Solutions

mean = 4 hours, standard deviation = 1.2 hours, sample size = 16

a. Check student's solution.
b. 3.5, 4.25, 0.2441

The fact that the two distributions are different accounts for the different probabilities.

0.3345

7833.92

0.0089

7326.49

77.45%

0.4207

3,888.5

0.8186

5

0.9772

The sample size, n, gets larger.

49

26.00

0.1587

1000

1. U(24, 26), 25, 0.5774
2. N(25, 0.0577)
3. 0.0416

0.0003

25.07

1. N(2500, 5.7735)
2. 0

2507.40

1. 10
2. $\frac{1}{10}$

N $\left(\text{10, }\frac{10}{8}\right)$

0.7799

1.69

0.0072

391.54

405.51

1. Χ = amount of change students carry
2. Χ ~ Exp(1/0.88) or approximately Χ ~ Exp(1.1364)
3. $\overline{X}$ = average amount of change carried by a sample of 25 students.
4. $\overline{X}$ ~ N(0.88, 0.176)
5. 0.0819
6. 0.4276
7. The probability in part (e) represents the probability of an individual value. In part (f), the probability describes the mean of a sample of 25. Part (f) relies on the central limit theorem, so the distributions are different. Part (e) is exponential and part (f) is normal.

1. length of time for an individual to complete IRS form 1040, in hours
2. mean length of time for a sample of 36 taxpayers to complete IRS form 1040, in hours
3. N$\left(\text{10}\text{.53, }\frac{1}{3}\right)$
4. Yes, I would be surprised, because the probability is almost 0.
5. No, I would not be totally surprised because the probability is 0.2312.

1. the length of a song, in minutes, in the collection
2. U(2, 3.5)
3. the average length, in minutes, of the songs from a sample of five albums from the collection
4. N(2.75, 0.0220)
5. 2.74 minutes
6. 0.03 minutes

1. True. The mean of a sampling distribution of the means is approximately the mean of the data distribution.
2. True. According to the central limit theorem, the larger the sample, the closer the sampling distribution of the means becomes normal.
3. The standard deviation of the sampling distribution of the means will decrease, making it approximately the same as the standard deviation of X as the sample size increases.

1. X = the yearly income of someone in a Third World country
2. the average salary from samples of 1,000 residents of a Third World country
3. $\overline{X}$ ∼ N$\left(\text{2,000, }\frac{\text{8,000}}{\sqrt{\text{1,000}}}\right)$
4. Very wide differences in data values can have averages smaller than standard deviations.
5. The distribution of the sample mean will have higher probabilities closer to the population mean.
   P(2,000 < $\overline{X}$ < 2,100) = 0.1537
   P(2,100 < $\overline{X}$ < 2,200) = 0.1317

b

1. the total length of time for nine criminal trials
2. N(189, 21)
3. 0.0432
4. 162.09; 90 percent of the total nine trials of this type will last 162 days or more.

1. X = the salary of one elementary school teacher in the district
2. X ~ N(44000, 6500)
3. ΣX ~ sum of the salaries of 10 elementary school teachers in the sample
4. ΣX ~ N(44,000, 20,554.80)
5. 0.9742
6. $52,330.09
7. 466,342.04
8. Sampling 70 teachers instead of 10 would cause the distribution to be more spread out. It would be a more symmetrical normal curve.
9. If every teacher received a $3,000 raise, the distribution of X would shift to the right by $3,000. In other words, it would have a mean of $47,000.

1. X = the closing stock prices for U.S. semiconductor manufacturers
2. i. $20.71, ii. $17.31, iii. 35
4. exponential distribution, Χ ~ Exp$\left(\frac{1}{20.71}\right)$
5. Answers will vary.
6. i. $20.71, ii. $11.14
7. Answers will vary.
8. Answers will vary.
9. Answers will vary.
10. N$\left(\text{20}\text{.71, }\frac{17.31}{\sqrt{5}}\right)$

b

b

a

1. 0
2. 0.1123
3. 0.0162
4. 0.0003
5. 0.0268

1. Check student’s solution.
2. $\overline{X}$ ~ N$\left(\text{60, }\frac{9}{\sqrt{25}}\right)$
3. 0.5000
4. 59.06
5. 0.8536
6. 0.1333
7. N(1500, 45)
8. 1530.35
9. 0.6877

1. $52,330
2. $46,634

• We have μ = 17, σ = 0.8, $\overline{x}$ = 16.7, and n = 30. To calculate the probability, we use normalcdf(lower, upper, μ, $\frac{\sigma }{\sqrt{n}}$) = normalcdf$\left(E–\text{99,16}\text{.7,17,}\frac{0.\text{8}}{\sqrt{\text{30}}}\right)$ = 0.0200.
• If the process is working properly, then the probability that a sample of 30 batteries would have at most 16.7 life span hours is only 2%. Therefore, the class was justified to question the claim.

1. For the sample, we have n = 100, $\overline{x}$ = 0.862, and s = 0.05.
2. $\Sigma \overline{x}$ = 85.65, Σs = 5.18
3. normalcdf(396.9,E99,(465)(0.8565),(0.05)($\sqrt{465}$)) ≈ 1
4. Because the probability of a sample of size of 465 having at least a mean sum of 396.9 is appproximately 1, we can conclude that the company is correctly labeling their candy packages.

Use normalcdf$$\left(E–\text{99,1}\text{.1,1,}\frac{1}{\sqrt{\text{70}}}\right)$$ = 0.7986. This means that there is an 80 percent chance that the service time will be less than 1.1 hours. It may be wise to schedule more time because there is an associated 20 percent chance that the maintenance time will be greater than 1.1 hours.

Because we have normalcdf$\left(5.\text{111,5}\text{.291,5}\text{.201,}\frac{0.\text{065}}{\sqrt{\text{280}}}\right)$ ≈ 1, we can conclude that practically all the coins are within the limits; therefore, there should be no rejected coins out of a well-selected sample size of 280.