Question

n All maximum possible random samples of size 4 are drawn from the finite population comprising of X=2,4 a) Verify central limit theorem (CLT) in this random experiment b) Between what two sample means would you expect the middle 75% of sample means to fall c) Estimate population mean at 90% C

Answer 1

a)

The central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution.

sample mean

2,2,2,2 2

2,2,2,4 2.5

2,2,4,4 3

2,4,4,4 3.5

4,4,4,4 4

mean of the sample means:

$\mu_{\overline{x}}=\frac{2+2.5+3+3.5+4}{5}=3$

standard deviation of the sample means:

$\sigma_{\overline{x}}=\sqrt{\frac{(2-3)^2+(2.5-3)^2+(3.5-3)^2+(4-3)^2}{5}}=1/\sqrt 2$

mean of population:

$\mu=(2+4)/2=3$

standard deviation of population:

$\sigma=\sqrt{\frac{(2-3)^2+(4-3)^2}{2}}=1$

so,

$\mu_{\overline{x}}=\mu=3$

$\sigma_{\overline{x}}=\sigma/\sqrt n=1/\sqrt 4=1/2$

b)

At least 75% of the data will be within two standard deviations of the mean (Chebychev’s rule)

then:

$\mu\pm 2\sigma_{\overline{x}}=3\pm 2\cdot1/ 2=(2,4)$

the middle 75% of sample means falls between 2 and 4

c)

for 90% confidence interval of population mean:

$z=\pm 1.645=\frac{x-\mu}{\sigma}=\frac{x-3}{1}$

$1.355<x<4.645$

sample means: