7.1

31.

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

Interpretation 1. The probability that a randomly selected phone plan is charged less than $44 is 0.1587.

Interpretation 2. 15.87% of phone plans are charged less than $44.

32.

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

Interpretation 1. The probability that a randomly selected refrigerator will last more than 17 years is 0.1151.

Interpretation 2. 11.51% of refrigerators will last more than 17 years.

33.

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

Interpretation 1. The probability that a randomly selected baby is greater than 4,410 grams is 0.0228.

Interpretation 2. 2.28% of full-term babies will be more than 4,410 grams.

34.

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

Interpretation 1. The probability that a randomly selected 10 year old boy is shorter than 46.5 inches is 0.0496.

Interpretation 2. 4.96% of 10 year old boys will be shorter than 46.5 inches.

35.

Interpretation 1. The probability that a randomly selected woman will have a pregnancy last longer than 280 days is 0.1908.

Interpretation 2. 19.09% of pregnant women will have a pregnancy longer than 280 days.

Interpretation 1. The probability that a randomly selected woman will have a pregnancy last between 230 and 260 days is 0.3416.

Interpretation 2. 34.16% of pregnant women will have a pregnancy between 230 and 260 days.

36.

Interpretation 1. The probability that a randomly selected data poin has a value greater than 26 is 0.3309.

Interpretation 2. 33.09% of data are greater than 26.

Interpretation 1. The probability that a randomly selected data poin has a value between 18 and 21 is 0.1107.

Interpretation 2. 11.07% of data are between 18 and 21.

7.2

5.

  1. P(z < -2.45) = 0.0071
  2. P(z < -0.43) = 0.3336
  3. P(z < 1.35) = 0.9115
  4. P(z < 3.49) = 0.9998

7.

  1. P(z > -3.01) = 0.9987
  2. P(z > -1.59) = 0.9441
  3. P(z > 1.78) = 0.0375
  4. P(z > 3.11) = 0.0009

9.

  1. P(-2.04 < z < 2.04) = 0.9586
  2. P(-0.55 < z < 0) = 0.2088
  3. P(-1.04 < z < 2.76) = 0.8489

11.

  1. P(z < -2) or P(z > 2) = 0.0456
  2. P(z < -1.56) or P(z > 2.56) = 0.0646
  3. P(z < -0.24) or P(z > 1.20) = 0.5203

13. z = -1.28

15. z = 0.67

17. -2.58 < z < 2.58

33. x = 40.62

35. x = 56.16

37.

  1. P(z < -1) = 0.1587
  2. P(z > 1) = 0.1587
  3. P(-2 < z < 0) = 0.4772
  4. P(z < -3) = 0.0013; very unusual - only 0.13% of eggs hatch in fewer than 18 days.

39.

  1. P(-2.22 ≤ z ≤ 1.17) = 0.8651
  2. P(z < -2.22) = 0.0139
  3. P(z > -0.53) = 0.7019
  4. P(z < -1.16) = 0.1230
  5. P(z < 1.81) = 0.9649; 96th Percentile
  6. P(z < -1.79) = 0.0367; 4th Percentile

41.

  1. P(z > 0.25) = 0.4013
  2. P(z < -1) = 0.1587
  3. P(-1.63 < z < 0.88) = 0.7590
  4. P(z > 0.88) = 0.1894
  5. P(z ≤ -1.31) = 0.0951
  6. P(z ≤ -2.63) = 0.0043; very unusual

43.

  1. P(z < -1.42) = 0.0778
  2. P(z < -2.14) or P(z > 2.14) = 0.0324
  3. 1,620 rods
  4. 11,843 rods

45.

  1. 32.28%
  2. 42.86%
  3. Teams are just as likely to win the bet as they are to lose.

47.

  1. 20 days
  2. Between 19 and 23 days

56. The SAT score has a z-score of 1.02, and the ACT score has a z-score of 0.96. You do better on the SAT because you are farther away from the mean on the positive side.

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, center: 80, spread: 2
  2. P(z > 1.50) = 0.0668
  3. P(z ≤ -2.10) = 0.0179
  4. P(-0.85 < z < 2.55) = 0.7969

17.

  1. normal, center: 64, spread: 4.91
  2. P(z < 0.67) = 0.7486
  3. P(z ≥ 0.24) = 0.4052

19.

  1. P(z < -0.38) = 0.3520
  2. Sample Deviation: 3.58
  3. P(z ≤ -1.68) = 0.0465
  4. P(z ≤ -2.65) = 0.0040
  5. Since the result is very unusual given the population mean gestation period and its standard deviation, so the sample likely consists mostly of pregnancies with a gestation period less than the population mean.
  6. 98.44%

21.

  1. P(z > 0.5) = 0.3085
  2. P(z > 1.73) = 0.0418
  3. P(z > 2.45) = 0.0071
  4. An increasing sample size decreases probability.
  5. P(z > 1.25) = 0.1056
  6. x = 93.69wpm

23.

  1. P(z > -0.18) = 0.5714
  2. P(z > -0.60) = 0.7257
  3. P(z > -0.86) = 0.8051
  4. P(z > -1.05) = 0.8531
  5. The probability of earning a positive return on stocks increases as time increases, but at a decreasing rate.

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” calculation is correct.

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, center: 0.8, spread: 0.0462
  2. P(z ≥ 0.87) = 0.1922
  3. P(z ≤ -0.12) = 0.4522

12.

  1. normal, center: 0.65, spread: 0.0337
  2. P(z ≥ 0.89) = 0.1867
  3. P(z ≤ -1.78) = 0.0375

13.

  1. normal, center: 0.35, spread 0.0151
  2. P(z ≥ 2.67) 0.0038
  3. P(z ≤ -2) = 0.0228

14.

  1. normal, center: 0.42, spread: 0.0129
  2. P(z ≥ 2.31) = 0.0104
  3. P(z ≤ -1.54) = 0.0618

15.

  1. normal, center: 0.47, spread: 0.0353
  2. P(z > 2.69) =
  3. P(z ≤ -1.98) =

16.

  1. normal, center: 0.82, spread: 0.0384
  2. P(z > 0.78) = 0.2177
  3. P(z ≤ -1.82) = 0.0344

17.

  1. normal, center: 0.39, spread: 0.0218
  2. P(z < -0.46) = 0.3228
  3. P(0.46 < z <2.75) = 0.6742
  4. P(z ≥ 1.38) = 0.0838