In each of the following settings, check whether the conditions for calculating a confidence interval for the population proportion p are met.
1. An AP Statistics class at a large high school conducts a survey. They ask the first students to arrive at school one morning whether or not they slept at least 8 hours the night before. Only 17 students say "Yes."
2. A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt.
No, the condition of the random sample is not satisfied.
Number of success (x)
Number of trials (n)
The sample was picked based on the school's convenience for children who arrive early, and using the basic random sampling technique is one of the conditions for generating the confidence interval for population percentage. As a result, in this circumstance, the criteria of a simple random sample is not met.
Equality for women? Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted adults. A newspaper article about the poll said, “Results have a margin of sampling error of plus or minus 3 percentage points.”
(a) The news article said that % of men, but only % of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give conﬁdence intervals for men and women separately.
(b) Would a % conﬁdence interval for women alone have a margin of error less than , about equal to , or greater than ? Why? (You see that the news article’s statement about the margin of error for poll results is a bit misleading.)
I collect an SRS of size n from a population and compute a confidence interval for the population proportion. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data?
(a) Use a larger confidence level.
(b) Use a smaller confidence level.
(c) Increase the sample size.
(d) Use the same confidence level, but compute the interval n times. Approximately of these intervals will be larger.
(e) Nothing can guarantee absolutely that you will get a larger interval. One can only say that the chance of obtaining a larger interval is .
The news article goes on to say: “The theoretical errors do not take into account a margin of additional error resulting from the various practical difficulties in taking any survey of public opinion.” List some of the “practical difficulties” that may cause errors in addition to the percentage point margin of error. Pay particular attention to the news article’s description of the sampling method.
Can you taste PTC? PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste PTC is inherited. About % of Italians can taste PTC, for example. You want to estimate the proportion of Americans who have at least one Italian grandparent and who can taste PTC.
(a) How large a sample must you test to estimate the proportion of PTC tasters within with % conﬁdence? Answer this question using the % estimate as to the guessed value for .
(b) Answer the question in part (a) again, but this time use the conservative guess . By how much do the two sample sizes differ?
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