Question

A random sample of size 36 is to be selected from a population that has a mean = 50 and a standard deviation of 10. a. This sample of 36 has a mean value of x, which belongs to a sampling distribution. Find the shape of this sampling distribution. b. Find the mean of this sampling distribution. c. Find the standard error of this sampling distribution. d. What is the probability that this sample mean will be between 45 and 55? 50 e. What is the probability that the sample mean will have a value greater than 48? f. What is the probability that the sample mean will be within 3 units of the mean?

a) Normal, 50, 10, 0.68, 0.5, 0.34
b) Uniform, 10, 1.67, 0.95, 0.68, 0.68
c) Positively Skewed, 36, 5, 0.5, 0.32, 0.95
d) Negatively Skewed, 5, 2.5, 0.34, 0.16, 0.5

Answer 1

a. The shape of the sampling distribution is Normal with a mean of 50 and a standard deviation of 10.

b. The mean of this sampling distribution is 50.

c. The standard error of this sampling distribution is 10/√36 = 1.67.

d. The probability that the sample mean will be between 45 and 55 is approximately 0.68.

e. The probability that the sample mean will have a value greater than 48 is approximately 0.5.

f. The probability that the sample mean will be within 3 units of the mean is approximately 0.68.

Step-by-step explanation:

a. The Central Limit Theorem ensures that the sampling distribution of the sample mean, for a sufficiently large sample size, becomes approximately Normal, irrespective of the shape of the population distribution. This justifies the choice of Normal for the shape of the sampling distribution.

b. The mean of the sampling distribution is the same as the population mean, which is 50 in this case.

c. The standard error (SE) of the sampling distribution is calculated as the population standard deviation divided by the square root of the sample size, which is 10/√36 = 1.67.

d. The probability that the sample mean will be between 45 and 55, within one standard deviation of the mean, is approximately 0.68 according to the empirical rule.

e. The probability that the sample mean will be greater than 48, within one standard deviation of the mean, is approximately 0.5.

f. The probability that the sample mean will be within 3 units of the mean, within two standard deviations, is approximately 0.68 based on the empirical rule.