A television sports commentator wants to estimate the proportion of citizens who​ "follow professional​ football." Complete parts​ (a) through​ (c).   (a) What sample size should be obtained if he wants to be within 3 percentage points with 96​% confidence if he uses an estimate of 48​% obtained from a​ poll?   (b) What sample size should be obtained if he wants to be within 3 percentage points with 96% confidence if he does not use any prior estimates?   (c) Why are the results from parts (a) and (b) so close?

A programmer plans to develop a new software system. In planning for the operating system that he will​ use, he needs to estimate the percentage of computers that use a new operating system. How many computers must be surveyed in order to be 99​% confident that his estimate is in error by no more than five percentage points? Complete parts​ (a) through​ (c) below.

In politics, marketing, etc. we often want to estimate a percentage or proportion p. One calculation in statistical polling is the margin of error - the largest (reasonble) error that the poll could have. For example, a poll result of 72% with a margin of error of 4% indicates that p is most likely to be between 68% and 76% (72% minus 4% to 72% plus 4%).   In a (made-up) poll, the proportion of people who like dark chocolate more than milk chocolate was 37% with a margin of error of 2.5%. Describe the conclusion about p using an absolute value inequality.   . That lets you type in a vertical bar | to represent absolute values. Also, when you type in  <  and then =, the symbolic entry option will automatically convert that to  ≤ . In the same way, if you type in  >  and then =, the symbolic entry option will automatically convert that to  ≥.   Be sure to use decimal numbers in your answer (such as using 0.40 for 40%).

In 1949​,an organization surveyed 1100 adults and​ asked, "Are you a total abstainer​ from, or do you on occasion​ consume, alcoholic​ beverages?" Of the 1100adults​ surveyed, 374indicated that they were total abstainers. In a recent​ survey, the same question was asked of 1100 adults and 319indicated that they were total abstainers. Complete parts​ (a) and​ (b) below. ​(a) Determine the sample proportion for each sample.   The proportions of the adults who took the 1949 survey and the recentrecent survey who were total abstainers are nothing and nothing​, respectively. ​(Round to three decimal places as​ needed.)