Question

Let X1, . . . , Xn, Xn+1 be a sample from a normal population having an unknown

mean μ and variance 1. Let ¯Xn =Σni=1 Xi /n be the average of the first n of them.
What is the distribution of Xn+1 − ¯Xn?

Answer 1

According to the Central Limit Theorem for Sample Means, when the sample size is large, the distribution of Xn+1 - ¯Xn approximates a normal distribution.

Step-by-step explanation:

The distribution of Xn+1 − ¯Xn can be described using the Central Limit Theorem for Sample Means. According to this theorem, if the sample size (n) is sufficiently large, the distribution of the sample means will be approximately normal. The mean of the sample means will equal the population mean, and the standard deviation of the distribution of the sample means (known as the standard error of the mean) is equal to the population standard deviation divided by the square root of the sample size (n).