Question

A simple random sample of 60 items resulted in a sample mean of 75. The population standard deviation is 17.

Compute the 95% confidence interval for the population mean (to 1 decimal).

Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean (to 2 decimals).

What is the effect of a larger sample size on the margin of error?

Answer 1

Solution :

Given that,

= 75

= 17

a ) n = 60

At 95% confidence level the z is ,

  = 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.960

Margin of error = E = Z_/2* (/n)

= 1.960 * (17 / 60 )

= 4.3

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

- E < < + E

75 - 4.3 < < 75 + 4.3

70.7 < < 79.3

(70.7 , 79.3)

b ) n = 120

At 95% confidence level the z is ,

  = 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.960

Margin of error = E = Z_/2* (/n)

= 1.960 * (17 / 120 )

= 3.0

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

- E < < + E

75 - 3.0 < < 75 + 3.0

72.0 < < 78.0

(72.0 , 78.0)

c ) The effect of a larger sample size on the margin of error is smaller