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. Point estimate = .1501
  2. np(1-p) = (3611)(.15)(1-.15)=460.4 The sample size is greather than 10 and is less then 5% of the population.
  3. Lower bound: .141, Upper bound: .159
  4. The confidence that adults have bought something with their smart phone is between .14 and .16.

26.

  1. P = 496/1153 = .43
  2. (1153)(.45)(1-.45) = 282.6 The sample size is greater than 10 and is less than 5% of the population
  3. Lower Bound: .42, Upper bound: .48
  4. The confidence that adults have less than 10,000$ in savings is between .42 and .48

27.

  1. P = 521/1003 = .52
  2. (1003)(.52)(1-.52) = 250.37 The sample size is greater than 10 and is less than 5% of the population
  3. Lower bound: .49, Upper bound: .55
  4. It is possible that more than .6 of adults believe that television is a luxury they can do without, but it isn’t likely because it is outside of the range of the confidence interval.
  5. Lower bound: .4497, Upper bound: .5115

28.

  1. p = 768/1024 = .75
  2. np(1-p) = (1024)(.75)(1-.75) = 192 The sample size is greater than 10 and more than 5% of the population.
  3. Lower bound: .7151, Upper bound: .7849
  4. It is possible, but not likely because it doesn’t fall in the range of the confidence interval.
  5. Lower bound: .2151, Upper bound: .2849

29.

  1. p = 944/1748 = .54
  2. np(1-p) = (1748)(.54)(1-.54) = 434.2 The sample size is greater than 10 and more than 5% of the population.
  3. Lower bound: .5204, Upper bound: .5596
  4. Lower bound: .5093, Upper bound: .5707
  5. Increasing the confidence level increases the width of the confidence 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. I am unsure where type 1/type 2 questions that professor blatt wrote for section 9.2 are supposed to be.

  1. Lower bound: 16.854 Upper bound: 19.945
  2. Lower bound: 16.14 Upper bound: 20.659 Increasing the sample size increases the margin of error
  3. Lower bound: 16.325 Upper bound: 20.475 Increasing the confidence level increases the margin of error
  4. The population must be normal.

23.

  1. Is inccorect because we aren’t looking for the probability that are dults worked between those hours, but the confidence level that adults worked between those hours.
  2. is correct because it states that it’s a confidence level, not a probability, but unlike (D), specifies that the data is looking at the general american adult population and not some random population.
  3. Same as (A), (C) is inccorect because we aren’t looking for the probability that are dults worked between those hours, but he confidence level that adults worked between those hours.
  4. is reasonable because like (B), (D) specifies that we are looking at a confidence level, not a probability. However, it is inccorect because it does not specify what population the data was collected from.

25.

There is a 90% confidence that the mean drive through service times at Taco Bell is between 161.5 seconds and 164.7 seconds.

27.

To narrow the confidence interval, one should increase the sample size and/or decrease the level of confidence.

29.

  1. The sample size needs dto be large to make the sample normal.
  2. The sample is .00204 (less than 5%)
  3. Lower bound: .165, Upper bound: .169
  4. Yes, it is possible that the drivers are less than the legal intoxication level, but it’s not likely because .08 g/dl is less then .165g/dl, which is the lower bound for the confidence interval.

31.

Lower bound: 13.318, Upper bound: 13.481 There is a 99% confidance that americans read between 12.05 and 14.75 bookd in the past year.

33.

Lower bound: 1.084, Upper bound: 8.116 There is a 95% confidence that the mean incubation period for a SARS patient is between 1.084 and 8.116 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

30.144, 10.117

7

40.289, 9.542

9

  1. sigma squared = (Lower bound: 7.94, Upper bound: 23.66) sigma = (lower bound 2.82, upper bound: 4.86)
  2. sigma squared = (Lower bound: 8.59, Upper bound: 20.63) sigma = (lower bound 2.93, upper bound: 4.53) increasing the sample size decreases the width of the confidence interval.
  3. sigma squared = (Lower bound: 6.61, Upper bound: 31.36) sigma = (lower bound 2.57, upper bound: 5.6) increasing the level of confidence increases the confidence interval.

11

sigma squared = (Lower bound: 2.597, Upper bound: 18.297) sigma = (lower bound 1.612, upper bound: 4.277) There is a 95% confidence that the population standard deviation of the cost of 4GB memory cards is between 1.612$ and 4.278$.

13

sigma squared = (Lower bound: 721978.5072, Upper bound: 2740136.351) sigma = (lower bound 849.693, upper bound: 165.336) There is a 95% confidence that the repair cost of low impact buper crash on a mini car or macro car is between 849.693$ and 165.336$.