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. p̂ = 0.150.
  2. np̂(1-p̂) = 460.40 > 10, and the sample is less than 5% of the population.
  3. Lower bound: 0.140. Upper bound: 0.160.
  4. 90% confident that the proportion of adult Americans 18 years and older who have used their smartphones to make a purchase between 0.140 and 0.160.

26.

  1. p̂ = 0.430.
  2. np̂(1-p̂) = 282.38 > 10, and the sample is less than 5% of the population.
  3. Lower bound: 0.401. Upper bound: 0.459
  4. 95% confident that the proportion of adult Americans 18 years and older who have used their smartphones to make a purchase between 0.401 and 0.459.

27.

  1. p̂ = 0.519.
  2. np̂(1-p̂) = 250.39 > 10, and the sample is less than 5% of the population.
  3. Lower boumd: 0.488. Upper bound: 0.550
  4. Yes; There is a possibility that the population proportion is more than 60% becuase its possible that the true proportion is not captured in the confidence interval. It is not likely.
  5. Lower bound: 0.450. Upper bound: 0.512

28.

  1. p̂ = 0.75.
  2. np̂(1-p̂) = 192 > 10, and the sample is less than 5%of the population.
  3. Lower bound: 0.715. Upper bound: 0.785
  4. Yes; it is possible that the population proportion is more than 70% because it is possible that the true proportion is not captured in the confidence interval. It is not likely.
  5. Lower bound: 0.215. Upper bound: 0.285

29.

  1. p̂ = 0.540
  2. np̂(1-p̂) = 432.20 > 10, and the sample is less than 5%of the population.
  3. Lower bound: 0.520. Upper bound: 0.560
  4. Lower bound: 0.509. Upper bound: 0.571
  5. Increasing confidence widens the interval.