In the setting of confidence intervals, the sample proportions are used to verify the success-failure condition and also compute standard error, just as was the case with a single proportion.
The way a question is phrased can influence a person’s response. For example, Pew Research Enter conducted a survey with the following question:3737www.people-press.org/2012/03/26/public-remains-split-on-health-care-bill-opposed-to-mandate/. Sample sizes for each polling group are approximate.
As you may know, by 2014 nearly all Americans will be required to have health insurance. [People who do not buy insurance will pay a penalty] while [People who cannot afford it will receive financial help from the government]. Do you approve or disapprove of this policy?
For each randomly sampled respondent, the statements in brackets were randomized: either they were kept in the order given above, or the two statements were reversed. Table 4.1 shows the results of this experiment. Create and interpret a 90% confidence interval of the difference in approval.
| Sample size () | Approve law (%) | Disapprove law (%) | Other | |
|---|---|---|---|---|
| ‘‘people who cannot afford it will receive financial help from the government’’ is given second | 771 | 47 | 49 | 3 |
| ‘‘people who do not buy it will pay a penalty’’ is given second | 732 | 34 | 63 | 3 |
Answer. First the conditions must be verified. Because each group is a simple random sample from less than 10% of the population, the observations are independent, both within the samples and between the samples. The success-failure condition also holds for each sample. Because all conditions are met, the normal model can be used for the point estimate of the difference in support, where corresponds to the original ordering and to the reversed ordering:
Answer. The standard error may be computed from Equation (4.2) using the sample proportions:
For a 90% confidence interval, we use :
We are 90% confident that the approval rating for the 2010 healthcare law changes between 9% and 17% due to the ordering of the two statements in the survey question. The Pew Research Enter reported that this modestly large difference suggests that the opinions of much of the public are still fluid on the health insurance mandate.