Question

1. The number of vehicles on a highway link is counted on each of 40 randomly chosen days. The mean number of vehicles is found to be 135, and the standard deviation is 90.

(a) Find a 90% confidence interval for the sample mean.
(b) Find a 90% confidence interval for the mean, assuming it had been based upon a sample of 15 days, instead of a sample 40 days.

Answer 1

Solution :

Given that,

a)

sample size = n = 40

Degrees of freedom = df = n - 1 = 39

At 90% confidence level the t is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

t _/2,df = t_0.05,39 = 1.685

Margin of error = E = t_/2,df * (s /n)

= 1.685 * (90 / 40)

= 23.978

The 90% confidence interval estimate of the population mean is,

- E < < + E

135 - 23.978 < < 135 + 23.978

111.022 < < 158.978

(111.022 , 158.978)

b)

sample size = n = 15

Degrees of freedom = df = n - 1 = 14

At 90% confidence level the t is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

t _/2,df = t_0.05,14 = 1.761

Margin of error = E = t_/2,df * (s /n)

= 1.761 * (90 / 15)

= 40.922

The 90% confidence interval estimate of the population mean is,

- E < < + E

135 - 40.922 < < 135 + 40.922

94.078 < < 175.922

(94.078 , 175.922)