Question

In a sample of 1000 cases, the mean of a certain test is 14 and the standard deviation is 2.5. Assume the distribution to be normal. The top 20% of the students will score how many points above the mean

Answer 1

The top 20% of the students will score at least 2.1 points above the mean.

Explanation:

When the distribution is normal, we use 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.

The mean of a certain test is 14 and the standard deviation is 2.5.

This means that
`\mu = 14, \sigma = 2.5`

The top 20% of the students will score how many points above the mean

Their score is the 100 - 20 = 80th percentile, which is X when Z has a pvalue of 0.8. So X when Z = 0.84.

Their score is:

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

`0.84 = (X - 14)/(2.5)`

`X - 14 = 0.84*2.5`

`X = 16.1`

16.1 - 14 = 2.1

The top 20% of the students will score at least 2.1 points above the mean.