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

Answer:

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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:

$Z = \frac{X - \mu}{\sigma}$

$0.84 = \frac{X - 14}{2.5}$

$X - 14 = 0.84*2.5$

$X = 16.1$

16.1 - 14 = 2.1

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