population proportion, we will use P * = 0.5. Then,You would need a minimum of 385 subjects for your sample. (Always round up for the minimum
sample size required.)
Which of the following statements is correct?
I. The t distribution has more area in its tails than the z distribution (normal).
II. When constructing a confidence interval for a population mean, you would always use z rather
than t if you have a sample size of at least 30 (n > 30).
III. When constructing a two-sample t interval, the “conservative” method of choosing degrees of
freedom (df = min {n 1 – 1, n 2 – 1}) will result in a wider confidence interval than other
methods.
Answer:
• I is correct. A t distribution, because it must estimate the population standard deviation, has more
variability than the normal distribution.
• II is not correct. It is definitely not correct that you would always use z rather than t in this
situation. A more interesting question is could you use z rather than t ? The answer to that
question is a qualified “yes.” The difference between z and t is small for large sample sizes
(e.g., for a 95% confidence interval based on a sample size of 50, z = 1.96 and t = 2.01) and,
while a t interval would have a somewhat larger margin of error, the intervals constructed would
capture roughly the same range of values. In fact, many traditional statistics books teach this as the
proper method. Now, having said that, the best advice is to always use t when dealing with a one-
sample situation when s is unknown (confidence interval or hypothesis test) and use z only when
you know, or have a very good estimate of, the population standard deviation.
• III is correct. The conservative method (df = min{n 1 – 1, n 2 – 1}) will give a larger value of t *,
which, in turn, will create a larger margin of error, which will result in a wider confidence interval
than other methods for a given confidence level.
Practice Problems
Multiple-Choice
You are going to create a 95% confidence interval for a population proportion and want the margin
of error to be no more than 0.05. Historical data indicate that the population proportion has remained
constant at about 0.7. What is the minimum size random sample you need to construct this interval?
a. 385
b. 322
c. 274
d. 275
e. 323