- Example: 99% CI for Proportion of Students Female

Scenario: Data were collected from a representative sample of 501 World Campus STAT 200 students. In that sample, 284 students were female and 217 were male. Construct a 99% confidence interval to estimate the proportion of all World Campus students who are female. 

StatKey was used to construct a sampling distribution using bootstrapping methods:

StatKey Bootstrap Distribution Plot

Because this distribution is approximately normal, we can approximate the sampling distribution using the z distribution. We will use the standard error, 0.022, from this distribution.

The original sample statistic was \(\widehat p =\frac{284}{501}=0.567\). 

We can find the \(z^*\) multiplier by constructing a z distribution to find the values that separate the middle 99% from the outer 1%:

Minitab Express output: z distribution showing the middle 99% versus the outer 1%

The \(z^*\) multiplier is 2.57583

Recall the general form of a confidence interval: sample statistic \(\pm\) \(z^*\) (standard error) where \(z^*\) is the multiplier. So in this case we have...

\(0.567 \pm 2.57583 (0.022)\)

\(0.567 \pm 0.057\)

\([0.510, 0.624]\)

I am 99% confident that the proportion of all World Campus students who are female is between 0.510 and 0.624