7.1

31.

shadenorm(mu = 62, sig = 18, below = 44, col = "blue", dens = 200)

Interpretation 1. There is a 15.87 percent proportion of the population that pay less than $44 a month.

Interpretation 2. There is a .1587 probability that a randomly selected individual from the population will pay less than $44 a month.

32.

shadenorm(mu = 14, sig = 2.5, above = 17, col = "blue", dens = 200)

Interpretation 1. There is an 11.51 percent proportion of the population that last more than 17 years.

Interpretation 2. There is an .1151 probability that a randomly selected individual from the population will last more than 17 years.

33.

shadenorm(mu = 3400, sig = 505, above = 4410, col = "blue", dens=200)

Interpretation 1. There is a 2.28 percent proportion of the population that has a birth weight greater than 4410 grams.

Interpretation 2. There is a .0228 probability that a randomly selected individual from the population will have a birth weight greater than 4410 grams.

34.

shadenorm(mu = 55.9, sig = 5.7, below = 46.5, col = "blue", dens=200)

Interpretation 1. There is a 4.96 percent proportion of the population that are less than 46.5 inches tall.

Interpretation 2. There is a .0496 probability that a randomly selected individual from the population will be less than 46.5 inches tall.

35.

Interpretation 1. There is a 19.08 percent proportion of the population that have gestation periods longer than 280 days.

Interpretation 2. There is a .1908 probability that a randomly selected individual from the population will have a gestation period longer than 280 days.

Interpretation 1. There is a 34.16 percent proportion of the population that have a gestation period between 230 and 260 days.

Interpretation 2. There is a .3416 probability that a randomly selected individual from the population will have a gestation period between 230 and 260 days.

36.

Interpretation 1. There is an 33.09 percent proportion of the population that will be greater than than 26 miles per gallon.

Interpretation 2. There is a .3309 probability that a randomly selected individual from the population will be greater than than 26 miles per gallon.

Interpretation 1. There is an 11.07 percent proportion of the population that will be greater between 18 and 21 mpg.

Interpretation 2. There is a .1107 probability that a randomly selected individual from the population will be greater between 18 and 21 mpg.

7.2

5.

  1. .0071
  2. .3336
  3. .9115
  4. .9998

7.

  1. .9987
  2. .9441
  3. .0375
  4. .0009

9.

  1. .9586
  2. .2088
  3. .8479

11.

  1. .0456
  2. .0646
  3. .5203

13. -1.28

15. .67

17. -2.58 and 2.58

33. 40.62

35. 56.16

37.

  1. .1587
  2. .1587
  3. .4772
  4. .0013 probability would be unusual.

39.

  1. .8658
  2. .0132
  3. .7019
  4. .1230
  5. 96th percentile
  6. 4th percentile

41.

  1. 40.13%
  2. 15.87%
  3. 75%
  4. .1894
  5. .0951
  6. yes, .0043 is unusual

43.

  1. .0764
  2. .0324
  3. 162
  4. 11804

45.

  1. .3228
  2. .4286
  3. Yes, because a team is equally likely to win or lose

47.

  1. 20 days
  2. 19-23 days

56. The SAT score is better with a probability of .8461 compared to that of the ACT score with a probability of .8315

8.1

## Here is the syntax you can use to check the probabilities you look up are correct.

## Say you want to know the Pr(X < 5) and X is Normal with a mean of 12 and standard deviation 4

pnorm(5, mean = 12, sd = 4 )
## [1] 0.04005916

15.

  1. Normal Distribution, mean=80 and spread=2
  2. 6.68%
  3. 1.79%
  4. .7969

17.

  1. The distribution of the population is normally distributed when the sampling distribution of the mean is also normally distributed with mean=64 and spread=4.91
  2. .7486
  3. .4052

19.

  1. .3520
  2. Normal with a mean=266 and spread=3.578
  3. .0465
  4. .0040
  5. I would conclude that it is unusual and the sample mean came from a population with a mean that had to be less than 266
  6. .9844

21.

  1. .3085
  2. .0418
  3. .0071
  4. Increased sample size effects the probability negatively causing it to decrease, because as n increases the std dev decreases.
  5. I would conclude that there isnt enough evidence that it was effective.
  6. 93.9

23.

  1. .5675
  2. .7291
  3. .8051
  4. .8531
  5. As the investment time horizon increases, the likelihood of earning a positive rate of return on stock increases.

Here is the syntax you can use to check your answers. (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. Normal Distribution, mean=.8 and spread=.0462
  2. .1922
  3. .0047

12.

  1. Normal Distribution, mean=.65 and spread=.0337
  2. .1867
  3. .0375

13.

  1. Normal Distribution, mean=.35 and spread=.01508
  2. .0040
  3. .0233

14.

  1. Normal, mean=.42 and spread=.012917
  2. .0102
  3. .0606

15.

  1. Normal, mean=.47 and spread=.0352
  2. .1977
  3. Yes, it would be unusual to have the probability be .0233.

16.

  1. Normal, mean=.82 and spread=.038418
  2. .2177
  3. Yes, it would be unusual to have the probabilitiy be .0344

17.

  1. Normal, mean=.39 and spread=.022
  2. .3228
  3. .3198
  4. No, it is not unusual to have the probability be .0838