Question

Researchers are interested in the average length of men in a large population. They randomly select 40 men from the population and measure their lengths to get
average of 175 (cm). Scientists also know in advance that the lengths are approximately normal in standard deviation σ = 20 (cm).
(a) Determine a 95% confidence interval for the average length of men for the entire population.
(b) Determine the corresponding estimate at 99% confidence level.

Answer 1

The formula for estimation is:

μ = M ± Z(s_M)

where:

M = sample mean
Z = Z statistic determined by confidence level
s_M = standard error = √(s²/n)

Calculation for 95% confidence interval

M = 175
t = 1.96
s_M = √(20²/40) = 3.16

μ = M ± Z(s_M)
μ = 175 ± 1.96*3.16
μ = 175 ± 6.2

You can be 95% confident that the population mean (μ) falls between 168.8 and 181.2

B) calculation of 99% confidence interval

M = 175
t = 2.58
s_M = √(20²/40) = 3.16

μ = M ± Z(s_M)
μ = 175 ± 2.58*3.16
μ = 175 ± 8.15

Result

M = 175, 99% CI [166.85, 183.15].

You can be 99% confident that the population mean (μ) falls between 166.85 and 183.15.