Home page for accesible maths 4.1 Inference for a single proportion 4.1.1 Identifying when the sample proportion is nearly normal 4.1.3 Hypothesis testing for a proportion

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4.1.2 Confidence intervals for a proportion

We may want a confidence interval for the proportion of Americans who approve of the job the Supreme Court is doing. Our point estimate, based on a sample of size $n=976$ from the NYTimes/CBS poll, is $\hat{p}=0.44$ . To use the general confidence interval formula from Section 2.11, we must check the conditions to ensure that the sampling distribution of $\hat{p}$ is nearly normal. We also must determine the standard error of the estimate.

The data are based on a simple random sample and consist of far fewer than 10% of the U.S. population, so independence is confirmed. The sample size must also be sufficiently large, which is checked via the success-failure condition: there were approximately $976\times\hat{p}=429$ ‘‘successes’’ and $976\times(1-\hat{p})=547$ ‘‘failures’’ in the sample, both easily greater than 10.

With the conditions met, we are assured that the sampling distribution of $\hat{p}$ is nearly normal. Next, a standard error for $\hat{p}$ is needed, and then we can employ the usual method to construct a confidence interval.

Example 4.1.1

Estimate the standard error of $\hat{p}=0.44$ using Equation (4.1). Because $p$ is unknown and the standard error is for a confidence interval, use $\hat{p}$ in place of $p$ .

Answer. $SE=\sqrt{\frac{p(1-p)}{n}}\approx\sqrt{\frac{0.44(1-0.44)}{976}}=0.016$ .

Example 4.1.2

Construct a 95% confidence interval for $p$ , the proportion of Americans who approve of the job the Supreme Court is doing.

Answer. Using the standard error estimate from Exercise 4.1.1, the point estimate 0.44, and $z^{\star}=1.96$ for a 95% confidence interval, the confidence interval may be computed as

\displaystyle\text{point estimate }\ \pm\ z^{\star}SE\quad\to\quad 0.44\ \pm\ % 1.96\times 0.016\quad\to\quad(0.409,0.471)

We are 95% confident that the true proportion of Americans who approve of the job of the Supreme Court (in June 2012) is between 0.409 and 0.471. If the proportion has not changed since this poll, than we can say with high confidence that the job approval of the Supreme Court is below 50%.

Constructing a confidence interval for a proportion • Verify the observations are independent and also verify the success-failure condition using $\hat{p}$ and $n$ . • If the conditions are met, the sampling distribution of $\hat{p}$ may be well-approximated by the normal model. • Construct the standard error using $\hat{p}$ in place of $p$ and apply the general confidence interval formula.