HELP CONFIDENCE INTERVALS

Question

A sample of size  n=46 is drawn from a population whose standard deviation is 21.

Find the margin of error for a 90% confidence interval for μ

Round the answer to at least three decimal places.

Answer 1

Because the population standard deviation is known and n>30, we can assume the sample to be normally distributed (NOTE: your class may opt for keeping the student t distribution instead, so I will include that calculation; if this is the case, degrees of freedom will be n-1=45). Then, because we are creating a 90% confidence interval (CI), we know that α=1-.9=.1, and this makes the the critical value be z_1-.1/2=z_.95=1.645 (t_{.95, 45}=1.68). The formula for Margin of Error (MOE) is as follows:

MOE = (crit. value)•(standard deviation of mean)

= 1.645•21/√46

= 5.0934

(Using student t distribution, MOE = 5.202)

From this point, the confidence interval is found by subtracting and adding the MOE from the mean:

(μ-MOE, μ+MOE).