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. .150
  2. 3611(.15)(1-.15) = 460.40 which is greater than 10. The sample size is less than 5% of the population
  3. (.1403, .1597)
  4. We are 90% confident the proportion is between .1403 and .1597

26.

  1. .43
  2. 1153(.43)(1-.43) = 282.60 which is greater than 10. The sample size is less than 5% of the population
  3. (.401, .459)
  4. We are 95% confident the proportion is between .401 and .459

27.

  1. .519
  2. 1003(.519)(1-.519) = 250.39 which is greater than 10. The sample size is less than 5% of the population
  3. (.488, .55)
  4. Yes technically it is possible but it’s not very likely
  5. (.45,512)

28.

  1. .75
  2. 1024(.75)(1-.75) = 192 which is greater than 10. The sample size is less than 5% of the population
  3. (.715, .785)
  4. Yes its possible but its not likely.
  5. (.215, .285).

29.

  1. .540
  2. 1748(.54)(1-.54) = 432.20 which is greater than 10. The sample size is less than 5% of the population
  3. (.520, .560)
  4. (. 509, .517)
  5. The width also increases