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. n= 3611; x= 542; p = x/n = 0.150
  2. 3611(0.150)(1-0.150) = 541.65(0.85) = 460.40 >10 and sample size is < than 5% of the population
  3. Z0.05 = 1.645; LOWER BOUND:0.140; UPPER BOUND: 0.160
  4. We’re 90% confident the proportion of adult Americans 18 yrs of age or older have used smartphones to make a purchase is between 0.140 and 0.160

26.

  1. n=1153; x = 469; p = x/n = 0.430.
  2. 1153(0.430)(1-0.430) = 495.79(0.57) = 282.6
  3. Z0.025 = 1.96; lower bound: 0.40; Upper bound: 0.46
  4. We’re 90% confident that the proportion of workers and retirees in teh US 25 yrs. old or older who has less than 10,000 in savings account is between 0.40 and 0.46

27.

  1. n=1003; x = 521; p = x/n = 0.519
  2. 1003(0.519)(1-0.519) = 520.55(0.481) = 250.39; sample size is less than 5% of population
  3. Z0.025 = 1.96; LOWER BOUND 0.488; UPPER BOUND: 0.550
  4. Yes, it’s possible because it’s possible that the true proportion will not be captured in the confident interval; it’s not likely
  5. 1003-521 = 482; p= x/n = 0.480; Lower bound: 0.450; Upper Bound: 0.511

28.

  1. n=1024; x = 768; p=x/n = 0.75
  2. 1024(0.75)(1-0.75) = 768(0.25) = 192 > 10 and the sample size is less than 5% of population
  3. Z0.005 = 2.575; Lower Bound: 0.71; Upper Bound: 0.78
  4. Yes, it’s possible, but not likely because there’s a possibility the true proportion will not be captured in the confidence interval
  5. 1024-768 =256; p= 256/1024 = 0.25; Lower Bound: 0.214; Upper Bound: 0.286

29.

  1. n=1748; x = 944; p= x/n = 0.540
  2. 1748(0.540)(1-0.540) = 943.92(0.46) = 434.20 >10 and the sample size is less than 5% of the population
  3. Z0.05 = 1.645; Lower BOUND2; UPPER BOUND: 0.56 (d)Z0.005 = 2.575; Lower Bound: 0.509; Upper Bound: 0.571
  4. Increasing the level of considence increases the interval