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. Point estimate = 0.1501
  2. Requirements are satisfied because sample is less than 5% of population and np(1-p)>10
  3. Critical value Za/2 = 1.645, upper bound = 0.1599, lower bound = 0.1403
  4. We are 90% confident the proportion of adult Americans who have used their smartphones to make a purchase is between 0.1403 and 0.1599

26.

  1. Point estimate = 0.4302
  2. Requirements are satisfied because sample is less than 5% of population and np(1-p)>10
  3. Critical value Za/2 = 1.96, upper bound = 0.4588, lower bound = 0.4016
  4. We are 95% confident the proportion of workers and retirees in the US 25 years and older who have less than $10,000 in savings is between 0.4016 and 0.4588

27.

  1. Point estimate = 0.5194
  2. Requirements are satisfied because sample is less than 5% of population and np(1-p)>10
  3. Critical value Za/2 = 1.96, upper bound = 0.5503, lower bound = 0.4885. We are 95% confident the proportion of adult Americans who think TV is a luxury they could live without is between 0.4885 and 0.5503.
  4. Yes it’s possible that the population proportion is more than 60% because it’s possible the true proportion is not within the confidence interval, however it’s not likely.
  5. Lower bound = 0.450, upper bound = 0.512

28.

  1. Point estimate = 0.75
  2. Requirements are satisfied because sample is less than 5% of population and np(1-p)>10
  3. Critical value Za/2 = 2.575, upper bound = 0.785, lower bound = 0.715
  4. Yes it’s possible that the population proportion is below 70% because it’s possible the true proportion is not within the confidence interval, however it’s not likely.
  5. Lower bound = 0.215, upper bound = 0.285

29.

  1. Point estimate = 0.540
  2. equirements are satisfied because sample is less than 5% of population and np(1-p)>10
  3. Critical value Za/2 = 1.645, upper bound = 0.560, lower bound = 0.520
  4. Critical value Za/2 = 2.575, upper bound = 0.571, lower bound = 0.509
  5. The higher the level of confidence, the wider the interval.