solution: Since we do not know the standard deviation for either population, we need to use a
two-sample t interval. The conditions necessary for using this interval are given in the
problem: SRSs from independent, approximately normal populations.
The 90% confidence interval isUsing t * for 35.9994 df as reported by the calculator.
(Note: You can calculate the interval on the calculator, and then use InvT to find t *. But that isn’t
actually necessary. You do need to report the degrees of freedom! Writing all the stuff above
shows clear communication, which can benefit you and can save you if you make a calculation
error. But writing something incorrect can cost you. At a minimum you must identify the
procedure (by name or by formula), report the degrees of freedom, and give the interval.)
We are 90% confident that the difference between the population means lies in the interval
from 0.227 to 5.25. If the true difference between the means is zero, we would expect to find 0
in the interval. Because it isn’t, this interval provides evidence of a difference between the
population means.
example: Construct a 95% confidence interval for p 1 – p 2 given that n 1 = 180, n 2 = 250, 1 =
0.31, 2 = 0.25. Assume that these are data from SRSs independently selected from two
populations.
solution: 180(0.31) = 55.8, 180(1 – 0.31) = 124.2, 250(0.25) = 62.5, and 250(0.75) = 187.5 are
all greater than or equal to 5 so, with what is given in the problem, we have the conditions
needed to construct a two-proportion z interval.We are 95% confident that the proportion of successes in population 1 is between 2.6
percentage points lower and 14.6 percentage points higher than that of population 2. (But be
sure to include context when describing the populations and what proportion you are
estimating.)Sample Size
It is always desirable to select as large a sample as possible when doing research because sample means