a) It means that the proportion answering "Yes, Should" in the sample is within 4% of the proportion of all adults nationwide who would answer "Yes". This can be verified by noting that the margin of error is 0.0394 or about 0.04.
b) 31% +/- 4% or 27% to 35%.
c) It probably represents a real difference. If the margin of error for each poll is 4%, then a confidence interval for the June poll is 15% to 23%, which does not overlap at all with the confidence interval given in part b.
Problem 4.
a) 0.0229
b) 2 standard deviations is about 4.5.
c) a 95% confidence interval is 70.5% to 79.5%; we are 95% confident that the percentage of adult Americans who think that Congress should maintain the ban is between 70.5% to 79.5%.
Problem 5.
a) 68%
a) 90%
a) 95%
a) 99%
Problem 6.
a) decrease
b) remain the same
c) decrease
Problem 11.
a) 56% to 64%
b) yes, we can be 95% sure that the true percentage of people who prefer the quarter system is 56% to 64% which is a majority.
c) 46% to 74%. Because the CI goes below 50% this does not provide convincing evidence that a majority prefer to remain on the quarter system.
d) The larger sample size in part a yielded a smaller standard deviation and thus a smaller CI. In general the larger the sample size the smaller the 95% CI we can get, so the sample size alone can determine whether we accept that the majority of a population indeed has a certain trait.
Problem 12.
a) A 95% CI is 54.6% to 63.4%. This means that we are 95% confident that the portion of American Catholics who believe that women should be allowed to be priests is between 54.6% to 63.4%.
b) The standard deviation is about 0.022, so a 95% CI is 0.546 to .6334, the same as in a.
c) Using the SD, a 95% confidence interval is about 86.2% to 91.8%. Using the margin of error, th CI is 84.6% to 93.4%, much wider than the first.
d) the intervals in parts a and b were quite similar because the proportion was close to 0.5. In part c the proportion was further away from 0.5, so 2*SD is smaller than the margin of error.
Problem 13.
a) sqrt(1/1000) = 0.03
b) We are 95% confident that the portion of American adults who feel that the world will come to an end is between 56% and 62%.
Problem 14.
a) The SD = 0.02, so the 95% CI is 29% to 37%.
b) You can't use the +/- 3% margin of error because this is based on the 1000 sample size.
Problem 15.
a) the 95% CI is 20% to 24%. This means that we can be 95% confident that the percentage of American college student binge drinkers having unprotected sex is between 20% to 24%.
b) The sample proportion is 0.02; the SD is 0.001. A 95% CI is 1.8 to 2.2%.
Problem 16.
The SD is about 0.04, so a 99.7% CI is about 0.37 +/- 3(0.04) or 25% to 48%. A critic would prefer to use this interval because it does include the 25% value that would be expected if there were no ESP at work.