The U.S. Census Bureau conducts a study to determine the time needed to complete the short form.

Question

The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 20 people. The sample mean is 8.2 minutes. There is a sample standard deviation of 2.2 minutes. The population distribution is assumed to be normal. Construct a 90% Confidence Interval for the true mean.

Answer 1

Hi Emma,

This is a fairly straightforward question. Given the sample mean xbar = 8.2, the sample standard deviation s = 2.2, a sample size of n = 20, we can construct a confidence interval using the following formula:

xbar ± t* (s / sqrt(n))

where t* is the critical value at the 90% confidence level with n-1 degrees of freedom. To find this value, you could use a t-table or your favorite software. I'll have that detailed down below, but the value is 1.729.

Thus, the formula would be:

8.2 ± 1.729 × (2.2 / √20) = (7.35, 9.05)

And we would say "We are 95% confident that the interval from 7.35 to 9.05 captures the true mean time needed to complete a short form."

In Python, it would look like this:

import scipy.stats as stats

import math

xbar = 8.2

s = 2.2

n = 20

t_crit = stats.t.ppf(q=0.95, df=19) # q = 1-(.1/2), df = 20-1

lower_bound = xbar - t_crit*(s/math.sqrt(n)) # 7.35

upper_bound = xbar - t_crit*(s/math.sqrt(n)) # 9.05