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. (a) 0.15 (b) 460.4 ≥ 10 (c) (0.14, 0.16) (d) We are 90% confident that the population proportion of 18+ y/o adult males who have used a smartphone to make a purchase is between 0.14 and 0.16.

26. (a) 0.43 (b) 282.6 ≥ 10 (c) (0.4, 0.46) (d) We are 95% confident that the population proportion of 25+ y/o who have less $10,000 in savings is between 0.4 and 0.6

27. (a) 0.52 (b) 250.35 ≥ 10 (c) (0.49, 0.55); we are 95% confident that the population of adult Americans who believe televisions are a luxury they could do without is between 0.49 and 0.55. (d) It is possible that the calculated confidence interval is does not include the true population proportion, but it is not likely. (e) (0.45, 0.51)

28. (a) 0.75 (b) 192 ≥ 10 (c) (0.715, 0.785) (d) It is possible that the calculated confidence interval is does not include the true population proportion, but it is not likely. (e) (0.215, 0.285)

29. (a) 0.54 (b) 434.2 ≥ 10 (c) (0.52, 0.56) (d) (0.51, 0.57) (e) Increasing confidence level increases margin of error, so the width of the interval increases.

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. (16.85, 19.95)
  2. (17.12, 19.68); increasing sample size decreases margin of error.
  3. (16.32, 20.48); increasing confidence level increases margin of error.
  4. Since n=15 is a relatively small sample size, the population distribution must be normal to compute the confidence interval.

23.

  1. Incorrect - confidence level is not a probability because it is computed based on a sample statistic.
  2. Correct
  3. Incorrect - confidence level is not the population proportion.
  4. Incorrect - the statistic is about all adult Americans, not Americans in Idaho.

25.

We are 90% confident that the mean service time of drive-thru windows in fast-food restaurants is between 161.5s and 164.7s.

27.

To increase precision of the confidence interval, you can increase sample size and decrease level of confidence.

29.

  1. Since the data is skewed, having a large sample size means the distribution of the sample means will be approximately normal; then we can compute the confidence interval.
  2. The sample size, n=51, is less than 5% of the population size.
  3. (0.165, 0.169); we are 90% confident that the mean positive BAC is between 0.165g/dL and 0.169 g/dL.
  4. It is always possible since we do not know for fact that the population mean is within the computed confidence interval, but it is unlikely.

31.

(12.05, 14.75); we are 99% confident that the mean number of books Americans read in the past year is between 12.05 and 14.75.

33.

(1.08, 8.12); we are 95% confident that the mean incubation period of SARS is between 1.08 days 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

Left: 10.117 Right: 30.144

7

Left: 9.542 Right: 40.289

9

  1. (7.94, 23.67)
  2. (8.59, 20.63); increasing sample size decreases width of confidence interval.
  3. (6.61, 31.36); increasing confidence level increases width of confidence interval.

11

(2.60, 18.30); we are 95% confident that the population standard deviation price of a 4GB flash drive is between $2.60 and $18.30.

13

(849.7, 1655.3); we are 90% confident that the population standard deviation repair costs will be between $849.70 and $1,655.30.