A Confidence Interval for p

How precise is sixty out of one hundred?

• In a fictional random sample of 100 customers, 60 support a new collection service. We want a confidence interval 置信区间 for the population support proportion.
• The observed estimate is $\hat p=0.60$. The interval should describe customers in the sampled population, not the proportion in every future sample.

Check before estimating uncertainty

• Check random sampling, $100\le0.10N$ for sampling without replacement, and the observed counts 60 and 40. Each count is at least ten.
• The standard error 标准误 estimates the sampling variability using $\hat p$, since the population proportion is unknown.

Build estimate plus or minus uncertainty

• Use the formula $\hat p\pm z^*SE$, with $SE=\sqrt{\hat p(1-\hat p)/n}$. A 95% normal interval uses $z^*=1.96$.
• Here $SE=\sqrt{0.60(0.40)/100}\approx0.04899$. Keep these extra digits until the endpoints are calculated.

Calculate the endpoints

• The margin of error 误差幅度 is $ME=z^*SE=1.96(0.04899)\approx0.09602$. The interval is $(0.50398,0.69602)$.
• The confidence level 置信水平 describes the repeated-sampling method. The shaded normal area at $\pm1.96$ is about 0.95; it is not a plot of individual customer answers.

Change one factor at a time

• With the same observed proportion and confidence level, quadrupling $n$ halves $SE$ and the margin of error. Raising confidence increases $z^*$ and widens the interval.
• These width comparisons hold the other inputs fixed. A new sample can also change $\hat p$, so its observed width need not follow a sample-size slogan exactly.

Interpret the estimate in context

• We are 95% confident that about 50.4% to 69.6% of customers in the sampled population support the service, using this checked procedure.
• The interval covers sampling uncertainty, not every source of error. Nonresponse, misleading questions and selection bias need separate attention.

Write from a sample proportion of 0.4

• 40 of 100 library users support a longer Sunday opening.
• The lesson margin inputs 0.05 and 0.1 are another poll.