Question

Random samples of size 3 are taken from a population of the numbers 3, 4, 5, 6 7,8, and 9.

3. Construct the histogram of the sampling distribution of the sample means. Describe the shape of the histogram.

2. Construct the sampling distribution of the sample means.

1. How many samples are possible? List them and compute the mean of each sample.

Answer 1

$\mu(3,4,5)=(3+4+5)/3=4$

$\mu(3,4,6)=(3+4+6)/3=4.3$

$\mu(3,4,7)=(3+4+7)/3=4.7$

$\mu(3,4,8)=(3+4+8)/3=5$

$\mu(3,4,9)=(3+4+9)/3=5.3$

$\mu(4,5,6)=(4+5+6)/3=5$

$\mu(4,5,7)=(4+5+7)/3=5.3$

$\mu(4,5,8)=(4+5+8)/3=5.7$

$\mu(4,5,9)=(4+5+9)/3=6$

$\mu(5,6,7)=(5+6+7)/3=6$

$\mu(5,6,8)=(5+6+8)/3=6.3$

$\mu(5,6,9)=(5+6+9)/3=6.7$

$\mu(6,7,8)=(6+7+8)/3=7$

$\mu(6,7,9)=(6+7+9)/3=7.3$

$\mu(7,8,9)=(7+8+9)/3=8$

Total 15 cases

Histogram has form of integer distribution.

f(4)=f(4.3)=f(4.7)=f(5.7)=f(6.3)=f(6.7)=f(7)=f(7.3)=f(8)=1/15

f(5)=f(5.3)=f(6)=2/15

$E(x)=\sum fx=2/15(4+4.3+4.7+5.7+6.3+6.7+7+7.3+8)+2/15(5+5.3+6)=3.6+2.2=5.8$

$\sigma^2=\sum fx^2-(\sum fx)^2=1/15(16+18.48+22.09+32.49+39.69+44.89+49+53.29+64)+2/15(25+28.09+36)-33.64=22.66+11.88-33.64=0.93$