Question

Construct an 80​% confidence interval to estimate the population mean when x overbar=131 and s​ = 28 for the sample sizes below. ​

a) n=20

​b) n=50 ​

c) n=80

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 131

sample standard deviation = s = 28

a) sample size = n = 20

Degrees of freedom = df = n - 1 = 20 - 1 = 19

At 80% confidence level

= 1 - 80%

=1 - 0.80 =0.20

/2 = 0.10

t/2,df = t_0.10,19 = 1.328

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

= 1.328 * (28 / 20)

Margin of error = E = 8.31

The 80% confidence interval estimate of the population mean is,

  ± E  

= 131  ± 8.31

= (122.69, 139.31)

b) sample size = n = 50

Degrees of freedom = df = n - 1 = 50 - 1 = 49

t/2,df = t_0.10,49 = 1.299

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

= 1.299 * (28 / 50)

Margin of error = E = 5.14

The 80% confidence interval estimate of the population mean is,

  ± E  

= 131  ± 5.14

= (125.86, 136.14)

c) sample size = n = 80

Degrees of freedom = df = n - 1 = 80 - 1 = 79

t/2,df = t_0.10,79 = 1.292

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

= 1.292 * (28 / 80)

Margin of error = E = 4.04

The 80% confidence interval estimate of the population mean is,

  ± E  

= 131  ± 4.04

= (126.96, 135.04)