Math 227 / Fall 2014
Instructor: David Soto
Name______________________________________________
Chapter 6
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Using the following uniform density curve, answer the question.
1) What is the probability that the random variable has a value greater than 5?
A) 0.375
B) 0.500
C) 0.325
1)
D) 0.250
Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread
evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of
pounds lost.
2) Between 8 pounds and 11 pounds
2)
A) 1
B) 1
C) 2
D) 1
2
3
3
4
Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.
3)
3)
A) 0.1788
B) 0.6424
C) 0.3576
D) 0.8212
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
4) Shaded area is 0.9599.
4)
A) 1.75
C) 1.82
B) -1.38
1
D) 1.03
If z is a standard normal variable, find the probability.
5) The probability that z lies between -1.10 and -0.36
A) 0.2239
B) -0.2237
5)
C) 0.4951
D) 0.2237
The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at
the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some
give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive
numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the
frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and
tested. Find the temperature reading corresponding to the given information.
6) A quality control analyst wants to examine thermometers that give readings in the bottom 4%.
6)
Find the reading that separates the bottom 4% from the others.
A) -1.48°
B) -1.63°
C) -1.75°
D) -1.89°
Provide an appropriate response.
7) Find the area of the shaded region. The graph depicts IQ scores of adults, and those scores are
normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).
A) 0.7938
B) 0.7303
C) 0.7745
D) 0.7619
Solve the problem. Round to the nearest tenth unless indicated otherwise.
8) The amount of rainfall in January in a certain city is normally distributed with a mean of 4.5
inches and a standard deviation of 0.3 inches. Find the value of the quartile Q1 .
A) 4.7
B) 1.1
C) 4.3
Assume that X has a normal distribution, and find the indicated probability.
9) The mean is µ = 22.0 and the standard deviation is = 2.4.
Find the probability that X is between 19.7 and 25.3.
A) 1.0847
B) 0.3370
C) 0.4107
2
7)
8)
D) 4.4
9)
D) 0.7477
Find the indicated probability.
10) The lengths of human pregnancies are normally distributed with a mean of 268 days and a
standard deviation of 15 days. What is the probability that a pregnancy lasts at least 300 days?
A) 0.0166
B) 0.9834
C) 0.0179
D) 0.4834
Solve the problem.
11) The amount of snowfall falling in a certain mountain range is normally distributed with a mean of
70 inches, and a standard deviation of 10 inches. What is the probability that the mean annual
snowfall during 25 randomly picked years will exceed 72.8 inches?
A) 0.0026
B) 0.4192
C) 0.5808
D) 0.0808
10)
11)
12) Suppose that replacement times for washing machines are normally distributed with a mean of
12)
13) A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet
13)
9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected
washing machines will have a mean replacement time less than 9.1 years.
A) 0.0643
B) 0.0714
C) 0.4286
D) 0.4357
Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics
are randomly selected, find the probability that their mean rebuild time exceeds 9.1 hours.
A) 0.1285
B) 0.0046
C) 0.1046
D) 0.0069
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
14) Three randomly selected households are surveyed as a pilot project for a larger survey to
be conducted later. The numbers of people in the households are 2, 4, and 8. Consider the
values of 2, 4, and 8 to be a population. Assume that samples of size n = 2 are randomly
selected with replacement from the population of 2, 4, and 8. The nine different samples
are as follows: (2, 2), (2, 4), (2, 8), (4, 2), (4, 4), (4, 8), (8, 2), (8, 4), and (8, 8). (i) Find the
variance of each of the nine samples, then summarize the sampling distribution of the
variances in the format of a table representing the probability distribution. (ii) Compare
the population variance to the mean of the sample variances. (iii) Do the sample variances
target the value of the population variance? In general, do variances make good
estimators of population variances? Why or why not?
3
14)
15) After constructing a new manufacturing machine, 5 prototype integrated circuit chips are
15)
16) Three randomly selected households are surveyed as a pilot project for a larger survey to
16)
produced and it is found that 1 is defective (D) and 4 are acceptable (A). Assume that two
of the chips are randomly selected with replacement from this population. (i) After
identifying the 25 different possible samples, find the proportion of circuits that are
acceptable in each of them, then use a table to describe the sampling distribution of the
proportions of circuits that are acceptable. (ii) Find the mean of the sampling distribution.
(iii) Is the mean of the sampling distribution equal to the population proportion of circuits
that are acceptable? (iv) Does the mean of the sampling distribution of proportions
always equal the population proportion?
be conducted later. The numbers of people in the households are 2, 4, and 10. Consider
the values of 2, 4, and 10 to be a population. Assume that samples of size n = 2 are
randomly selected with replacement from the population of 2, 4, and 10. The nine
different samples are as follows: (2, 2), (2, 4), (2, 10), (4, 2), (4, 4), (4, 10), (10, 2), (10, 4), and
(10, 10). (i) Find the mean of each of the nine samples, then summarize the sampling
distribution of the means in the format of a table representing the probability distribution.
(ii) Compare the population mean to the mean of the sample means. (iii) Do the sample
means target the value of the population mean? In general, do means make good
estimators of population means? Why or why not?
4
Answer Key
Testname: CHAPTER 6 PRACTICE
1)
A
2) A
3) B
4) A
5) D
6) C
7) A
8) C
9) D
10) A
11) D
12) A
13) D
14) (i)
s
Probability
0
3/9
2.000
2/9
8.000
2/9
18.000
2/9
(ii) The population variance is 6.222, and the mean of the sample variances is also 6.222. The values are equal.
(iii) The sample variances target the population variance, so sample variances make good estimators of population
variances.
15) (i)
proportion of
acceptable
circuits
probability
0
1/25
0.5
8/25
1
16/25
(ii) The mean of the sampling distribution is 4/5, or 0.8.
(iii) Yes. The proportion of circuits that are acceptable in the population is 4/5, and the sample proportions also have a
mean of 4/5.
(iv) Yes. The mean of the sampling distribution of proportions always equals the population proportion.
16) (i)
x
2
4
10
3
6
7
Probability
1/9
1/9
1/9
2/9
2/9
2/9
(ii) The mean of the population is 5.33 and the mean of the sample means is also 5.33.
(iii) The sample means target the population mean. Sample means make good estimators of population means because
they target the value of the population mean instead of systematically underestimating or overestimating it.
5