Answer to Question #261401 in Statistics and Probability for Ayana Prica

Question

The time for a major exam to be completed is normally distributed with an average of 55 minutes and a standard deviation of 9 minutes. If 92% of the students completed the exam, when should the test be terminated?

Answer 1

The average length of time to complete an test In a given course Is 55 minutes, with a standard deviation of 9 minutes.

Let, X be the random variable denoting the length of time for a randomly selected Individual to complete the test.

Then, X follows normal with mean 55 and standard deviation 9 .

Then, we can also say that,

$Z=\frac{(X-55)}{9}$ follows standard normal with mean 0 and variance of 1 .

Now, the Instructor wants to terminate the test, allowing sufficient time for 92% of the students to complete their exam.

So, basically we have to find a value for m, such that

$P(X \leq m)=0.92$

Now this means,

$P\left(Z \leq \frac{m-55}{9}\right)=0.92 \\$

$\text { or, } p h i\left(\frac{m-55}{9}\right)=0.92$

Where, Z is the standard normal variate and phi is the distribution function for the standard normal variate.

Now, from the standard normal table, we note that,

$p h i(1.41)=0.92$

Comparing, we can say that,

$\frac{m-55}{9}=1.41 \\ m=67.69$

The test should be terminated after 67.69 minutes if the instructor wants to allow sufficient time for 92% of the students.