Download Ch. 5-6 Review Exercises 1. Prenatal care. Results of a 1996

Ch. 5-6 Review Exercises
1. Prenatal care. Results of a 1996 American Medical
Association report about the infant mortality rate for
twins carried for the full term of a normal pregnancy
are shown below, broken down by the level of
prenatal care the mother had received.
Full-Term
Infant Mortality Rate
Pregnancies, Level of
among Twins (deaths
Prenatal Care
per thousand live births)
Intensive
5.4
Adequate
3.9
Inadequate
6.1
Overall
5.1
a) Is the overall rate the average of the other three
rates? Should it be? Explain.
b) Do these results indicate that adequate prenatal care
is important for pregnant women? Explain.
c) Do these results suggest that a woman pregnant
with twins should be wary of seeking too much
medical care? Explain.
2. Singers. The boxplots shown display the heights (in
inches) of 130 members of a choir.
It appears that the median height for sopranos is
missing, but actually the median and the upper
quartile are equal. How could that happen?
3. Beanstalks. Beanstalk Clubs are social clubs for very
tall people. To join, a man must be over 6’2” tall, and
a woman must be over 5’10”. The National Health
Survey suggests that height of adults may be normal
distributed with mean heights of 69.1’ for men and
64’ for women. The respective standard deviations
are 2.8” and 2.5”.
a) You are probably not surprised to learn that men
are generally taller than women, but what does the
greater standard deviation for men’s height indicate?
b) Who are more likely to qualify for Beanstalk
membership, men or women?
4. Bread. Clarksburg Bakery is trying to predict how many
loaves to bake. In the last 100 days, they have sold
between 95 and 140 loaves per day. Here is a
histogram of the number of loaves they sold for the
last 100 days.
a) Describe the distribution.
b) Which should be larger, the mean number of sales
or the median? Explain.
c) Here are the summary statistics for Clarksburg
Bakery's bread sales. Use these statistics and the
histogram above to create a boxplot. You may
approximate the values of any outliers.
Summary of Sales
Median
100
Min
95
Max
140
25th %tile
97
75th %tile 105.5
5. Acid Rain. Based on long-term investigation,
researchers have suggested that the acidity (pH) of
rainfall in the Shenandoah Mountains can be
described by the Normal model N(4.9, 0.6).
a) Draw and carefully label the model.
b) What percent of storms produce rain with pH over
6?
c) What percent of storms produce rain with pH under
4?
d) The lower the pH, the more acidic the rain. What is
the pH level for the most acidic 20% of all storms?
e) What is the pH level for the least acidic 5% of all
storms?
f) What is the IQR for pH of rainfall?
6. Fraud detection. A credit card bank is investigating the
incidence of fraudulent card use. The bank suspects
that the type of product bought may provide clues to
the fraud. To examine this situation, the bank looks at
the Standard Industrial Code (SIC) of the business
related to the transaction. This is a code that was used
by the U.S. Census Bureau and Statistics Canada to
identify the type of business of every registered
business in North America. For example, 1011
designates Meat and Meat Products (except Poultry),
1012 is Poultry Products, 1021 is Fish Products, 1031
is Canned and Preserved Fruits and Vegetables, and
1032 is Frozen Fruits and Vegetables.
A company intern produces the following histogram
of the SIC codes for 1536 transactions:
He also reports that the mean SIC is 5823.13 with a
standard deviation of 488.17.
a) Comment on any problems you see with the use of
the mean and standard deviation as summary
statistics.
b) How well do you think the Normal model will
work on these data? Explain.
7. Hard water II. The data set from England and Wales
also notes for each town whether it was south or north
of Derby. Here are some summary statistics and a
comparative boxplot for the two regions.
Summary of
Mortality
Group Count Mean Median StdDev
North 34 1531.59 1631 138.470
South 27 1388.85 1369 151.114
a) What is the overall mean mortality rate for the two
regions?
b) Do you see evidence of a difference in mortality
rates? Explain.
8. Seasons. Average daily temperatures in January and
July for 60 large U.S. cities are graphed in the
histograms below.
a) What aspect of these
histograms makes it
difficult to compare the
distributions?
b) What differences do you
see between the
distributions of January
and July average
temperatures?
c) Differences in
temperatures (JulyJanuary) for each of the
cities are displayed in the
boxplot above- Write a
few sentences describing
what you see.
9. Liberty's nose. Is the Statue of Liberty's nose too long?
Her nose measures 4'6", but she is a large statue after
all. Her arm is 42 feet long. That means her arm is
42/4.5 = 9.3 times as long as her nose. Is that a
reasonable ratio? Shown in the table are arm and nose
lengths of 18 girls in a Statistics class, and the- ratio
of arm to nose length for each.
Arm
Nose
Arm/Nose
(cm)
(cm
Ratio
)
73.8
5
14.8
74
4.5
16.4
69.5
4.5
15.4
62.5
4.7
13.3
68.6
4.4
15.6
64.5
4.8
13.4
68.2
4.8
14.2
63.5
4.4
14.4
63.5
5.4
11.8
67
4.6
14.6
67.4
4.4
15.3
70.7
4.3
16.4
69.4
4.1
16.9
71.7
4.5
15.9
69
4.4
15.7
69.8
4.5
15.5
71
4.8
14.8
71.3
4.7
15.2
a) Make an appropriate plot and describe the
distribution of the ratios.
b) Summarize the ratios numerically, choosing
appropriate measures of center and spread.
c) Is the ratio of 9.3 for the Statue of Liberty
unrealistically low? Explain.
10. Sluggers. Roger Maris's 1961 home run record stood
until Mark McGwire hit 70 in 1998. Listed below are
the home run totals for each season McGwire played.
Also listed are Babe Ruth's home run totals.
McGwire: 3*, 49, 32, 33, 39, 22, 42, 9*, 9*, 39, 52,
58, 70, 65, 32*, 27*
Ruth: 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41,
34, 22
a) Find the five-number summary for McGwire's
career.
b) Do any of his seasons appear to be outliers?
Explain.
c) McGwire played in only 18 games at the end of his
first big league season, and missed major portions of
some other seasons because of injuries to his back and
knees. Those seasons might not be representative of
his abilities. They are marked with asterisks in the list
above. Omit these values and make parallel boxplots
comparing McGwire's career to Babe Ruth's.
d) Write a few sentences comparing the two sluggers.
e) Create a side-by-side stem-and-leaf display
comparing the careers of the two players.
f) What aspects of the distributions are apparent in the
stem-and-leaf displays that did not clearly show in the
boxplots?
11. Be Quick! Avoiding an accident when driving can
depend on reaction time. That time, measured from
the moment the driver first sees the danger until he or
she gets the foot on the brake pedal, is thought to
follow a normal model N(1.5, 0.18).
a) Use the Empirical Rule to draw the Normal model
b) What percent of drivers have a reaction time less
than 1.25 seconds?
c) What percent of drivers have a reaction time
between 1.6 and 1.8 seconds?
d) What is the IQR of reaction times?
e) Describe the reaction time of the slowest 1/3 of all
drivers.
12. Music and memory. Is it a good idea to listen to music
when studying for a big test? In a study conducted by
some Statistics students, 62 people were randomly
assigned to listen to rap music, Mozart, or no music
while attempting to memorize objects pictured on a
page. They were then asked to list all the objects they
could remember. Here are the five-number summaries
for each group:
n Min Q1 Median Q3 Max
Rap
29 5 8
10
12 25
Mozart 20 4 7
10
12 27
None 13 8 9.5 13
17 24
b) Name the variables and classify each as categorical
or quantitative.
c) Create parallel boxplots as best you can from these
summary statistics to display these results.
d) Write a few sentences comparing the performances
of the three groups.
13. Mail. Here are the number of pieces of mail received
at a school office for 36 days.
123 70 90
80 78 72
52 103 138
112 92 93
118 118 106
95 131 59
151 115 97
100 128 130
66 135 76
143 100 88
110 75 60
115 105 85
a) Plot these data.
b) Find appropriate summary statistics.
c) Write a brief description of the school's mail
deliveries.
14. Pay. According to the 1999 National Occupational
Employment and Wage Estimates for Management
Occupations, the mean hourly wage for Chief
Executives was $48.67 and the median hourly wage
was $52.08. By contrast, for General and Operations
Managers, the mean hourly wage was $31.69 and the
median was $27.23. Are these wage distributions
likely to be symmetric, skewed left, or skewed right?
Explain. (Bureau of Labor Statistics)
15. Engines. One measure of the size of an automobile
engine is its "displacement," the total volume (in liters
or cubic inches) of its cylinders. Summary statistics
for several models of new cars are shown. These
displacements were measured in cubic inches.
Summary of Displacement
Count
38
Mean
177.289
Median
148.500
StdDev
88.8767
Range
275
25th %tile
105
75th %tile
231
a) How many cars were measured?
b) Why might the mean be so much larger than the
median?
c) Describe the center and spread of this distribution
with appropriate statistics.
d) Your neighbor is bragging about the 227-cubic-inch
engine he bought in his new car. Is that engine
unusually large? Explain.
e) Are there any engines in this data set that you
would consider to be outliers? Explain.
16. Engines, again. Horsepower is another measure
commonly used to describe auto engines. Here are the
summary statistics and histogram displaying
horsepowers of the same group of 38 cars.
Summary of Horsepower
Count
39
Mean
101.737
Median
100
StdDev
26.4449
Range
90
25th %tile
78
75th %tile
125
a) Describe the shape,
center, and spread of
this distribution.
b) What is the
interquartile range?
c) Are any of these
engines outliers in
terms of horsepower? Explain.
17. Bike safety. The Massachusetts Governor's Highway
Safety Bureau's report on bicycle injuries for the years
1991-2000 included the counts shown in the table.
Year Bicycle Injuries Reported
1991
1763
1992
1522
1993
1452
1994
1370
1995
1380
1996
1343
1997
1312
1998
1275
1999
1030
2000
1118
a) Display the data in a stem-and-leaf display.
b) Display the data in a timeplot.
c) What is apparent in the stem-and-leaf display that is
hard to see in the timeplot?
d) What is apparent in the timeplot that is hard to see
in the stem-and-leaf display?
e) Write a few sentences about bicycle injuries in
Massachusetts.
18. Profits. Here is a stem-and-leaf display showing
profits as a percent of sales for 29 of the Forbes 500
largest U.S. corporations. The stems are split; each
stem represents a span of 5%, from a loss of 9% to a
profit of 25%.
a) Find the five-number summary.
b) Draw a boxplot for these data.
c) Find the mean and standard deviation.
d) Describe the distribution of profits for these
corporations.
19. The ages of people in a college class (to the nearest
year) are as follows
Age
18 19 20 21 22 23 24 25 32
# of students 14 120 200 200 90 30 10 2 1
Determine the mean, median, and interquartile range
of the data
20. The following are parallel boxplots showing the daily
fluctuations of a certain common stock over the
course of 5 years What trends do the boxplots show?
21. From September 2, 1977, to May 29, 1987, Edwin
Moses set one of the most incredible records in sports
history by winning 122 consecutive races in the 400meter hurdles. The graph below shows a cumulative
relative frequency curve (vertical axis is the percent)
of the times of those 122 races.
a) Approximately what percent of the times were
under 48 seconds?
b) What is the median time recorded by Moses?
c) Approximately what is the fastest time recorded by
Moses?
d) What time exceeds approximately 75% of the
remaining times?
Multiple Choice
22. The average salary of all female workers is $35,000.
The average salary of all male workers is $41,000.
What must be true about the average salary of all
workers?
a) It must be $38,000.
b) It must be larger than the median salary.
c) It could be any number between $35,000 and
$41,000?
d) It must be larger than $38,000.
23. The mean age of five people in a room is 30 years.
One the people whose age is 50 leaves the room. The
mean age of the remaining four people in the room is
a) 40 b) 30 c) 25 d) cannot be determined from
the information.
24. A sample of data contains the numbers 1000, 600,
800, and 1000. The median of the data is
a) 1000 b) 850
c) 700
d) 900
25. The number of new projects started each month at an
advertising agency for the last six months are
2 5 3 3 6 3
The interquartile range for the above data is
a) 1
b) 4
c) 5
d) 2
26. This is a standard deviation contest. Which of the
following sets of four numbers has the largest possible
standard deviation?
a) 7, 8, 9, 10
b) 5, 5, 5, 5
c) 0, 0, 10, 10
d) 0, 1, 2, 3
27. When a set of data has suspected outliers, which of the
following are preferred measures of central tendency
and of variability?
a) mean and standard deviation
b) mean and variance
c) mean and range
d) median and range
e) median and interquartile range
28. In the second game of the 1989 World Series between
Oakland and San Francisco, ten players went hitless,
eight players had one hit apiece, and one player had
three hits. What were the mean and median numbers
of hits?
a) 11/19, 0
b) 11/19, 1
c) 1, 1
d) 0, 1
e) 1, 0
PART I REVIEW ANSWERS
1. a) It is (rounded to 1 decimal place), but there's no
reason it should be unless the number of women
receiving each type of cart' was roughly the same.
b) Yes, but they do not prove that adequate prenatal
care is important for pregnant women. The mortality
rate is quite a bit lower for women with adequate care
than for other women, but there may be a lurking
variable.
c) Intensive care is given for emergency conditions.
The data do not suggest that the care is the cause of
the higher mortality.
2. If enough sopranos have a height of 65 inches, this
can happen.
3. a) It means their heights are also more spread out
(more variable)
b) The z-score for women to qualify is 2.4 compared
with 1.75 for men, so it is harder for women to
qualify.
4. a) The distribution is unimodal and skewed to the
right and values range from 95 to 145.
b) The mean will be larger than the median, since the
distribution is right skewed.
c) Create a boxplot with quartiles at 97 and 105.5,
median at 100. The IQR is 8.5 so the upper fence is at
(1.5 X 8.5) + 105.5 = 118.25. There are several
outliers to the right. There are no outliers to the left
because the minimum at 95 lies well within the left
fence at 97- (1.5 x 8.5) = 84.25.
5. a)
b) 3.3%
c) 6.7% d) pH 4.40
e) pH 5.89
f) Quartiles are 4.5 and 5.3, so IQR is 0.8
6. a) These are categorical data, so mean and standard
deviation are meaningless,
b) Not appropriate. Even if it fits well, the Normal
model is meaningless for categorical data.
7. a) Overall mean is (34 x 1531.59 + 27 X
1388.85)/(34+27) = 1524.15 deaths per 100,000.
b) Yes. The distribution for the Northern towns is
higher than that for the South. Fully half of the towns
in the South have mortality rates lower than any of the
Northern towns. Some Northern towns have rates
higher than all of the Southern towns. It seems clear
that mortality is higher, in general, in the North.
8. a) They are on different scales.
b) January's values are lower and more spread out.
c) Roughly symmetric but slightly skewed to the left.
There are more low outliers than high ones. Center is
around 40 degrees with an IQR of around 7.5 degrees.
9. a)
The distribution is left skewed with a center of about
15. It has an outlier near 11.
b) Even though the distribution is somewhat skewed,
the mean and median are close. The mean is 15.0 and
the SD is 1.25.
c) Yes. 11.8 is already an outlier. 9.3 is more than 4.5
SDs below the mean. It is a very low outlier.
10. a) 3, 25.5, 36, 50.5, 70
b) Because the IQR is so c)
large, none are
technically outliers, but
the seasons with fewer
than 20 homeruns stand
out as a separate group.
d) Without the injured seasons, McGwire and Ruth's
home run production distributions look similar.
(Ruth's seasons as a pitcher were not included as
well.) Ruth's median is a little higher, and he was a
little more consistent (less spread), but McGwire had
the two highest season totals.
e)
f) Now we can see how much more consistent Ruth
was. Most of Ruth's seasons had home run totals in
the 40s or 50s. McGwire's seasons are much more
spread out (not even including the three at the
bottom).
11. a)
b) 8.2%
c) 24.1%
d) Quartiles are 1.38 and 1.62, so IQR is 0.24
e) The slowest 1/3 of all drivers have reaction times of
1.58 or more.
12. b) Type of music (categorical) and number of items
remembered (quantitative)
c) Because we do not have all the data, we can't know
exactly how the boxplots look, but we do know that
the minimums are all within the fences and that two
groups have at least one outlier on the high side.
d) All three distributions are right skewed. Mozart and
rap had very similar distributions. The scores for the
None groups are, if anything, slightly higher than
those for the other two groups. It is clear that groups
listening to music did not score higher than those who
heard none.
13. a)
b) Mean 100.25, SD 25.54 pieces of mail.
c) The distribution is somewhat symmetric and
unimodal, but the center is rather flat, almost uniform.
14. Chief executives have a mean salary less than the
median, so the distribution is likely to be skewed to
the left. General and operations managers have a mean
salary larger than the median, so their distribution are
likely to be skewed to the right.
15. a) 38 cars
b) Possibly, the distribution is skewed to the right.
c) Center—median is 148.5 cubic inches. Spread—
IQR is 126 cubic inches.
d) No It's bigger than average, but smaller than more
than 25% of cars. The upper quartile is at 231 inches.
e) No. 1.5 IQR is 189, and 105 - 189 is negative, so
there can't be any low outliers. 231 + 189 = 420.
There aren't any cars with engines bigger than this,
since the maximum has to be at most 105 (the lower
quartile) + 275 (the range) = 380.
16. a) Fairly symmetric, almost uniform except for the
right tail.
b) 47 horses
c) IQR is 47, so fences are at 7.5 and 195.5. Since the
range is only 90, we know there can't be any
observations lower than 35 (upper quartile — range)
or greater than 168 (lower quartile + range), so no.
17. a)
Stem Leaf
10 3
11 2
12 8
13 1478
14 5
15 2
16
17 6
10 | 3 = 1030 injuries
b)
c) In the stemplot, it is clear that some years may be
unusual.
d) The downward trend is visible in the timeplot.
e) In the decade 1991 to 2000, reported bicycle
injuries decreased steadily from about 1800 per year
to around 1100 per year.
18. a) -9, 1, 4, 9, 25
b)
c) Mean 4.72% of sales,
SD 7.55% of sales
d) Fairly symmetric and
unimodal, centered
around 4% of sales.
50% of the companies
report % profit
between 1% and 9%.
There is one outlier at
25% of sales.
19. Mean = 20.58; Median = 20; IQR = 1
20. The median has increased during the time while the
spread has decreased.
21. a. 30%
b. 48.5
c. 47
d. 49
22. C
23. C
24. D
25. D
26. C
27. A
28. A