Question

Let the following sample of 8 observations be drawn from a normal population with unknown mean and standard deviation:

21, 14, 13, 24, 17, 22, 25, 12

Required:
a. Calculate the sample mean and the sample standard deviation.
b. Construct the 90% confidence interval for the population mean.
c. Construct the 95% confidence interval for the population mean

Answer 1

a

`\= x = 18.5` ,
`\sigma = 5.15`

b

`15.505 < \mu < 21.495`

c

`14.93 < \mu < 22.069`

Explanation:

From the question we are are told that

The sample data is 21, 14, 13, 24, 17, 22, 25, 12

The sample size is n = 8

Generally the ample mean is evaluated as

`\= x = (\sum x )/(n)`

`\= x = ( 21 + 14 + 13 + 24 + 17 + 22+ 25 + 12 )/(8)`

`\= x = 18.5`

Generally the standard deviation is mathematically evaluated as

`\sigma = \sqrt{(\sum (x- \=x )^2)/(n)}`

`\sigma = \sqrt{(\sum ((21 - 18.5)^2 + (14-18.5)^2+ (13-18.5)^2+ (24-18.5)^2+ (17-18.5)^2+ (22-18.5)^2+ (25-18.5)^2+ (12 -18.5)^2 ))/(8)}`

`\sigma = 5.15`

considering part b

Given that the confidence level is 90% then the significance level is evaluated as

`\alpha = 100-90`

`\alpha = 10\%`

`\alpha = 0.10`

Next we obtain the critical value of
`( \alpha )/(2)` from the normal distribution table the value is

`Z_{( \alpha )/(2) } = 1.645`

The margin of error is mathematically represented as

`E = Z_{( \alpha )/(2) } * (\sigma )/(√(n) )`

=>
`E =1.645 * (5.15 )/(√(8) )`

=>
`E = 2.995`

The 90% confidence interval is evaluated as

`\= x - E < \mu < \= x + E`

substituting values

`18.5 - 2.995 < \mu < 18.5 + 2.995`

`15.505 < \mu < 21.495`

considering part c

Given that the confidence level is 95% then the significance level is evaluated as

`\alpha = 100-95`

`\alpha = 5\%`

`\alpha = 0.05`

Next we obtain the critical value of
`( \alpha )/(2)` from the normal distribution table the value is

`Z_{( \alpha )/(2) } = 1.96`

The margin of error is mathematically represented as

`E = Z_{( \alpha )/(2) } * (\sigma )/(√(n) )`

=>
`E =1.96 * (5.15 )/(√(8) )`

=>
`E = 3.569`

The 95% confidence interval is evaluated as

`\= x - E < \mu < \= x + E`

substituting values

`18.5 - 3.569 < \mu < 18.5 + 3.569`

`14.93 < \mu < 22.069`