You can use the following syntax to check your answers. Note the answers you get in R will not be EXACTLY the same you get by hand but they should be pretty close.

If your \(x = 542\) (ie the number of “successes” and your \(n = 3611\), this is how you can have R calculate a 90% Confidence Interval for you.

binom.test(x = 542, n = 3611,  conf.level = .90)[["conf.int"]]
## [1] 0.1403939 0.1602214
## attr(,"conf.level")
## [1] 0.9

25.

  1. 542/3611 = 0.15
  2. Yes (460 is greater than 10); sample size is no more than 5% of the population size.
  3. p^±0.0098= upper bound of 0.1598 and lower bound of 14.02
  4. I can be 90% confident that p is between 14.02 and 0.1598.

26.

  1. 0.43
  2. Yes, 282 is greater than 10; sample size is no more than 5% of the population size.
  3. p-hat +/- 0.029; upper bound of 0.46 and lower bound of 0.4
  4. I can be 95% confident that p is between 0.4 and 0.46.

27.

  1. 0.52
  2. Yes, 250 is greater than 10; sample size is no more than 5% of the population size.
  3. p-hat +/- 0.03; upper bound of 0.55 and lower bound of 0.49
  4. It is possible but not likely (would be in remaining 5% of occurances)
  5. Upper bound of 0.51 and lower bound of .45

28.

  1. 0.75
  2. Yes, 192 is greater than 10; sample size is no more than 5% of the population size.
  3. p-hat +/- 0.035; upper bound of 0.785 and lower bound of 0.715
  4. It is possible but extremely unlikely (only possible in 1% of occurances).
  5. Upper bound of 0.285 and lower bound of 0.215

29.

  1. 0.54
  2. Yes, 434 is greater than 10; sample size is no more than 5% of the population size.
  3. p-hat +/- 0.02; upper bound of 0.56 and lower bound of 0.52
  4. p-hat +/- 0.03; upper bound of 0.57 and lower bound of 0.51
  5. Increasing the level of confidence increases the width of the interval.