Question

A simple random sample of 60 items from a population with σ = 7 resulted in a sample mean of 38.

If required, round your answers to two decimal places.

a. Provide a 90% confidence interval for the population mean.
_____ to _____

b. Provide a 95% confidence interval for the population mean.
_____ to _____

c. Provide a 99% confidence interval for the population mean.
_____ to _____

Answer 1

Solution:

Given,

= 38 ....... Sample mean

= 7 ........Sample standard deviation

n = 60 ....... Sample size

Note that, Population standard deviation() is known. So we use z distribution.   

a.)c = 90% = 0.90

= 1- c = 1- 0.90 = 0.10

  /2 = 0.10 2 = 0.05 and 1- /2 = 0.950

Search the probability 0.950 in the Z table and see corresponding z value

  = 1.645   

The margin of error is given by

E =  /2 * ( / n )

= 1.645 * (7 / 60)

= 1.49

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(38 - 1.49)   <   <  (38 + 1.49)

36.51 <   < 39.49

Answer : 36.51 to 39.49

b.)

c = 95% = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.05 2 = 0.025 and 1- /2 = 0.975

Search the probability 0.975 in the Z table and see corresponding z value

= 1.96   

The margin of error is given by

E =  /2 * ( / n )

= 1.96 * (7 / 60)

= 1.77

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(38 - 1.77)   <   <  (38 + 1.77)

36.23 <   < 39.77

Answer : 36.23 to 39.77

c.)

c = 99% = 0.99

= 1- c = 1- 0.99 = 0.01

  /2 = 0.01 2 = 0.005 and 1- /2 = 0.995

Search the probability 0.995 in the Z table and see corresponding z value

= 2.576   

The margin of error is given by

E =  /2 * ( / n )

= 2.576 * (7 / 60)

= 2.33

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(38 - 2.33)   <   <  (38 + 2.33)

35.67 <   < 40.33

Answer : 35.67 to 40.33