Question

Suppose the distribution of weekly study times among FIU students has mean 20 hours and standard deviation 8 hours.

1. In the sample of size 75, determine the probability that the average study time is more than 21.5 hours. Round to 3 decimal places.

2. Would the probability in question 1 increase, decrease or stay the same if you had selected a sample of size 250 instead of 75?

Answer 1

Random variable X: Weekly study time among FIU students

Mean =

Standard deviation =

1)

According to central limit theorem, if sample size n is large (n > 30) then the sampling distribution of sample mean is approximately normally distributed with mean = and standard deviation =

irrespective of distribution of random variable X.

Here n = 75 > 30

So we can say that the sampling distribution of sample mean is approximately normally distributed with mean = = 20 and standard deviation is

Here we have to find

   where z is standard normal variable.

= 1 - P(z < 1.62) (Round to 2 decimal)

= 1 - 0.9474 (From statistical table of z values)

= 0.0526

Probability that the average study time is more than 21.5 hours is 0.0526

2) For n = 250

   where z is standard normal variable.

= 1 - P(z < 2.96) (Round to 2 decimal)

= 1 - 0.9985 (From statistical table of z values)

= 0.0015

Probability that the average study time is more than 21.5 hours is 0.0015

So the probability in question 1 decrease if we had selected a sample of size 250 instead of 75