Here is the syntax you can use to check your answers for Section 8.1. (Forward and Backward)

Say you want to know the \(Pr(\bar{X} < 5)\) and \(\bar{X} \sim \mathcal{N}(6,1.5)\)

pnorm(5, mean = 6, sd = 1.5 )
## [1] 0.2524925

Here is the syntax you can use to check if a “Backward” calculation is correct.

Say you know the probability to the left of \(\bar{x}\) = .04 and you want to know what the appropriate \(\bar{x}\) is. You also know that \(\bar{X} \sim \mathcal{N}(6,1.5)\)

qnorm(.04, mean = 6, sd = 1.5)
## [1] 3.373971

15.

  1. xbar is normal with μ xbar = 80 and σ xbar = 2
  2. P(xbar>83)=0.0668
  3. P(xbar < 75.8)=0.0179
  4. P(78.3 < xbar < 85.1)=0.7969

17.

  1. the population and sampling distributions must be both normally distributed
  2. P(xbar < 67.3)=0.7486
  3. P(xbar > 65.2)=0.4052

19.

  1. P(x < 260)=0.3520
  2. the sampling distribution of xbar is normal when μ xbar = 266 and σ xbar = 3.578
  3. P(xbar < 260)=0.0465
  4. P(xbar < 260)=0.0040
  5. then it would probably come from a population whose mean gestation period is less than 266 days
  6. P(256 < xbar < 276)=0.9844

21.

  1. P(X>95)=0.3085
  2. P(xbar>95)=0.0418
  3. P(xbar>95)=0.0071
  4. increasing the sample size decreases both σ xbar and P(xbar > 95)
  5. that is not an unusual mean reading speed since it is within one standard deviation, so the program is not that much more effective than the previous one
  6. 93.9 words per minute

23.

  1. P(X>0)=0.5675
  2. P(xbar>0)=0.7291
  3. P(xbar>0)=0.8051
  4. P(xbar>0)=0.8531
  5. the likelihood of earning a positive return rate increases as the investment time horizon increases

Here is the syntax you can use to check your answers for Section 8.2. (Forward and Backward)

Say you want to know the \(Pr (\hat{P} < .35)\) and \(\hat{P} \sim \mathcal{N}(.4,.07)\)

pnorm(.35, mean = .4, sd = .07 )
## [1] 0.2375253

Here is the syntax you can use to check if a “Backward” calcuation is corect.

Say you know the probability to the left of \(\hat{p}\) = .04 and you want to know what the appropriate \(\hat{p}\) is. You also know that \(\hat{P} \sim \mathcal{N}(.4,.07)\)

qnorm(.05, mean = 12, sd = 4)
## [1] 5.420585

Section 8.2

11.

  1. distribution is normal with μ p-hat = 0.8 and σ p-hat = 0.046
  2. P(p-hat > 0.84)=0.1922
  3. P(p-hat < 0.68)=0.0047

13.

  1. distribution is normal with μ p-hat = 0.35 and σ p-hat = 0.015
  2. P(p-hat > 0.39)=0.0040
  3. P(p-hat < 0.32)=0.0233

15.

  1. distribution is normal with μ p-hat = 0.47 and σ p-hat = 0.035
  2. P(p-hat > 0.5)=0.1977
  3. P(p-hat < 0.4)=0.0239

17.

  1. distribution is normal with μ p-hat = 0.39 and σ p-hat = 0.022
  2. P(p-hat < 0.38)=0.3228
  3. P(0.40 < p-hat < 0.45)=0.3198
  4. P(p-hat > 0.42)=0.0838