Scores on an english test are normally distributed with a mean of 31.5 and a standard deviation of 7.3. find the score that separates the top 59% from the bottom 41%.

Let X be the score on an english test which is normally distributed with mean of 31.5 and standard deviation of 7.3

μ = 31.5 and σ = 7.3

Here we have to find score that separates the top 59% from the bottom 41%

So basically we have to find here x value such that area above it is 59% and below it is 49%

This is same as finding z score such that probability below z score is 0.49 and above probability is 0.59

P (Z < z) = 0.49

Using excel function to find the z score for probability 0.49 we get

z = NORM. S. INV (0.49)

z = - 0.025

It means for z score - 0.025 area below it is 41% and above it is 59%

Now we will convert this z score into x value using given mean and standard deviation

x = (z * standard deviation) + mean

x = (-0.025 * 7.3) + 31.5

x = 31.6825 ~ 31.68

The score that separates the top 59% from the bottom 41% is 31.68