Question

A simple random sample of 40 items resulted in a sample mean of 80. The population standard deviation is s=20.

a. Compute the 95% confidence interval for the population mean. Round your answers to one decimal place.

b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean. Round your answers to two decimal places.

c. What is the effect of a larger sample size on the interval estimate?

Larger sample provides a _______ (larger/smaller) margin of error.

Answer 1

Solution :

Given that,

= 80

s = 20

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 * (20 / 40)

= 6.4

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

- E < < + E

80 - 6.4 < < 80 + 6.4

73.6 < < 86.4

(73.6, 86.4 )

B) n = 120

Degrees of freedom = df = n - 1 = 120 - 1 = 119

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

  / 2 = 0.05 / 2 = 0.025

t _/2,df = t_0.025,119 = 1.980

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

= 1.980 * (20 / 120)

= 3.61

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

- E < < + E

80 - 3.61 < < 80 + 3.61

76.39 < < 83.61

(76.39, 83.61 )

C ) The effect of a larger sample size on the interval estimate

smaller margin of error.

The effect of a smaller sample size on the interval estimate

larger margin of error.