Interval Estimate of Population Proportion
After we found a point sample estimate of the population proportion, we would need to estimate its confidence interval.
Let us denote the 100(1 −α∕2) percentile of the standard normal distribution as zα∕2. If the samples size n and population proportion p satisfy the condition that np ≥ 5 and n(1 − p) ≥ 5, than the end points of the interval estimate at (1 − α) confidence level is defined in terms of the sample proportion as follows.
Compute the margin of error and estimate interval for the female students proportion in survey at 95% confidence level.
We first determine the proportion point estimate. Further details can be found in the previous tutorial.
> gender.response = na.omit(survey$Sex)
> n = length(gender.response) # valid responses count
> k = sum(gender.response == "Female")
> pbar = k/n; pbar
Then we estimate the standard error.
Since there are two tails of the normal distribution, the 95% confidence level would imply the 97.5th percentile of the normal distribution at the upper tail. Therefore, zα∕2 is given by qnorm(.975). Hence we multiply it with the standard error estimate SE and compute the margin of error.
Combining it with the sample proportion, we obtain the confidence interval.
At 95% confidence level, between 43.6% and 56.3% of the university students are female, and the margin of error is 6.4%.
Instead of using the textbook formula, we can apply the prop.test function in the built-in stats package.