Download Probability and Statistics Chapter 6: Normal Probability Distributions

Probability and Statistics
Chapter 6: Normal Probability Distributions
Answers to Even Problems
6-2 The Standard Normal Distribution
2. See figure 6-1 on page 244
4. The notation 𝑧𝛼 represents the z score that has an area of 𝛼 to its right.
6. 0.15
8. 0.6
10. 0.8508
12. 0.6063
16. 0.82
18. 0.4247
20. 0.9750
22. 0.0344
26. 0.1004
28. 0.2996
30. 0.6949
32. 0.9771
36. 0.5000
38. -1.645
40. -2.575, 2.575
42. 2.33
46. 95.44%
48. 99.98%
14.
24.
34.
44.
-0.51
0.7995
0.9999
1.88
6-3 Applications of Normal Distributions
2. a. 1 b. 100 c. 100 d. 225
4. No. Randomly generated digits have a uniform distribution, not a binomial distribution. The
probability of a digit less than 3 is 0.3.
6. 0.7257
8. 0.1571
10. 115
12. 120
14. 0.9772
16. 0.1596
18. 90
20. 131
22. a. 46.80% b. 98.81% c. 58.9 in. to 74.0 in.
24. a. 0.01%, practically no men can fit without bending b. 0.01%, practically no women can fit
without bending c. The door design is very inadequate, but the jet is small and only seats six people.
A much higher door would require such major changes in the design and cost of the jet, that the greater
height is not practical.
26. a. 23.4 in. If there us clearance for 95% of males, there will certainly be clearance for all women
in the bottom 5% b. Men: 95.99%. Women 99.99%. The table will fit almost everyone except about
4% of the men with the largest sitting knee heights.
28. a. 0.01%; yes b. 99.22º
30. 16.7 in.
32. a. 61.70% of legal quarters are rejected. That percentage is too high because most quarters will be
rejected. b. Accept quarters with weights between 5.55 g and 5.79 g.
6-4 Sampling Distributions and Estimates
2. a. Without replacement b. (1) When selecting a relatively small sample from a large population, it
makes no significant different whether we sample with replacement or without replacement. (2)
Sampling with replacement results in independent events that are unaffected by previous outcomes,
and independent events are easier to analyze and they result in simpler calculations and formulas.
4. No. The data set is only one sample, but the sampling distribution of the mean is the distribution of
the means from all samples, not the one sample mean obtained from this single sample.
6. a. Normal b. 4.5 c. 0.5
8. a. 2.160
b.
c. 1.571
Sample Standard Deviation s
Probability
0.000
3/9
0.707
2/9
2.828
2/9
3.536
2/9
d. No. The mean of the sampling distribution of the sample deviation is 1.571, and it is not equal to the
value of the population standard deviation (2.160), so the sample standard deviations do not target the
value of the population standard deviation.
10. a. 2/3 or 0.7
b.
c. 2/3 or 0.7
Sample Proportion Probability
0
1/9
0.5
4/9
1
4/9
d. Yes. The mean of the sampling distribution of the sample proportion of odd number is 2/3, and it is
equal to the value of the population proportion of odd numbers (2/3), so the sample proportions target
the value of the population proportion.
12. a. Same as Exercise 11 part (a) b. The median of the population is 52.5, but the mean of the
sample medians is 52.25, so those values are not equal. c. The sample medians do not target the
population median of 52.5, so sample medians do not make good estimators of population medians.
14. a.
s2
Probability
0
4/16
2
2/16
4.5
2/16
24.5
2/16
40.5
2/16
50
2/16
72
2/16
b. The population variance is 24.1875, and the mean of the sample variances is also 24.1875. Those
values are equal.
c. The sample variances do target the population variance, so sample variances do make good
estimators of the population variance.
16.
Proportion of Girls Probability
0
1/8
1/3
3/8
2/3
3/8
1
1/8
Yes. The proportion of girls in 3 births is 0.5, and the mean of the sample proportions is 0.5. The
result suggests that a sample proportion is an unbiased estimator of a population proportion.
18. a. The proportions of 0, 0.5, 1 have the corresponding probabilities of 9/25, 12/25, 4/25.
b. 0.04 c. Yes; yes
6-5 The Central Limit Theorem
2. No. Because the original population is normally distributed, the sample means will be normally
distributed for any sample size, not just n > 30.
4. Because the digits are equally likely, they have a uniform distribution. Because the sample means
are based on samples is size n = 3 drawn from a population that does not have a normal distribution,
we should not treat the sample means as having a normal distribution.
6. a. 0.1587 b. 0.3632
8. a. 0.8888 b. 0.9265 c. Because the original population has a normal distribution, the distribution of
sample means is normal for any sample size.
10. a. 0.2596 b. 0.6590
12. 0.0183. The elevator appears to be relatively safe because there is a very small chance (0.0183) of
overloading. Using an outdated mean that is too low has the effect of making the elevator appear to be
much safer than it really is.
14. a. 99.99% b. 0.9641. No, when considering the diameters of manholes, we should use a design
based on individual men, not samples of 36 men.
16. a. 0.5239 b. 0.8944 c. Instead of filling each bag with exactly 465 M&Ms, the company probably
fills the bags so that the weight is as stated. In an event, the company appears to be doing a good job
of filling the bags.
18. a. P1 = 50.5 beats per minute and P99 = 104.5 beats per minute b. 0.9988 c. Instead of the mean
pulse rate from patients in a day, the cutoff values should be based on individual patients, so it would
be better to use the pulse rates of 50.5 beats per minute and 104.5 beats per minute.
20. a. 0.9999 b. 0.7939. Because there is a high probability (0.7939) of overloading, the new ratings
do not appear to be safe when the boat is loaded with 14 male passengers.
22. There is a 0.9887 probability that the aircraft is overloaded. Because that probability is so high,
the pilot should take action, such as removing excess fuel and/or requiring that some passengers
disembark and take a later flight.
6-6 Assessing Normality
2. Either the points are not reasonably close to a straight line pattern or there is some systematics
pattern that is not a straight-line pattern.
4. Because the histogram is roughly bell shaped, conclude that the data are from a population having a
normal distribution.
6. Normal. The points are reasonably close to a straight-line pattern, and there is no other pattern that
is not a straight-line pattern.
8. Not normal. The points are not reasonably close to a straight-line pattern, and there appears to be a
pattern that is not a straight-line pattern.
6-7 Normal as Approximation to Binomial
2. The continuity correction is used to compensate for the fact that a continuous distribution (normal)
is used to approximate a discrete distribution (binomial). The discrete number of 13 is represented by
the interval from 12.5 to 13.5.
4. Yes. The circumstances correspond to 25 independent trials of a binomial experiment in which the
probability of success is 0.2. Also, with n = 25, p = 0.2, and q = 0.8, the requirements of np ≥ 5 and
nq ≥ 5 are both satisfied because 5 ≥ 5 and 20 ≥ 5 are both true.
6. Normal approximation should not be used.
8. 0.5793
10. 0.2743
12. 0.0738
14. a. 0.0012 b. 0.0060 c. The result from part (b) is useful. We want the probability of getting a
result that is at least as extreme as the one obtained. d. If the 33% rate is correct, there is a very small
chance (0.0060) of getting 172 or fewer calls overturned, so there is strong evidence against the 33%
rate.
16. a. 0.0020 b. Because the probability of getting 291 or more with the value of 25% is so small, the
result of 291 is unusually high. c. The results do suggest that the rate is greater than 25%.
18. 0.0001. It appears that many adult males say that they wash their hands in a public restroom when
they actually do not.
20. 0.2709. Media reports appear to be wrong.
22. Probability of three or fewer: 0.0107. Because that probability is so small, the evidence against
the rate of 20% is very strong. It appears that the rate of smoking among statistics students is lower
than the 20% rate for the general population.
24. Probability of 175 or more households with Internet access: 0.1736. If the Internet access rate is
67%, there is a relatively high probability (0.1732) of getting 175 or more households with Internet
access when 250 households are surveyed. It does not appear that the 67% rate is too low.