Question

A population consists of six scores: 11,13,15,16,17 and 19. If a sampling distribution of size 3 is drawn from the population. Compute for the population of mean, variance,variance of the sampling distributions of the sample mean and standard error of the mean.

Answer 1

Given the population with $N=6$ values given as, 11,13,15,16,17,19. The population mean is $\mu=15.166667$

The population variance, $\sigma^2={\sum X^2-{(\sum X)^2\over N}\over N}={1421-1380.1667\over6}=6.805555$

The number of possible samples of size $n=3$ from a population of size $N=6$ is,

$\binom{6}{3}=20$

The sample means are derived from the formula, $\bar x_i={\sum x_i\over3}$

The list below gives the possible samples with their corresponding means.

Sample Mean

(11,13,15) 13

(11,13,16) 13.33

(11,13,17) 13.67

(11,13,19) 14.33

(11,15,16) 14

(11,15,17) 14.33

(11,15,19) 15

(11,16,17) 14.67

(11,16,19) 15.33

(11,17,19) 15.67

(13,15,16) 14.67

(13,15,17) 15

(13,15,19) 15.67

(13,16,17) 15.33

(13,16,19) 16

(13,17,19) 16.33

(15,16,17) 16

(15,16,19) 16.67

(15,17,19) 17

(16,17,19) 17.33

The mean of the sampling distributions of the mean is the population mean $\mu=15.66667$

The variance of the sampling distribution of the means is $var(\bar x_i)=({\sigma^2\over n})={6.805555\over 3}=2.26851$

The standard deviation of the sampling distribution of the means is $sd(\bar x_i)=\sqrt{var(\bar x_i)}=\sqrt{2.26851}=1.50616$