Question

The length of time needed to complete a certain test is normally distributed with mean 77 minutes and standard deviation 11 minutes. Find the probability that it will take between 74 and 80 minutes to complete the test.

Answer 1

In Normal distribution first, we calculate the z score with the below formula

After calculating the Z score, we look for the z table below to get the probability value.

In the equation

  

We have = mean = 77 minutes , = standard deviation = 11 minutes, is the random value for which the probability is calculated.

We need to Find the probability that it will take between 74 and 80 minutes to complete the test.

case 1 : Take x = 74 mins

   =>  

=>   

We got , from the z table below find the probability value corresponding to a z value of -0.27 is 0.3936

Let p₁ = 0.3936

case 2 : take Take x = 80 minutes

=>  

=> =>  

We got from the z table below find the probability value corressponding to a z value of 0.27 is 0.6064

let p₂ = 0.6064

  

Now observe the normal curve below

the total area under the bell curve is 1 which represents the total probability, the area between x =74 and x = 80, that is the

shaded region represents the probability that it will take between 74 and 80 minutes to complete the test.

The area to the left of x =74 is p₁ = 0.3936, and the area to the left of x = 80 is 0.6064

Since we want to Find the probability that it will take between 74 and 80 minutes to complete the test., for this we need to subtract the values

   Now    

and the answer is 0.2128