9.1

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

26.

  1. 0.430
  2. np(1-p)=282.60 is greater than or equal to 10; and the sample is 5% less than the population
  3. Lower bound=0.402 and upper bound= 0.459
  4. We are 95% confident that the population proportion of workers and retirees in the United States 25 years of age and older who have less than 100,000 in savings is between 0.402 and 0.459

27.

  1. p hat= 0.519
  2. n (p hat) (1-p hat)= 250.39 >/= to 10, and the sample is less than 5% of the population.
  3. Lower bound: 0.488; Upper bound: 0.550
  4. Yes, it is possible that the population proportion is more than 60%, because it is 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. 0.75
  2. np(1-p)=192 is greater than or equal to 10; and the sample is less
  3. The lower bound is =0.715 and the upper bound is 0.785
  4. Yes, this is very likely because even at 99% confidence, we know that the lowest is already above 70%
  5. The lower bound is =0.285 and the upper bound is 0.215

29.

  1. p hat= 0.540
  2. n (p hat) (1-p hat)=434.20 >/= to 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 the level of confidence widens the interval.

9.2

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 \(\bar{x} = 18.4\), sample standard deviation is 4.5, sample size = 35, and your confidence level is 95% this is how you can have R calculate a Confidence Interval for you.

conf_int(xbar = 18.4, size = 35, conf = .95, s = 4.5)
## [1] 16.8542 19.9458

21.

  1. Lower bound: 16.85; Upper bound: 19.95
  2. Lower bound: 17.12; Upper bound: 19.68–> increasing the sample size decreases the margin of error
  3. Lower bound: 16.32; Upper bound: 20.48–> increasing the level of confidence increases the margin of error
  4. If n=15, the population must be normal

23.

  1. Flawed; this interpretatioin implies that the population mean varies rather than the interval
  2. Correct
  3. Flawed; this interpretation makes an implication about individuals rather than the mean
  4. Flawed; the interpretation should be about the mean number of hours worked by adult Americans, not about adults in Idaho.

25.

We are 90% confident that the mean drive through service time of Taco Bell restaurants is between 161.5 and 164.7 seconds.

27.

Increase the sample size, and decrease the level of confidence to narrow the confidence interval.

29.

  1. Since the distribution of blood alcohol concentrations is not normally distributed (highly skewed right), the sample must be large so that the distribution of the sample mean will be approximately normal.
  2. The sample size is less than 5% of the population
  3. Lower bound: 0.1647; Upper bound: 0.1693. We are 90% confident that the mean BAC in fatal crashes where the drives had a positive BAC is between 0.1647 and 0.1693 g/dL
  4. Yes, it is possible that the mean BAC is less than 0.08 g/dL because it is possible that the true mean is not captured in the confidence interval, but it is not likely.

31.

Lower bound: 12.05 books; Upper bound: 14.75 books. We can be 99% confident that the mean number of books read by Americans in the past year was between 12.05 and 14.75.

33.

Lower bound: 1.08 days; Upper bound: 8.12 days. We can be 95% confident that the mean incubation period of patients with SARS is between 1.08 and 8.12 days.

9.3

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 you sample standard deviation s = 2, the sample size = 35, and your confidence level is 95% this is how you can have R calculate a Confidence Interval for \(\sigma\).

conf_sig(s = 2, size = 35, conf = .95)
## [1] 1.617744 2.620404

5

x^2 0.95= 10.117, x^2 0.05= 30.144

7

x^2 0.99= 9.542, x^2 0.01= 40.289

9

  1. Lower bound: 7.94, Upper bound: 23.66
  2. Lower bound: 8.59, Upper bound: 20.63. The width of the interval decreases.
  3. Lower bound: 6.61, Upper bound: 31.36. The width of the interval increases.

11

Lower bound: 1.612, Upper bound: 4.278. We can be 95% confident that the population standard deviation of the prices of 4GB flash memory cards at online retailers is between 1.612 and 4.278 dollars.

13

Lower bound: 849.7, Upper bound: 1655.3. Wwe can be 90% confident that the population standard deviation of repair costs of a low impact bumper crash on a mini or micro car is between 849.7 and 1655.3 dollars.