9.2

21

a.Lower bound : 16.85, upper bound: 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(16.85,19.95))
## [1] 16.85 19.95
  1. Lower bound : 17.12, upper bound : 19.68, increasing the sample size would decrease the margin of error.
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(17.12,19.68))
## [1] 17.12 19.68

c.Lower bound: 16.32, upper bound : 20.48

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(16.32,20.48))
## [1] 16.32 20.48

It appears the margin of error increases.

  1. The population must be normal.

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

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

27

1 Increase the sample size 2 decreases the level of confidence, which causes to narrow the confidence interval.

29

  1. Since the distribution blood alcohol concentrations is not normally distributed, the sample should be large. The distribution of the sample mean will be normal(approximately)

  2. Sample size is less than 5% of population

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(0.1647,0.1693))
## [1] 0.1647 0.1693
  1. Yes.

31

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(12.05,14.75))
## [1] 12.05 14.75

99% confident.

33

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(1.08,8.12))
## [1] 1.08 8.12

95% confident

9.3

5

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

7

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

9

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(7.94,23.66))
## [1]  7.94 23.66
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(8.59,20.63))
## [1]  8.59 20.63

The width of the interval decreases.

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(6.61,31.36))
## [1]  6.61 31.36

The width of interval increases.

11

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(1.612,4.278)))
## [1] 1.269646 2.068333

95% confident that it is between 1.612 and 4.278.

13

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(849.7,1655.3)))
## [1] 29.14961 40.68538

90% confident that it is between 849.7 and 1655.3