Question

Construct a 95​% confidence interval to estimate the population mean when Mean=125 and s​ = 26 for the sample sizes below.

​a)N=40       

​b)N=70       

​c) N=100

A.)The 95​% confidence interval for the population mean when N=40is from a lower limit of_____to an upper limit of ______.

B.) The 95​% confidence interval for the population mean when N=70is from a lower limit of _____to an upper limit of ______.

​C.) The 95​% confidence interval for the population mean when N=100is from a lower limit of_____to an upper limit of ______.

​

Answer 1

Solution :

Given that,

= 125

s = 26

(A)

n = 40

Degrees of freedom = df = n - 1 = 40 - 1 = 39

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t _/2,df = t_0.025,39 = 2.023

Margin of error = E = t_/2,df * (s /n)

= 2.023 * (26 / 40)

= 8.3

The 95% confidence interval estimate of the population mean is,

- E < < + E

125 - 8.3 < < 125 + 8.3

116.7 < < 133.3

The 95​% confidence interval for the population mean when N = 40 is from

a lower limit of 116.7 to an upper limit of 133.3 .

(B)

n = 70

Degrees of freedom = df = n - 1 = 70 - 1 = 69

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t _/2,df = t_0.025,69 = 1.995

Margin of error = E = t_/2,df * (s /n)

= 1.995 * (26 / 70)

= 6.2

The 95% confidence interval estimate of the population mean is,

- E < < + E

125 - 6.2 < < 125 + 6.2

118.8 < < 131.2

The 95​% confidence interval for the population mean when N = 70 is from

a lower limit of 118.8 to an upper limit of 131.2

(C)

n = 100

Degrees of freedom = df = n - 1 = 100 - 1 = 99

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t _/2,df = t_0.025,99 = 1.984

Margin of error = E = t_/2,df * (s /n)

= 1.984 * (26 / 100)

= 5.2

The 95% confidence interval estimate of the population mean is,

- E < < + E

125 - 5.2 < < 125 + 5.2

119.8 < < 130.2

The 95​% confidence interval for the population mean when N = 100 is from

a lower limit of 119.8 to an upper limit of 130.2