Question

Let X be the sample mean of a random sample of size n obtained from a normal distribution with mean μ = θ/3 - 1, where θ is a parameter. Which of the following statements is TRUE?

Answer 1

The distribution for ΣX is normal with the same mean but a smaller standard deviation than the distribution for X.

Step-by-step explanation:

The statement that is TRUE is D. The distribution for ∑X is normal with the same mean but a smaller standard deviation than the distribution for X.

The central limit theorem states that as the sample size increases, the distribution of the sample mean approaches a normal distribution. In this case, we are considering the sum of random variables, which also follows the central limit theorem. The distribution for ∑X will be normal with a mean equal to the sum of the means of individual random variables, and a standard deviation equal to the sum of the standard deviations of individual random variables. Since the sample mean X has a smaller standard deviation than X, the same applies to ∑X.

For example, if a random variable X has a mean of 5 and a standard deviation of 2, and we take a sample of size n=50, the distribution for the sum of X's will be normal with a mean of 5 * 50 = 250 and a standard deviation of 2 * √50 = 14.14.