Question

In a random sample of 8 people, the mean commute time to work was 36.5 minutes and the standard deviation was 7.3

minutes. A 90​% confidence interval using the​ t-distribution was calculated to be (31.6,41.4).

After researching commute times to​ work, it was found that the population standard deviation is 9.3.

minutes. Find the margin of error and construct a 90​% confidence interval using the standard normal distribution with the appropriate calculations for a standard deviation that is known. Compare the results.

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 36.5

Population standard deviation =    = 9.3

Sample size = n = 8

At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_{0.05 = 1.645}

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

= 1.645 * (9.3 /  8 )

= 5.41

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

  ± E

36.5  ± 5.41

( 31.1 , 41.9)

t- distribution confidence interval is smaller than z -distribution confidence interval.