Question

A random sample of size 36 is selected from a population with a mean μ=50 and a standard deviation σ=10.

a. The shape of the sampling distribution is approximately normal.
b. The mean of the sampling distribution is μ=50.
c. The standard error of the sampling distribution is σ/n
d. The probability that the sample mean will be between 45 and 55 is about 0.6827.
e. The probability that the sample mean will be greater than 48 is about 0.3085.
f. The probability that the sample mean will be within 3 units of the mean is about 0.9973.

Answer 1

The mean of the sampling distribution is equal to the mean of the population. The standard error of the sampling distribution is σ/√n. The probability of the sample mean falling within a certain range can be calculated based on the standard deviation of the population.

Step-by-step explanation:

b. The mean of the sampling distribution is not necessarily μ=50. The mean of the sampling distribution is equal to the mean of the population, which is μ=50.

c. The standard error of the sampling distribution is not σ/n. The standard error of the sampling distribution is σ/√n, where σ is the standard deviation of the population and n is the sample size.

d. The probability that the sample mean will be between 45 and 55 is about 0.6827. This probability is based on the fact that approximately 68% of the sample means will fall within one standard deviation of the population mean.

e. The probability that the sample mean will be greater than 48 is not about 0.3085. To find this probability, we need to calculate the z-score for 48 and use a z-table to find the corresponding probability.

f. The probability that the sample mean will be within 3 units of the mean is about 0.9973. This probability is based on the fact that approximately 99.73% of the sample means will fall within three standard deviations of the population mean.