Answer to Question #169415 in Statistics and Probability for JEREMIAH UBALDO

Question

. Random sample of size 4 are drawn from the finite population which consists of the numbers 2,3,7,8,and 10.

a. What is the population mean, population variance and population standard deviation of the given data?

b. What is the sampling distribution of the sample means for a sample of size 2 which can be drawn without replacement from the given population?

c. What is the mean, variance and standard deviation of the sampling distribution?

Answer 1

a). Suppose that we have chosen numbers: $x_1,x_2,x_3,x_4$. Then the mean is: $\mu=\frac{x_1+x_2+x_3+x_4}{4}$. The variance is: $\sigma^2=\frac{(x_1-\mu)^2+(x_2-\mu)^2+(x_3-\mu)^2+(x_4-\mu)^2}{4}$ . The standard deviation is $\sigma$. For 5 numbers 2,3,7,8,10 we have: $\mu=\frac{2+3+7+8+10}{5}=6$ , The variance is: $\sigma^2=\frac{(2-6)^2+(3-6)^2+(7-6)^2+(8-6)^2+(10-6)^2}{5}=\frac{16+9+1+4+16}{5}=9.2$. The standard deviation is: $\sigma=\sqrt{9.2}\approx3.03$.

b). We have the discrete distribution. We can obtain $5+4+3+2+1=15$ different pairs of numbers. We enumerate them and their means:

(2,2) $\mu=2$ , (2,3) $\mu=2.5$, (2,7) $\mu=4.5$ , (2,8) $\mu=5$, (2,10) $\mu=6$; (3,3) $\mu=3$ , (3,7) $\mu=5$, (3,8) $\mu=5.5$ , (3,10) $\mu=6.5$ ; (7,7) $\mu=7$ , (7,8) $\mu=7.5$ , (7,10) $\mu=8.5$ ; (8,8,) $\mu=8$ ,(8,10) $\mu=9$ ; (10,10) $\mu=10$; For all means the probability is $p=\frac{1}{15}$ except of $\mu=5$. For the latter we have: $p=\frac{2}{15}$.We received a discrete probability distribution with a finite probability space.

c). The mean is: $E[X]=\frac{2+2.5+4.5+2\cdot5+6+3+5.5+6.5+7+7.5+8.5+8+9+10}{15}=6$ . The variance is: $\sigma^2=Var[X]=E[X^2]-(E[X])^2=\frac{2^2+2.5^2+4.5^2+2\cdot5^2+6^2+3^2+5.5^2+6.5^2+7^2+7.5^2+8.5^2+8^2+9^2+10^2}{15}-6^2\approx41.367-36\approx5.37$

The standard deviation is: $\sigma=\sqrt{5.37}\approx2.32$