Difference between probability of an individual and of a population?

Im a little confused as to finding the probability of an individual to a population. My answers don't look correct. Could anyone help me out?

A study was conducted to examine how much sleep per night was achieved. Previous investigations have shown that in the population the average amount of sleep achieved per night follows a Normal model with a mean of 7.1 hours and a standard deviation of 2 hours. A SRS of 19 people were examined and the average sleep per night was recorded.

a. What is the probability that an individual student will sleep for more than 8 hours
per night?

b. What is the probability that the mean hours slept per day for the 19 students will be
more than 8 hours?

a. z score for = (8-7.1)/2 = 0.45
P= 1 - 0.6736 = 0.3264
Probability of an individual student sleeping for more than 8 hours per night is 32.6%

b. z score = (9-7.1)/.0459 = 1.961
p= 1-0.9750 = 0.025
Probability of 19 students sleeping for more than 8 hours per night is 2.5%

Re: Difference between probability of an individual and of a population?

1 a is correct. What does not look correct about 32.6%?

Think about it this way... What is the probability an individual will sleep more than 7.1 hours? In a normal distribution, half will be above 7.1 and half will be below. SO the probability an individual sleeps more than 7.1 hours is .5.

What is the probability an individual will sleep more than 9.1 hours? For a normal distribution about .68 will be within 1 standard deviation, so .68 will be between 5.1 and 9.1 hours. .32 will be less than 5.1 hours OR greater than 9.1 hours. By symmetry of the normal curve, .16 will be greater than 9.1 hours.

If the probability of sleeping more than 7.1 hours is 50%, and the probability of sleeping more than 9.1 hours is 16%, then it is reasonable that the probability of sleeping more then 8 hours is 32% .

b is correct also. Not sure how you calculated z but I got the same answer.