Statistics - Sampling distribution

Question

Could someone help me understand what this question is asking in my homework?

Answer 1

Let Xbar = (X_1 + X_2 + ... + X_25)/25, where the X_i's are the weights of the boxes in the sample. We're told that the X_i's are normally distributed with mean μ = 368 grams and std deviation σ = 15 grams. Therefore the variance σ² = 225.

Xbar itself is a random variable, so we can look at its distribution. The mean of Xbar is the same as that of the individual X_i's- i.e., 368 grams.

The problem statement does not say whether the X_i's are independent. We need independence for the rest of the problem solution, so I assume that below.

Xbar is normal, since it is the sum of independent normals multiplied by a constant.

Since the X_i's are independent, then var(Xbar) = (variance of a single X_i)/(sample size) = 225/25 = 9.

Therefore the standard deviation of Xbar is 3 grams.