Question

Times spent studying by students in the week before final exams follow a normal distribution with standard deviation 8 hours. A random sample of 4 students was taken in order to estimate the mean study time for the population of all students a. Find the standard error of the mean b. What is the probability that the sample mean exceeds the population mean by more than 2 hours? c. What is the probability that the sample mean is more than 3 hours below the population mean? d. What is the probability that the sample mean differs from the population mean (on either side) by more than 4 hours? e. Suppose that a second (independent) random sample of ten students was taken. Without doing the calculations, state whether the probabilities in b, c, and d would be higher, lower, or the same for the second sample.

Answer 1

The sample mean X¯

X

¯

has normal distribution, mean the population mean μ

μ

, and standard deviation τ=84√

τ

=

8

4

. We want Pr(X¯−μ>2)

Pr

(

X

¯

−

μ

>

2

)

, which is Pr(Z>2τ)

Pr

(

Z

>

2

τ

)

, where Z

Z

is standard normal.