Context

A Gallup poll conducted in November of 2011 asked the following question, “What would you say is the most urgent health problem facing this country at the present time?” 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. “Results 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 ±4 percentage points.”

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.