Answer to Question #272555 in Statistics and Probability for Shah

Question

(A).If a population is given as (23 6 8 11), after selecting all possible samples of size 2 with replacement, find out and standard error ofx ̅

(B). Discuss the relationship between x ̅ and μ

Answer 1

A)

sample mean

23,23 23

23,6 14.5

23,8 15.5

23,11 17

6,6 6

6,8 7

6,11 8.5

8,8 8

8,11 9.5

11,11 11

The standard deviation of the distribution of the sample means is called the standard error of the mean:

standard error of $\overline{x}:$

$SE=s=\sqrt{\frac{(\overline{x}_i-\overline{x})^2}{n}}=5.1$

B)

mean of samples averages:

$\overline{x}=\frac{\sum\overline{x}_i}{n}=12$

n is number of samples

mean of population:

$\mu=\frac{\sum x_i}{n}=12$

n is number of items in population

$\overline{x}=\mu$

The mean of the sample means will equal the population mean.

The central limit theorem for sample means says that if you keep drawing larger and larger samples and calculating their means, the sample means form their own normal distribution (the sampling distribution). The normal distribution has the same mean as the original distribution.