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. p hat = 417/2309 =0.181

  2. np(1-p) = 2306 x 0.181 (1-0.181); = 341.84 greater than or equal to 10 The sample is less than 5% of the population

  3. Lower bound: 0.181 - 1.645 x the square root of 0.18(1-0.181)/2306 = 0.168

Upper bound: 0.181 + 1.645 x the square room of 0.181(1-0.181)/2306 = 0.194

  1. From this we are 90% confident that the population proportion of adult Americans 18 years and older who have donated blood in the past two years is between 0.168 and 0.194.

26.

  1. p hat = 496/1153 = 0.430

  2. np(1-p)=1153 x 0.430(1-0.430) = 282.60 is greater than or equal to 10; The sample is less than 5% of the population

  3. Lower bound: 0.430-1.96 x the square root of 0.430(1-0.430)/1153=0.401

Upper bound: 0.430+1.96 x the square root of 0.430(1-0.430)/1153 = 0.459

  1. We are 95% confident that the population porportion of workers and retirees int he US 25 years of age and older who have less that $10,000 in savings is between 0.401 and 0.459.

27.

  1. p=521/1003 = 0.519

  2. np(1-p)=1003x0.519(1-0.519) = 250.39 is greater than or equal to 10; The sample is less than 5% of the population

  3. Lower bound: 0.519-1.96 x the square room of 0.519(1-0.519)/1003=0.488

Upper bound: 0.519+1.96 x the square root of 0.519(1-0.519)/1003 = 0.550

  1. It is possible that the population proportion is more than 60% because the true proportion is not captured in the confidence interval. 0.6 is outside the confidence interval.

  2. Lower bound: 1-0.550=0.450 Upper bound: 1-0.488=0.512

28.

  1. p=768/1024 = 0.75

  2. np(1-p)=1024x0.75(1-0.75) = 192

  3. Lower bound: 0.75-2.575 x the square root of 0.75(1-0.75)/1024=0.715

    Upper bound: 0.75+2.575 x the square root of 0.75(1-0.75)/1024=0.785

  4. It could be that the population proportion is less than 70% because it is possible that the true proportion is not captured in the confidence interval. However, it is not likely because 0.7 is outside of the confidence interval.

  5. Lower bound: 1-0.785=0.215 Upper bound: 1-0.715=0.285

29.

  1. p=26/234 = 0.111

    Lower bound: 0.111-1.96 x the square root of 0.111(1-0.111)/234=0.071

    Upper bound: 0.111+1.96 x the square root of 0.111(1-0.111)/234 = 0.151

  2. Lower bound: 0.111-2.575 x the square root of 0.111(1-0.111)/234=0.058

    Upper bound: 0.111+2.575 x the square root of 0.111(1-0.111)/234=0.164

  3. Increasing the confidence level increases the margin of error.