Question

A random sample of 91 observations produced a mean x̄ = 25.6 and a standard deviation s = 2.6.

a. find a 95% confidence interval for μ.

b. find a 90% confidence interval for μ.

c. find a 99% confidence interval for μ.

Answer 1

Given:

n - random sample size = 91

a) 95% confidence interval for population mean()

The formula of confidence interval is

Here the population standard deviation is unknown, so t interval is applicable.

where t is the critical value using t distribution table for given confidence level.

c = Confidence level = 95% = 0.95

Degrees of freedom = n - 1 = 91 - 1 = 90

The critical value for degrees of freedom 90 and area in two tail with 0.05 using t table is 1.987

Plug all the values in the formula of confidence interval,

The 95% confidence interval for population mean is (25.058, 26.142)

b)

90% confidence interval for population mean()

The formula of confidence interval is

where t is the critical value using t distribution table for given confidence level.

c = Confidence level = 90% = 0.90

Degrees of freedom = n - 1 = 91 - 1 = 90

The critical value for degrees of freedom 90 and area in two tail with 0.10 using t table is 1.662

Plug all the values in the formula of confidence interval,

The 90% confidence interval for population mean is (25.147, 26.053)

c)

99% confidence interval for population mean()

The formula of confidence interval is

where t is the critical value using t distribution table for given confidence level.

c = Confidence level = 99% = 0.99

Degrees of freedom = n - 1 = 91 - 1 = 90

The critical value for degrees of freedom 90 and area in two tail with 0.01 using t table is 2.632

Plug all the values in the formula of confidence interval,

The 99% confidence interval for population mean is (24.883, 26.317)