Reading 9.2:

21

  1. (16.85,19.95)
n = 35
xbar = 18.4
s = 4.5
t_critical = qt(.975, n - 1)
lower = xbar - t_critical*s/sqrt(n)
upper = xbar + t_critical*s/sqrt(n)

(answer = c(lower,upper))
## [1] 16.8542 19.9458
  1. Lower bound is 17.12 and upper bound is 19.68. If you increased the sample, you will decrease the margin of error in the situation.
n = 50
xbar = 18.4
s = 4.5
t_critical = qt(.975, n - 1)
lower = xbar - t_critical*s/sqrt(n)
upper = xbar + t_critical*s/sqrt(n)

(answer = c(lower,upper))
## [1] 17.12111 19.67889
  1. Lower bound is 16.32 and upper bound is 20.48. If you increased the confidence level, you would increase the margin of error.
n = 35
xbar = 18.4
s = 4.5
t_critical = qt(.995, n - 1)
lower = xbar - t_critical*s/sqrt(n)
upper = xbar + t_critical*s/sqrt(n)

(answer = c(lower,upper))
## [1] 16.32468 20.47532
  1. The population will be normal if n is 15.

23

  1. A confidence interval is NOT a probability interval.
  2. Correct
  3. A confidence interval is NOT a census.
  4. We are making a statement about the population parameter of the whole country, NOT just Idaho.

25

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

27

  1. They should increase the sample size.
  2. They should decrease the level of confidence.

29

  1. The sample size needs to be large enough for the interval to be correct.
  2. The sample size is less than 5% of the population.
  3. Lower bound is 0.1647 and upper bound is 0.1693. Having a 90% confidence interval implies that we are 90% confident that the average driver’s blood alcohol concentration in fatal car crashes is between 0.1647 g/dL and 0.1693 g/dL.
n = 51
xbar = .167
s = .01
t_critical = qt(.95, n - 1)
lower = xbar - t_critical*s/sqrt(n)
upper = xbar + t_critical*s/sqrt(n)

(answer = c(lower,upper))
## [1] 0.1646533 0.1693467
  1. It is possible for the mean blood alcohol concentration to be less than 0.08 d/L because there is always a chance that the actual average is not measured by the confidence interval.

31

Lower bound is 12.05 and upper bound is 14.75. These numbers indicate that we are 99% confident that the average amount of books Americans have read this year is between 12.05 books and 14.75 books.

n = 1006
xbar = 13.4
s = 16.6
t_critical = qt(.995, n - 1)
lower = xbar - t_critical*s/sqrt(n)
upper = xbar + t_critical*s/sqrt(n)

(answer = c(lower,upper))
## [1] 12.04932 14.75068

33

Lower bound is 1.08 and upper bound is 8.12. These numbers indicate that we are 95% confident that the average incubation period for patients with severe acute respiratory syndrome is between 1.08 days and 8.12 days.

n = 81
xbar = 4.6
s = 15.9
t_critical = qt(.975, n - 1)
lower = xbar - t_critical*s/sqrt(n)
upper = xbar + t_critical*s/sqrt(n)

(answer = c(lower,upper))
## [1] 1.084221 8.115779

Reading 9.3:

5

The chi square (x^2) of 0.95 is 10.117 and the chi square (x^2) of 0.05 is 30.144.

n = 20
(small_value = qchisq(.05, n-1))
## [1] 10.11701
(large_value = qchisq(.95, n-1))
## [1] 30.14353

7

The chi square (x^2) of 0.99 is 9.542 and the chi square (x^2) of 0.01 is 40.289.

n = 23
(small_value = qchisq(.01, n-1))
## [1] 9.542492
(large_value = qchisq(.99, n-1))
## [1] 40.28936

9

  1. Lower bound is 7.94 and upper bound is 23.66.
n = 20
ssquared = 12.6
small_value = qchisq(.05, n-1)
large_value = qchisq(.95, n-1)

lower = (n-1)*ssquared/large_value
upper = (n-1)*ssquared/small_value

(answer = c(lower,upper))
## [1]  7.942004 23.663111
  1. Lower bound is 8.59 and upper bound is 20.63. Here, we see that the width of the interval decreases.
n = 30
ssquared = 12.6
small_value = qchisq(.05, n-1)
large_value = qchisq(.95, n-1)

lower = (n-1)*ssquared/large_value
upper = (n-1)*ssquared/small_value

(answer = c(lower,upper))
## [1]  8.586138 20.634315
  1. Lower bound is 6.61 and upper bound is 31.36. Here, we see that the width of the interval increases.
n = 20
ssquared = 12.6
small_value = qchisq(.01, n-1)
large_value = qchisq(.99, n-1)

lower = (n-1)*ssquared/large_value
upper = (n-1)*ssquared/small_value

(answer = c(lower,upper))
## [1]  6.614928 31.364926

11

Lower bound is 1.612 and upper bound is 4.278. According to these numbers, we can be 95% confident that the population standard deviation for a 4 GB memory card for online sellers is between 1.612 dollars and 4.278 dollars.

n = 10
ssquared = (2.343)^2
small_value = qchisq(.025, n-1)
large_value = qchisq(.975, n-1)

lower = (n-1)*ssquared/large_value
upper = (n-1)*ssquared/small_value

(answer = sqrt(c(lower,upper)))
## [1] 1.611598 4.277405

13

Lower bound is 849.7 and upper bound is 1655.3. According to these numbers, we can be 90% confident that the population standard deviation for the cost of repairing a bumper on a car is between 849.7 dollars and 1655.3 dollars.

n = 14
ssquared = (1114.412)^2
small_value = qchisq(.05, n-1)
large_value = qchisq(.95, n-1)

lower = (n-1)*ssquared/large_value
upper = (n-1)*ssquared/small_value

(answer = sqrt(c(lower,upper)))
## [1]  849.6926 1655.3548