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( (x-\mu)/(\sigma) )=1-P( (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( (S-12.98)/(1.95) )=0.98 \\ (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.