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 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?
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
To determine the sampling distribution of the sample mean age, we need to consider the properties of the population distribution and the sample size.
In this case, the population distribution of age for UVA undergraduate and graduate students has a mean of 23 and a standard deviation of 2 years. Since the population is assumed to be normally distributed, the sampling distribution is also expected to be approximately normal due to the Central Limit Theorem.
The Central Limit Theorem states that for a random sample of sufficiently large size (in this case, 100), the sampling distribution of the sample mean will be approximately normally distributed regardless of the shape of the population distribution. The mean of the sampling distribution will be equal to the population mean (23), and the standard deviation of the sampling distribution (also known as the standard error) will be equal to the population standard deviation divided by the square root of the sample size.
To calculate the standard error, we divide the population standard deviation by the square root of the sample size:
Standard error = Population standard deviation / √(sample size)
= 2 / √(100)
= 2 / 10
= 0.2
Therefore, the sampling distribution of the sample mean age is a normal distribution with a mean of 23 (same as the population mean) and a standard deviation of 0.2.
Hence, the correct answer is option c: A Normal distribution with a mean of 23 and a standard deviation of 0.2.