Question

A teacher gives her class a particular examination on mathematics at the end of each year. The results are normally distributed with a mean test score of 60 and a standard deviation of 8. The teacher has told the class that students with a test score in the top 2.5% will win a special prize. Calculate the test score above which 2.5% of all test scores lie. Give your answer to 1 decimal place. Test score

Answer 1

Answer:

Test score of 75.7.

Step-by-step explanation:

Problems of normal distributions can be 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:

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

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 results are normally distributed with a mean test score of 60 and a standard deviation of 8.

This means that $\mu = 60, \sigma = 8$

Calculate the test score above which 2.5% of all test scores lie.

Above the 100 - 2.5 = 97.5th percentile, which is the value of X when Z has a pvalue of 0.975, so X when Z = 1.96.

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

$1.96 = \frac{X - 60}{8}$

$X - 60 = 8*1.96$

$X = 75.7$

Test score of 75.7.