**Question 1: **Find the z test for the following series: 5,10,15, 20, 25 for x = 20?

** Solution: **

The given series is 5,10,15,20,25 and variable x = 20

The population mean

$\mu$ = $\frac{\Sigma x_i}{n}$

= $\frac{5 + 10 + 15 + 20 + 25}{5}$

= $\frac{75}{5}$

= 15

The variance is

For x_{i} = 5, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (5 - 15)^{2} = 100

For x_{i} = 10, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (10 - 15)^{2} = 25

For x_{i} = 15, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (15 - 5)^{2} = 100

For x_{i} = 20, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (20 - 15)^{2} = 25

For x_{i} = 25, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (25 - 15)^{2} = 100

The standard deviation is

$\sigma$ = $\sqrt{\frac{100 + 25 + 100 + 25 + 100}{5}}$

= 8.36.

The z test is

z = $\frac{x - \mu}{\sigma}$

= $\frac{20 - 15}{70}$

= 0.6.

**Question 2: **Calculate the z test for the following series: 2,4,6,8 for x = 4.5

** Solution: **

The given series is 2,4,6,8 for x = 5.5 .The population mean is given by

$\mu$ = $\frac{\Sigma x_i}{n}$

= $\frac{2 + 4 + 6 + 8}{4}$

= 5

The variance is

For x_{i }= 2, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (2 - 5)^{2} = 9

For x_{i} = 4, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (4 - 5)^{2} = 1

For x_{i} = 6, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (6 - 5)^{2} = 1

For x_{i }= 8, $\sigma^2$ = $\sqrt{(x_i - \mu)^2}$ = (8 - 5)^{2} = 9_{}

The standard deviation is

$\sigma$ = $\sqrt{\frac{9+1+1+9}{4}}$

= $\sqrt{5}$
= 2.236.

The z test is

z = $\frac{x - \mu}{\sigma}$

= $\frac{5.5 - 5}{2.236}$

= 0.224.