Question

A simple random sample of 70 items resulted in a sample mean of 90. The population standard deviation is

σ = 5.

(a)

Compute the 95% confidence interval for the population mean. (Round your answers to two decimal places.)

to

(b)

Assume that the same sample mean was obtained from a sample of 140 items. Provide a 95% confidence interval for the population mean. (Round your answers to two decimal places.)

to

(c)

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

A larger sample size provides a smaller margin of error

.A larger sample size does not change the margin of error.    

A larger sample size provides a larger margin of error.

Answer 1

Solution :

Given that,

Point estimate = sample mean =     = 90

Population standard deviation =    = 5

Sample size n =70

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96   ( Using z table )

Margin of error = E =   Z/2    * ( /n)
= 1.96* ( 5/ 70 )

E= 1.17
At 95% confidence interval estimate of the population mean
is,

- E < < + E

90 - 1.17 <   < 90+ 1.17

88.83 <   < 91.17

( 88.83 , 91.17 )

(B)

Solution :

Given that,

Point estimate = sample mean =     = 90

Population standard deviation =    = 5

Sample size n =140

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96   ( Using z table )

Margin of error = E =   Z/2    * ( /n)
= 1.96* ( 5/ 140 )

E= 0.83
At 95% confidence interval estimate of the population mean
is,

- E < < + E

90 - 0.83 <   < 90+ 0.83

89.17 <   < 90.83

( 89.17 ,90.83 )

A larger sample size provides a smaller margin of error