# introduction to confidence intervals about 100 words

### Context

A Gallup poll conducted in November of 2011 asked the following question, â€œWhat would you say is the most urgent health problem facing this country at the present time?â€ The choices were access, cost, obesity, cancer, government interference, or the flu. The responses were access (27%), cost (20%), obesity (14%), cancer (13%), government interference (3%), or the flu (less than 0.5%).

The following is an excerpt from the *Survey Methods *section. â€œResults for this Gallup poll are based on telephone interviews conducted Nov. 3-6, 2011, with a random sample of 1,012 adults ages 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on a total sample of national adults, one can say with 95% confidence that the maximum margin of sampling error is Â±4 percentage points.â€

### Prompt

If we accept the maximum margin of sampling error provided above, find a 95% confidence interval to estimate the percentage of U.S. adults who feel that access to healthcare is the most urgent health problem facing this country. Interpret your interval in context.

# ANSWER(S){ Hint }

Answers will vary (a little). But we expect 95% (19 out of 20) of the intervals to contain the population proportion. So we expect 5% (1 out of 20) will not contain the population proportion.

Of course, the confidence level is a long run probability, so we may not get exactly these results. But in the long run with many, many repetitions, we expect 95% of the confidence intervals to contain the population proportion.

another example :

5% (1 out of 20) of the random samples do not contain the population proportion and also since only 1 out of 2 did not contain it, we expect that only 95% ( 19/20) to contain the sample proportion. Yes, this is what I expected because every time you select a random sample it is not always going to be within 2 standard error, it has to shift one way or another.