Answer to Question #292294 in Statistics and Probability for Yaku

Question

1. A population random variable is normally distributed with a mean of

50 and a variance of 16. If a random sample of size n = 25 is drawn

from this population, what are the following probabilities?

a) that the sample mean will exceed 51;

b) that the sample mean will be between 48.5 and 51.5.

2. Repeat Exercise 1, but now with n = 36. What is the effect on the

probabilities of increasing the sample size?

3. Random samples of size n were selected from populations with the

means and variances given here. Find the mean and standard deviation

of the sampling distribution of the sample mean in each case:

a) n = 36, $\mu$ = 10, $\sigma$²= 9;

b) n = 100, $\mu$= 5, $\sigma$²= 4;

c) n = 8, $\mu$= 120, $\sigma$²= 1.

4. Refer to Exercise 3.

a) If the sampled populations are normal, what is the sampling distribution of X \bar for parts a), b), and c)?
b) According to the Central Limit Theorem, if the sampled populations are not normal, what can be said about the sampling distribution of X \bar for parts a), b), and c)?