4 and
`\sigma` = 7.9

Now, the z score probability distribution is given by;

Z =
`(X - \mu)/(\sigma)` ~ N(0,1)

The bottom 41% area is given by the critical z value of -0.2278 (from z% table)

So, P(Z <
`(X-37.4)/(7.9)` ) = 0.41

which means
`(X-37.4)/(7.9)` = -0.2278

X - 37.4 = -0.2278 * 7.9

X = 37.4 - 1.79962 = 35.6

Therefore, score of 35.6 separates the top 59% from the bottom 41%.

Answer 2

The score that separates the top 59% from the bottom 41% is 35.6225

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 = 37.4, \sigma = 7.9`

Find the score that separates the top 59% from the bottom 41%"

This is the value of X when Z has a pvalue of 0.41. So it is X when Z = -0.225.

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

`-0.225 = (X - 37.4)/(7.9)`

`X - 37.4 = -0.225*7.9`

`X = 35.6225`

The score that separates the top 59% from the bottom 41% is 35.6225