Question

Scores on a statewide standardized test are normally distributed with a mean of 12.89 and a standard deviation of 1.95. Certificates are given to students whose scores are in the top 2% of those who took the test. This means that they scored better than 98% of the other test takers. Marcus received his score of 13.7 on the exam and is wondering why he didn’t receive a certificate. Show all work to determine whether Marcus’ score was high enough to earn a certificate. Write a letter to Marcus explaining whether or not he will be receiving a certificate. Include a brief summary of your statistical analysis in your letter.

Answer 1

Given that the scores on a statewide standardized test are normally distributed with a mean of 12.89 and a standard deviation of 1.95 and given that certificates are given to students whose scores are in the top 2% of those who took the test.

Let S be the score needed for a certificate to be awarded and X be a randomly chosen score, then
$P(X \ \textgreater \ S)=2%=0.02$
Recall that the probability of a normally distributed random variable is given by
$P(X\ \textgreater \ x)=1-P(X\ \textless \ x)=1-P( \frac{x-\mu}{\sigma} )=1-P( \frac{x-mean}{standard \, deviation} )$
$P(X \ \textgreater \ S)=0.02 \\ 1-P(X \ \textless \ S)=0.02 \\ P(X \ \textless \ S)=1-0.02=0.98 \\ P( \frac{S-12.98}{1.95} )=0.98 \\ \frac{S-12.98}{1.95} =2.054 \\ S=2.054(1.95)+12.98=4.0053+12.9\approx16.99$

Therefore, for anybody to be awarded a certificate, that person has to score 16.99 or above.

Marcus did not get a certificate becaude his score of 13.7 on the exam was not good enough for a certificate.