3. An epidemiologist is planning a study on the prevalence
of oral contraceptive use in a certain population. She
plans to choose a random sample of n women and to use
the sample proportion of oral contraceptive users ($\hat{p}$) as
an estimate of the population proportion ($p$). Suppose
that in fact $p = 0.12$. Use the normal approximation to
determine the probability that $\hat{p}$ will be within $\pm 0.03$ of
p if
(a) $n = 100$, without continuity correction
(b) $n = 100$, with continuity correction
(c) $n = 200$, without continuity correction
(d) $n = 200$, with continuity correction
4. Over 9 months, a random sample of 50 women were
asked to record their average menstrual cycle length (in
days). The sample average was 28.86 days, with a sample
standard deviation of 3.24 days.
(a) Calculate the 90% confidence interval for the true
average menstrual cycle length.
(b) Interpret the confidence interval found in (a) in terms
of the problem.
(c) A researcher hypothesized that women's menstrual
cycles are typically the same length as a lunar month
- 29.5 days. Does your interval from (a) support this
hypothesis?
(d) Are the assumptions for a confident interval met for
this problem?
