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. 15%
  2. np(1-p) = 460
  3. .140 and .160
  4. We are 90% confident that the proportion of Americans 18+ who have bought something with their smartphone is between .140 and .160.

26.

  1. 43%
  2. np(1-p) = 283
  3. .401 and .459
  4. We are 95% confident that the proportion of 25+ workers/retirees that have less than 10k in savings is between .401 and .459.

27.

  1. 52%
  2. np(1-p) = 250
  3. .489 and .551
  4. Possible but super unlikely.
  5. .449 and .511

28.

  1. 75%
  2. np(p-1) = 192
  3. .715 and .785
  4. Possible, not super likely.
  5. .215 and .285

29.

  1. 54%
  2. np(p-1) = 434
  3. .520 and .560
  4. .509 and .571
  5. It widens it.