H0: p = p0An alternative hypothesis can be one-sided.
HA: p < p0or
HA: p > p0Otherwise, an alternative hypothesis can be two-sided
HA: p does not equal p0In any case, the hypothesis is tested by considering the sample proportion and its sampling distribution.
For large samples, the shape of the sampling distribution is approximately normal. A rule of thumb is that if both np and n(1-p) are larger than 5, the sample is sufficiently large for the normal approximation. The result can be compared to the standard normal distribution. This depends on the central limit theorem.
Since np = 1000(.02) = 20 > 5, we may use the normal approximation.
We state hypotheses as
H0: p = .02 HA: p < .02Our sample proportion is 18/1000 = .018. If the null hypothesis is true, the SE is
sqrt( (.02)(.98)/1000 ) = .00443The z statistic for the z-test is
(.018 - .020) / .0043 = -4.65The p-value is essentially 0. This is very strong evidence that the proportion of women is below 2%. However, it is only below 2% by a small amount. The large sample size allows us to conclude with high confidence that the population proportion is in a small interval close to 1.8%.
Bret Larget, email@example.com