Question

A simple random sample of 90 items resulted in a sample mean of 60. The population standard deviation is

σ = 10.

(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 180 items. Provide a 95% confidence interval for the population mean. (Round your answers to two decimal places.)

______ to _______

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 60

Population standard deviation = = 10

a) Sample size = n = 90

At 95% confidence level

= 1-0.95% =1-0.95 =0.05

/2 =0.05/ 2= 0.025

Z/2 = Z_{0.025 = 1.960}

Z/2 = 1.960  

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

= 1.960 *( 10 /90 )

= 2.07

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

- E < < + E

60 - 2.07 <  < 60 + 2.07

57.93 <   < 62.07

( 57.93 ,62.07 )

A 95% confidence interval for the population mean is 57.93 to 62.07

b)

Sample size = n = 180

At 95% confidence level

= 1-0.95% =1-0.95 =0.05

/2 =0.05/ 2= 0.025

Z/2 = Z_{0.025 = 1.960}

Z/2 = 1.960  

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

= 1.960 *( 10 /180 )

=1.46

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

- E < < + E

60 - 1.46 <  < 60 + 1.46
58.54 <   < 61.46

( 58.54 ,61.46 )

A 95% confidence interval for the population mean is 58.54 to 61.46