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?
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 \\\\\n\n 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.
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