Question

An engineer has carefully measured the spacing between anchor bolts (in mm) as:
19   23   21   21
19   13   26   15
30   30   17   24
20   16   24   20
Find:
a. The 90 percent CI for µ
b. The 80 percent CI for µ
c. The 99 percent CI for µ assuming σ = 4.1 mm is known

Answer 1

Given:

n = 16 observations given

a. 90% confidence interval for

The formula to find the confidence interval for population mean, when population standard deviation know is

Where

Z - z critical value at given confidence level

n - sample size = 16

The formula to find the sample mean is,

c = confidence level = 90% = 0.90

alpha = 1 - c = 0.10

1 - (alpha/2) = 1 - (0.10/2) = 1 - 0.05 = 0.95

By using z table the z critical value at 0.95 is 1.645

Plug all the values in the formula of confidence interval

Therefore, (19.44, 22.81) is the 90% confidence interval for

b. 80% confidence interval for

Formula is same as above, just the z critical value will change

Here c = confidence level = 80% = 0.80

alpha = 1 - c = 1 - 0.80 = 0.20

1 - (alpha/2) = 1 - (0.20/2) = 1 - 0.10 = 0.90

The z critical value using z table for area 0.90 is 1.28

The confidence interval is

The 80% confidence interval for is (19.81, 22.44)

c. 99% confidence interval for

Formula is same, just z changes

c = confidence level = 99% = 0.99

alpha = 1 - 0.99 = 0.01

1 - (alpha/2) = 1 - (0.01/2) = 0.995

The z critical value using z table for area 0.995 is 2.58

The confidence interval is,

Therefore, the 99% confidence interval for is (18.48, 23.77)