Suppose that, for the population of UVA undergraduate and graduate students, the distribution of age has a mean of 23 and a standard deviation of 2 years. We take a simple random of size 100 from this population to survey their attitudes towards course enrollment. In this sample, the sample mean age is x ̅ = 22.5. of course, a different random sample of 100 UVA students would likely return a different sample mean age. What is the sampling distribution x ̅?
a. A Normal distribution with a mean of 22.5 and a standard deviation of 0.2
b. A Normal distribution with a mean of 22.5 and a standard deviation of 2
c. A Normal distribution with a mean of 23 and a standard deviation of 0.2
d. A Normal distribution with a mean of 23 and a standard deviation of 2
c. A Normal distribution with a mean of 23 and a standard deviation of 0.2
why would the mean not be 22.5
The mean of the sampling distribution, denoted by μx̅, is equal to the population mean, which in this case is 23. The sample mean age of 22.5 is just one particular value obtained from a specific random sample, and it is likely to differ from the population mean. The sampling distribution, however, will have a mean equal to the population mean.
To determine the sampling distribution of x-bar (sample mean), we need to consider the properties of the population distribution and the sample size.
In this case, the population distribution of age follows a normal distribution with a mean of 23 and a standard deviation of 2 years.
According to the Central Limit Theorem, when the sample size is sufficiently large (typically n > 30), the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.
Since the sample size in this case is 100, we can assume that the sampling distribution of x-bar will be approximately normal.
The mean of the sampling distribution of x-bar is equal to the population mean, which is 23.
The standard deviation of the sampling distribution of x-bar, known as the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size. Therefore, the standard error of the mean for this case is 2 / sqrt(100) = 0.2.
So, the correct answer is:
c. A Normal distribution with a mean of 23 and a standard deviation of 0.2