Question

You may need to use the appropriate appendix table or technology to answer this question. A population of 1,000 students spends an average of $11.60 a day on dinner. The standard deviation of the expenditure is $4. A simple random sample of 64 students is taken. (a) What is the expected value (in dollars) of the sampling distribution of the sample mean? $ What is the standard deviation (in dollars) of the sampling distribution of the sample mean? (Round your answer to four decimal places.) $ What is the shape of the sampling distribution of the sample mean? exponential O uniform O skewed to the left skewed to the right o normal

Answer 1

The expected value of the sampling distribution of the sample mean is $11.60, and the standard deviation of the sampling distribution of the sample mean is $0.50. The shape of the sampling distribution of the sample mean is normal.

Step-by-step explanation:

To find the expected value (mean) of the sampling distribution of the sample mean, we use the same mean as the population, which is $11.60. For the standard deviation of the sampling distribution of the sample mean, we employ the formula:

population standard deviation

sample size

sample size

​

population standard deviation

​

. Substituting the values, we get

4

64

64

​

4

​

, which simplifies to 0.50.

The sampling distribution of the sample mean tends to be normally distributed due to the Central Limit Theorem, especially with a sample size of 64, indicating a bell-shaped curve for the distribution of sample means. This theorem asserts that as the sample size increases, regardless of the shape of the population distribution, the sampling distribution of the sample mean will become normal.

This normal distribution suggests that a larger number of sample means will cluster around the population mean, and as the sample size increases, the variability of the sample means' distribution around the population mean decreases, forming a symmetrical, bell-shaped curve.