Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1)
A quantitative data set has size 60. At least how many observations lie within 2 standard deviations
to either side of the mean?
1)
A)
At least 14
B)
At least 60
C)
At least 46
D)
At least 57
Use the empirical rule to solve the problem.
2)
A data set has mean 14 and standard deviation 2. Approximately 95% of the observations lie
between ____ and ____ .
2)
A)
10, 20
B)
10, 18
C)
12, 16
D)
8, 20
3)
A data set has 50 observations and has mean 55 and standard deviation 10. Approximately how
many observations lie between 25 and 85?
3)
A)
50
B)
48
C)
34
D)
35
Solve the problem.
4)
A quantitative data set of size 100 has mean 20 and standard deviation 3. At least how many
observations lie between 11 and 29?
4)
A)
At least 100
B)
At least 89
C)
At least 95
D)
At least 75
Use the empirical rule to solve the problem.
5)
A data set has size 70. Approximately how many observations lie within 2 standard deviations to
either side of the mean?
5)
A)
4
B)
48
C)
67
D)
70
Solve the problem.
6)
A quantitative data set has mean 23 and standard deviation 2. At least what percentage of the
observations lie between 19 and 27?
6)
A)
75%
B)
68%
C)
25%
D)
95%
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
7)
Does the mode of a data set always lie near the center? Explain your answer and give an
example of a data set to illustrate your answer.
7)
8)
A teacher records the test scores for the 30 students in her class. Corresponding to each
score, she calculates a z–score. If you were the parents of a girl in the class, would you be
more interested in knowing your daughter’s test score or z–score? Explain your reasoning.
8)
9)
In Mary’s first math test she scored 87%. The mean and standard deviation for the class
were 71% and 18% respectively. In her second math test, Mary scored 66%. The mean and
standard deviation for the class were 53% and 14% respectively. In which test did Mary do
better relative to the rest of the class? Explain your reasoning. (Hint: find the z–scores
corresponding to her two test scores.)
9)
10)
Roughly speaking, the standard deviation indicates how far, on average, the observations
are from the mean. Do you think that for the data set below the standard deviation will
give a good indication of the typical deviation from the mean?
2, 3, 4, 4, 5, 5, 6, 6, 100
What drawback of the standard deviation is illustrated by this example?
10)
11)
We want to compare two different groups of students, students taking Composition 1 in a
traditional lecture format and students taking Composition 1 in a distance learning format.
We know that the mean score on the research paper is 85 for both groups. What additional
information would be provided by knowing the standard deviation?
11)
12)
Discuss the differences between the distributions represented by the two boxplots below.
Assume the two boxplots have the same scale. Explain your reasoning.
12)
13)
The table below provides a frequency distribution for the winner of the Davis Cup during
the period 1977–1994.
Winner of
Davis Cup Frequency
United States 6
Germany 3
Czechoslovakia 1
Australia 3
France 1
Sweden 4
Which measure of center, the mean, the median, or the mode is most appropriate here?
Why?
13)
14)
The data set below consists of the scores of 15 students on a quiz. For this data set, which
measure of variation do you think is more appropriate, the range or the standard
deviation? Explain your thinking.
90 90 91 91 89
90 89 91 91 90
60 90 89 90 91
14)
15)
Explain how two data sets could have equal means and modes but still differ greatly. Give
an example with two data sets to illustrate.
15)
16)
Explain how to find the adjacent values when constructing a modified boxplot. When are
the adjacent values equal to the minimum and maximum observations?
16)
17)
Without calculating the standard deviation, compare the standard deviation for the
following data sets. (Note: All data sets have a mean of 30.) Which do you expect to have
the largest standard deviation and which do you expect to have the smallest standard
deviation? Explain your answers in terms of the formula
s =
(x –x)2
n – 1 .
Data set 1: 30, 30, 30, 30, 30, 30, 30, 30, 30, 30
Data set 2: 20, 25, 25, 30, 30, 30, 30, 35, 35, 40
Date set 3: 20, 20, 20, 25, 25, 35, 35, 40, 40, 40
17)
18)
In the Florida lottery, the numbers (between 1 and 49) are generated randomly with the
expectation that each number has an equal chance of winning. Draw a boxplot which
should illustrate the data set of all numbers picked for the lottery during the past year.
18)
19)
The median of a data set is always/sometimes/never (select one) one of the data points in a
set of data. Explain your answer with brief examples.
19)
20)
A teacher records the test scores for the 40 students in her class. Corresponding to each
score, she calculates a z–score. Julia’s z–score is –2.8. Should she be concerned? Explain
your reasoning.
20)
21)
When finding the 5–number summary, it becomes easy to identify the outliers of a data set.
Under what conditions should outliers be deleted, and when should they be further
investigated?
21)
22)
The range and standard deviation of the data set below are 35 and 12.47 respectively.
5, 24, 25, 26, 40
If the 26 is replaced with 39, how will this affect the range? How will this affect the
standard deviation. Use your answers to explain why the standard deviation is preferable
to the range as a measure of variation.
22)
23)
For a particular population, the _____________ mean is a constant and the ____________
mean is a variable.
23)
24)
A census bureau collects information about the household income (total income of
everyone living in one residence) of people in a certain country.
(i) Identify the variable and population under consideration.
(ii) By consulting the most recent census data, it was found that the mean household
income of all residents in the country is $35,352. Decide whether this descriptive measure
is a parameter or a statistic and use the correct notation to express the result.
24)
25)
Suppose that a state introduces a state income tax which will be at a flat rate of 3%. The
state legislature wishes to estimate how much money they will receive in taxes, and to do
this they need to know the average income of residents of the state. Which information
would be most useful, the mean income, the median income, or the mode of the incomes?
Why?
25)
26)
Explain what each symbol represents: (i) , (ii) n, (iii) x
26)
27)
Heights of adult women are known to have a bell–shaped distribution. Draw a boxplot to
illustrate the results.
27)
28)
Which boxplot shape (uniform, bell–shaped, or skewed) best matches the boxplot for the
first 100 digits of ? (Below is the frequency table for the first 100 digits of .)
x
f
0 1 2 3 4 5 6 7 8 9
8 8 12 11 10 8 9 8 12 14
28)
29)
Fill in the blank. If both the sample standard deviation, s, and the population standard
deviation, , are computed for the same data set, they will tend to be closer together if the
data set is ___________.
29)
30)
A population consists of 100 professional gymnasts and 100 professional basketball
players. For this group, the average height is 70 inches. However, most of the gymnasts are
between 57 and 61 inches tall while most of the basketball players are between 78 and 82
inches tall. For this group, observations far from the mean are more common than
observations close to the mean. Describe what a boxplot for the heights of this group
would look like. Discuss, in particular, the lengths of the whiskers relative to the width of
the box and explain your reasoning.
30)
31)
Dave is a college student contemplating a possible career option. One factor that will
influence his decision is the amount of money he is likely to make. He decides to look up
the average starting salary of graduates in that profession. Which information would be
most useful to him, the mean starting salary, the median starting salary, or the mode of the
starting salaries? Why?
31)
32)
Do you think it is possible to find two data sets such that the first data set has a smaller
range but a larger standard deviation than the second set? If so, give an example of two
such data sets. If it is not possible, explain why not.
32)
33)
A company advertises an average of 42,000 miles for one of its new tires. In the
manufacturing process there is some variation around that average. Would the company
want a process that provides a large or a small variance? Justify your answer.
33)
34)
A group of medical researchers is interested in knowing the mean cholesterol level for all
men in the U.S. aged between 70 and 80. They pick a sample of 5,000 men and measure
their cholesterol levels. They then calculate the mean and standard deviation of these
cholesterol levels. Do the mean and standard deviation obtained in this way represent
parameters or statistics? Why? What symbols could you use to denote the mean and
standard deviation of the 5,000 cholesterol levels?
34)
35)
Draw one boxplot to illustrate bell–shaped data, another for uniform data, and a third for
skewed data.
35)
36)
The two most frequently used measures of central tendency are the mean and the median.
Compare these two measures for the following characteristics: Takes every score into
account? Affected by extreme scores? Advantages.
36)
37)
Explain in your own words the difference between a parameter and a statistic. Give
examples of two statistics and two parameters.
37)
38)
Explain the difference between the interquartile range and the range. Which is more
sensitive to extreme values? Explain your thinking.
38)
39)
A machine fills bottles with juice. The average amount filled in the bottles is 16 ounces;
however this amount varies slightly from bottle to bottle. The manufacturer is interested in
knowing how much the amount of juice varies from bottle to bottle. In this context, what is
the population of interest and what does represent?
39)
40)
Can the sample variance ever be a negative number? If so, for what types of data? If not,
why not? Can the sample variance ever be zero? If so, for what types of data? If not, why
not? Explain your reasoning.
40)
41)
A machine fills bottles with juice. The average amount filled in the bottles is 16 ounces;
however this amount varies slightly from bottle to bottle. The manufacturer is interested in
knowing how much the amount of juice varies from bottle to bottle. How could the
manufacturer obtain an estimate of the population standard deviation ?
41)
42)
Describe any similarities or differences in the two distributions represented by the
following boxplots. Assume the two boxplots have the same scale. Explain your reasoning.
42)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the range for the given data set.
43)
The manager of an electrical supply store measured the diameters of the rolls of wire in the
inventory. The diameters of the rolls (in m) are listed below.
0.545 0.663 0.166
0.417 0.296 0.12
43)
A)
0.12 m
B)
0.545 m
C)
0.543 m
D)
0.13 m
44)
Fred, a local mechanic, gathered the following data regarding the price, in dollars, of an oil and
filter change at twelve competing service stations.
32.95 24.95 26.95 28.95
18.95 28.95 30.95 22.95
24.95 26.95 29.95 28.95
44)
A)
$12
B)
$8
C)
$10
D)
$14
Solve the problem. If necessary, round your answer to one more decimal place than that used for the observations.
45)
Let x1=5, x2=17, x3=7, x4=7, and x5=4. Find n.
45)
A)
40
B)
4
C)
5
D)
8.0
46)
The students in Hugh Logan‘s math class took the Scholastic Aptitude Test. Their math scores are
shown below. Find xi.
474 669 526 624
343 373 651 573
390 407 482
46)
A)
5512
B)
5402
C)
5622
D)
11
Identify potential outliers, if any, for the given data.
47)
The National Education Association collects data on the number of years of teaching experience of
high–school teachers. A sample taken this year of 19 high–school teachers yielded the following
data on number of years of teaching experience.
16 24 1 32 15
6 18 8 20 14
17 19 16 10 21
26 14 38 18
47)
A)
32, 38
B)
1, 32, 38
C)
1, 38
D)
None
Find the median for the given sample data.
48)
15, 15, 27, 29, 44, 47
48)
A)
29.5
B)
27
C)
29
D)
28
Determine the quartile or interquartile range as specified.
49)
Determine the interquartile range.
1, 3, 5, 7, 10, 11, 1, 3, 5, 7, 10, 11
49)
A)
7.5
B)
3
C)
6.5
D)
7
Construct and interpret a boxplot or a modified boxplot as specified.
50)
The ages of the 21 members of a track and field team are listed below. Construct a modified boxplot
for the data.
15 18 18 19 22 23 24
24 24 24 25 26 26 27
28 28 30 32 33 40 42
50)
A)
The data is roughly symmetrical, possibly bell–shaped, with no outliers.
B)
The data is roughly symmetrical with two potential outliers.
C)
The data is roughly symmetrical with one potential outlier.
D)
The data is roughly symmetrical with three potential outliers.
Find the mode(s) for the given sample data.
51)
The weights (in ounces) of 14 different apples are shown below.
4.1 5.0 5.9 6.0 6.2 4.1 5.0
6.7 4.7 6.0 4.1 6.7 6.0 5.8
51)
A)
4.1 oz
B)
4.1 oz, 6.0 oz
C)
5.05 oz
D)
6.0 oz
Provide an appropriate response.
52)
Find the z–score corresponding to the given value and use the z–score to determine whether the
value is unusual. Consider a score to be unusual if it is at least three standard deviations above or
below the mean. Round the z–score to one decimal place, if necessary.
A time for the 100 meter sprint of 12.6 seconds at a school where the mean time for the 100 meter
sprint is 17.6 seconds and the standard deviation is 2.1 seconds.
52)
A)
2.4; not unusual
B)
–5.0; unusual
C)
–2.4; not unusual
D)
–2.4; unusual
Find the range for the given data set.
53)
The owner of a small manufacturing plant employs six people. As part of their personnel file, she
asked each one to record to the nearest one–tenth of a mile the distance they travel one way from
home to work. The six distances are listed below.
2.4 5.4 1.1 4.5 6.6 3.9
53)
A)
1.5 mi
B)
1.1 mi
C)
5.5 mi
D)
5.4 mi
Find the median for the given sample data.
54)
The distances traveled (in miles) to 7 different swim meets are given below:
14, 22, 35, 53, 65, 75, 82
54)
A)
49 mi
B)
53 mi
C)
35 mi
D)
65 mi
Construct and interpret a boxplot or a modified boxplot as specified.
55)
The test scores of 40 students are listed below. Construct a boxplot for the data.
25 35 43 44 47 48 54 55 56 57
59 62 63 65 66 68 69 69 71 71
73 73 74 76 77 77 78 79 80 81
81 82 83 85 89 92 93 94 97 98
55)
A)
The data is slightly left–skewed.
B)
The data is slightly left–skewed.
C)
The data is left–skewed.
D)
The data is fairly symmetrical.
Find the mode(s) for the given sample data.
56)
–20, –34, –46, –34, –49, –34, –49
56)
A)
–49
B)
–34
C)
–38
D)
–46
Find the mean for the given sample data. Unless otherwise specified, round your answer to one more decimal place than
that used for the observations.
57)
13, 15, 11, 13, 10
57)
A)
11
B)
12.4
C)
15.5
D)
13
Determine the quartile or interquartile range as specified.
58)
The weekly salaries (in dollars) of sixteen government workers are listed below. Find the first
quartile, Q1.
690 591 813 652
728 562 489 635
529 670 685 464
554 787 496 826
58)
A)
$489
B)
$529
C)
$535.25
D)
$541.50
Find the mean for the given sample data. Unless otherwise specified, round your answer to one more decimal place than
that used for the observations.
59)
The students in Hugh Logan‘s math class took the Scholastic Aptitude Test. Their math scores are
shown below. Find the mean score.
580 533 348 340 499
342 350 575 470 482
59)
A)
476
B)
451.9
C)
443
D)
461.1
Determine the quartile or interquartile range as specified.
60)
The weights (in pounds) of 17 randomly selected adults are given below. Find the interquartile
range.
144 165 187 143 119 132
127 156 179 159 180 202
114 146 151 168 173
60)
A)
37 lb
B)
38 lb
C)
30 lb
D)
37.5 lb
Find the range for the given data set.
61)
27,37,16,43,58
61)
A)
16
B)
42
C)
58
D)
10
Use the empirical rule to solve the problem.
62)
Consider the following sample of exam scores, arranged in increasing order:
22 35 44 55 67 70
78 81 81 82 84 88
88 89 90 90 91 92
92 93 93 94 94 95
95 96 96 97 99 100
Note: The sample mean and sample standard deviation of these weights are, respectively, 82.4 and
19.5.
(i) Use the Empirical rule to estimate the percentages of the observations that lie within 3 standard
deviation(s) to either side of the mean.
(ii) Use the data to obtain the exact percentages of the observations that lie within 3 standard
deviation(s) to either side of the mean.
62)
A)
(i) Assuming that the scores have a roughly bell–shaped distribution, approximately 99.7% of
the observations should lie within 19.5 points of the mean 82.4 or within the interval from 23.9
to 140.9;
(ii) 29 of the 30 observations (96.7%) lie between 23.9 and 140.9.
B)
(i) Assuming that the scores have a roughly bell–shaped distribution, approximately 99.7% of
the observations should lie within 19.5 points of the mean 82.4 or within the interval from 23.4
to 141.4;
(ii) 29 of the 30 observations (96.7%) lie between 23.4 and 141.4.
C)
(i) Assuming that the scores have a roughly bell–shaped distribution, approximately 99.7% of
the observations should lie within 19.5 points of the mean 82.4 or within the interval from 22.7
to 142.7;
(ii) 30 of the 30 observations 100% lie between 22.7 to 142.7.
D)
(i) Assuming that the scores have a roughly bell–shaped distribution, approximately 99.7% of
the observations should lie within 19.5 points of the mean 82.4 or within the interval from 23.9
to 140.9;
(ii) 27 of the 30 observations 90% lie between 23.9 and 140.9.
Solve the problem.
63)
Here are boxplots of the points scored during the first 10 games of the basketball season for both
Caroline and Alexandra. Summarize the similarities and differences in their performance so far.
63)
A)
Both girls have a median score of about 18 points per game. Alexandra is much more
consistent, because her IQR is about 15 points, while Caroline’s is over 3. In other words,
Alexandra has less variation in her scores than does Caroline. The distribution of scores for
both women is symmetric.
B)
Both girls have a median score of about 18 points per game. Caroline is much more consistent,
because her IQR is about 6 points, while Alexandra’s is over 20. In other words, Alexandra
has more variation in her scores than does Caroline. The distribution of scores for Caroline is
right skewed, while the distribution of scores for Alexandra is bell–shaped.
C)
The girls have a different average score per game. Caroline is much more consistent, because
her IQR is about 4 points, while Alexandra’s is over 15. In other words, Alexandra has more
variation in her scores than does Caroline. The distribution of scores for Caroline is
symmetric, while the distribution of scores for Alexandra is left–skewed.
D)
Both girls have a median score of about 18 points per game. Caroline is much more consistent,
because her IQR is about 4 points, while Alexandra’s is over 15. In other words, Alexandra
has more variation in her scores than does Caroline. The distribution of scores for both
women is symmetric.
Provide an appropriate response.
64)
A company’s raw–data sample of weekly salaries (in dollars) is shown below.
210 330 510 330 730 510
330 510 210 330 1130 210
A frequency distribution of this data set, based on single–value grouping, is presented below, with
a third column showing the xf–values– – the class midpoint (which here is the same as the class)
times the class frequency.
Salary
x
Frequency
f
Salary * Frequency
xf
210
330
510
730
1130
3
4
3
1
1
630
1320
1530
730
1130
64)
17
(i) Use the raw data to obtain the sample standard deviation of the ungrouped data. Round your
answer to two decimal places.
(ii) Use the grouped–data formula to obtain the sample standard deviation of the grouped data in
the frequency distribution. Round your answer to two decimal places.
(iii) Compare your answers in parts (i) and (ii).
A)
(i) The sample standard deviation of the ungrouped data is 266.58;
(ii) The sample standard deviation of the grouped data is 283.5;
(iii) The results in parts (i) and (ii) are different. This discrepancy occurs because in the
grouped data formulas, every actual data value in a given class is replaced by the class
midpoint even though most values in the class are not equal to the midpoint.
B)
(i) The sample standard deviation of the ungrouped data is 283.5;
(ii) The sample standard deviation of the grouped data is 296.11;
(iii) The results in parts (i) and (ii) are different. This discrepancy occurs because in the
grouped data formulas, every actual data value in a given class is replaced by the class
midpoint even though most values in the class are not equal to the midpoint.
C)
(i) The sample standard deviation of the ungrouped data is 266.58;
(ii) The sample standard deviation of the grouped data is 266.58;
(iii) The results in parts (i) and (ii) are the same. The grouped data formula will always
provide the actual standard deviation when the data are grouped in classes each based on a
single value because the class midpoint is the same as each observation in each class.
D)
(i) The sample standard deviation of the ungrouped data is 283.5;
(ii) The sample standard deviation of the grouped data is 283.5;
(iii) The results in parts (i) and (ii) are the same. The grouped data formula will always
provide the actual standard deviation when the data are grouped in classes each based on a
single value because the class midpoint is the same as each observation in each class.
Solve the problem.
65)
The mean of a set of data is 116.53 and its standard deviation is 116.22. Find the z–score for a value
of 395.29. Round your final answer to two decimal places.
65)
A)
2.64
B)
2.40
C)
2.70
D)
2.16
Determine the quartile or interquartile range as specified.
66)
The weekly salaries (in dollars) of sixteen government workers are listed below. Find the
interquartile range.
787 627 820 668
475 618 541 645
565 693 875 504
587 460 558 490
66)
A)
$158.00
B)
$173.50
C)
$297
D)
$164
Find the mean for the given sample data. Unless otherwise specified, round your answer to one more decimal place than
that used for the observations.
67)
Frank’s Furniture employees earned $364.73, $191.73, $167.32, $315.63, $454.14, and $168.46 last
week. Find the mean wage of the employees. Round your answer to the nearest cent.
67)
A)
$277.00
B)
$332.40
C)
$320.40
D)
$415.50
D)
Solve the problem.
68)
Scores on a test have a mean of 70 and a standard deviation of 8. Michelle has a score of 78.
Convert Michelle’s score to a z–score.
68)
A)
8
B)
1
C)
–8
D)
–1
D)
69)
The mean of a set of data is –2.65 and its standard deviation is 3.89. Find the z–score for a value of
3.52.
Round your final answer to two decimal places.
69)
A)
1.59
B)
1.89
C)
1.75
D)
1.43
D)
D)
Find the range for the given data set.
70)
Jeanne is currently taking college economics. The instructor often gives quizzes. On the past five
quizzes, Jeanne got the following scores.
817 114 11
70)
A)
3
B)
1
C)
17
D)
16
Find the mean for the given sample data. Unless otherwise specified, round your answer to one more decimal place than
that used for the observations.
71)
17, 8, 22, 17
71)
A)
24
B)
14.5
C)
16
D)
16.5
Provide an appropriate response.
72)
Find the z–score corresponding to the given value and use the z–score to determine whether the
value is unusual. Consider a score to be unusual if it is at least three standard deviations above or
below the mean. Round the z–score to one decimal place, if necessary.
The mean height of a basketball team is 6 feet with a standard deviation of 0.2 feet. The team’s
center is 6.7 feet tall.
72)
A)
3.5, unusual
B)
2.98, not unusual
C)
3.85, unusual
D)
3, not unusual
Find the mode(s) for the given sample data.
73)
The blood types for 30 people who agreed to participate in a medical study were as follows.
O A A O A AB O B A O
A O A B O O O AB A A
A B O A A O O B O O
Find the mode of the blood types.
73)
A)
A
B)
13
C)
O
D)
O, A