Math 140 Review Chapter 1-3 KEY
Make sure you write your answers in complete sentences and in context when applicable.
1. We collect these data from 50 male students. Which variable is categorical (C) and which is
quantitative (Q)?
a. eye color C
b. head circumference Q
c. marital status C
d. number of cigarettes smoked daily Q
e. number of TV sets at home Q
f. temperatures in Southern California for the past year Q
g. weather conditions in Southern California in past year C
2.
A survey asked 200 people if they thought women in the armed forces should be permitted to
participate in combat. The following table summarizes the responses.
Yes
No
Total
Male
72
28
100
Female
8
92
100
Total
80
120
200
a) What percent of the females answered yes? 8/100= 8% of the females answered yes
b) What percent of the males answered yes? 72/11 = 72% of the females answered yes
c) Does there appear to be a difference in gender regarding opinion on whether women should be
permitted to participate in combat? Explain. Yes, a greater percentage of men (72%) thought women
should be permitted to participate in combat compared to women (8%).
3. A survey of an introductory statistics class in the Fall of 2003 asked students whether or not they ate
breakfast the morning of the survey. Results are as follows:
Sex
Breakfast
Yes
No
Male
66
67
Female
125
74
191
141
Total
133
199
332
a. Was this a categorical or quantitative study? Categorical (gender & whether they ate breakfast)
b. What is the variable (variables)? gender & whether they ate breakfast
c. What percent of females ate breakfast? 125/199 = 62.8% of the females ate breakfast
d. What percent of males ate breakfast? 66/133 = 49.6% of the males ate breakfast
e. Does there appear to be a difference in gender regarding eating breakfast? Explain. Yes, there is
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about a 13% difference. More females tended to eat breakfast compared to males. However, the
question is: is this a significant difference? 13% is not that significant in this context.
4. The population in 2000 of the U.S. was about 282,000,000. Of these 282,000,000 people, 34992 were of
age 65 or older. What percent of the U.S. population was considered elderly (65 or older) in 2000?
34992/282000000 = 0.01% of the U.S. population was considered elderly in 2000. (This is less than 1% of the
population.)
5. Identify the following research studies as observational or a controlled experimental. Explain why.
a. Data from the Motorcycle Industry Council stated that “Motorcycle owners are getting older and
richer.” Data were collected on the ages and incomes of motorcycle owners for the years 1980 and 1998
and then compared. The findings showed considerable differences in the ages and incomes of
motorcycle owners for the two years. Observational, no treatment assigned.
b. A study conducted at Virginia Polytechnic Institute and presented by Psychology Today divided female
athletes into two groups and had the students perform as many sit-ups as possible in 90 seconds. The
first group was told only to ‘do your best,’ while the second group was told to try to increase the
number of sit-ups they did each day by 10%. After 4 days, the first group averaged 43 sit-ups while the
second group averaged 56 sit-ups. The conclusion was that athletes who were given specific goals
performed better than those who were not given specific goals. Experimental, a treatment was assigned
(group encouranged to increase number by 10%)
c. A recent study showed that eating garlic can lower blood pressure. Researchers prescribed garlic pills
to high blood pressure patients and monitored their results over a 6 month period. These results were
then compared to high blood pressure patients who had received a placebo. The doctors administering
the pills were not aware of which patients had received the treatment. Experimental, a treatment was
assigned (garlic pills)
6. In problem 5a above, what might be a confounding variable? Answers may vary: example: Older people
tend to be more economically stable.
7. In problem 5c above, why is it important that the patients and doctors not know who received the
treatment? What is the name of this technique? One wants to avoid lurking variables (patients or doctors
changing their behavior and therefore influencing the outcome). This is called double blind since both
patients and doctors were not aware.
8. The dot plot below shows the ages for about 108 people in three community college math classes.
a. Any age 26 and over is considered unusually high for this sample. How many student ages are
considered unusual for this sample? 21 students had ages that were unusual for this group
b. What percent of the sample was this? 21/108= 19.4% of students’ ages are unusual
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9. Answer the following questions given the distribution of following exam scores.
Histogram of Chapter 3 Exam
16
14
Frequency
12
10
8
6
4
2
0
40
50
60
70
80
90
100
110
Chapter 3 Exam
a) How many students took the chapter 3 exam? 37 students
b) What is the shape of the distribution of exam scores? Roughly symmetric (or slightly skewed
right)
c) What was a typical score for this class (center)? Around 75 points
d) What was the typical spread for this class? Around 65 to 85
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e) How many students got at least an 80 on the exam? 8+2+1 = 11 students
f) What percentage of students got at least an 80 on the exam? 11/37 = 29.7% of students got at
least an 80 on the exam.
g) How many students scored less than an 80 on the exam? 7+4+15 = 26 students
h) What percentage of students scored less than an 80 on the exam? 26/37 = 70.2% of students
received less than 80 on the exam.
i) What percentage of students scored below 70 on the exam? 7 + 4 = 11; 11/37 = 29.7% of
students scored below 70 on the exam.
j) Approximately what percentage of students scored from 70 to 90 on the exam? 15 + 8 = 23;
23/37 = 62.1% scored from 70 to 90 on the exam.
10. Which is true of the data whose distribution is shown?
I. The distribution is skewed to the right. T
II. The mean is smaller than the median. F
III. We should summarize with mean and standard deviation. F
11. Answer the following questions given the distribution of salaries of a random company.
Salary
Relative Frequency (%)
40
30
20
10
0
40000
60000
80000
Salary (In U.S. Dollars)
4
100000
a)
b)
c)
d)
What percentage of employees made a salary of less than $35,000? 25%
What percentage of employees made a salary of more than $80,000? 5%
60% of employees made a salary of less than $45000
How many employees made a salary of less than $35,000? Cannot be determined. The
number of employees is not given.
12. All students in the physical education class completed a basketball free-throw shooting event and the
highest number of shots made was 32. The next day, the PE teacher realized that he had made a mistake.
The best student had actually made 38 shots (not 32). Indicate whether changing the student’s score made
each of these summary statistics increase, decrease, or stay about the same:
a) Mean increase
b) Median about the same
c) Range increase
d) IQR about the same
13. The mean and median scores of a recent math 075 exam were close to 68%. The instructor decided not
to count one score of zero that was from an absent student to get a better representation of the class
average and then recalculated the new mean and median.
a) Will the new mean increase, decrease or remain about the same? Explain. Since a very low score
was dropped, the new mean will now be higher (class average will go up). The mean is sensitive
to outliers.
b) Will the new median increase, decrease or remain about the same? Explain. The new median
will be roughly the same since the person with the middle score is roughly in the same position.
(The only time it would increase would be if the two middle people had scores that were far
apart from each other.) The median is not sensitive to outliers.
c) True or false: The overall range increased. False, since the minimum changes from zero to the
next lowest score in the class, the overall range will get smaller. The variability will decrease.
d) True or false: The IQR remained about the same. True, the scores at the 25th and 75th percentile
are roughly in the same position. Therefore, the IQR will be close to the same amount.
14. The following boxplots compare the ages of all the Oscar Winners from 1970 to 2001. Use this to
answer the following questions.
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Consider the distributions of ages for Oscar winning
actors and actresses. around
a.
50% of winners were below what age?
Actor: 43
Actress: 35
b.
75% of winners were below what age?
Actor: 51
Actress: 42
c.
75% of winners were above what age?
Actor: 37
Actress: 32
d.
25% of winners were above what age?
Actor: 51
Actress: 42
Actor 5 Number Summary: 31 , 37.25 , 42.5 , 50.25 , 76
Actress 5 Number Summary: 21 , 32 , 35 , 41.5 , 80
a. How many outliers are there for each gender and what are they?
Actor: 1, around 75 years old
Actress: 3, around 60, 73, & 80 years old
b. What are the shapes of the distributions?
Actor: right skewed
Actress: right skewed
c. Did a typical actor or actress win at a younger age? Explain. The typical age for an actor to win
an Oscar is around 43 years old versus the typical age for an actress is around 35 years old.
Therefore, actresses tend to win Oscars at a younger typical age.
d. What are the IQRs for actors and actresses? Interpret these IQRs. Actors IQR = 51-37 = 14 years,
Actress IQR = 42-32 = 10 years There is more variability in typical ages for men compared to
women. The typical age to win an Oscar for women is more consistent.
e. Based on the IQRs, did actors or actresses win at a younger age? Explain. Typical ages to win an
Oscar for men is around 37 to 51 years old. Typical ages for women is around 32 to 42. This
shows that women tend to win this award at a younger age.
f. Which data set is more consistent and why? The female group is more consistent since the IQR
is smaller. This means that it is easier to predict a typical age for the female group
g. Did actors or actresses win at a younger age? Utilize percentages from the Boxplot of the
distributions above to support your answer. Typical ages to win an Oscar for men is around 37
to 51 years old. Typical ages for women is around 32 to 42. This shows that women tend to win
this award at a younger age. Half (50%) of the male winners were below 43 years old compared
to half of the female winners who were below 35 years old.
15. The following data represent the annual chocolate sales (rounded to nearest billions of dollars) for a
sample of seven countries in the world. Round answers to nearest tenths.
2, 5, 7, 2, 5, 3, 18
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a. Find the mean for the data. Write the answer in a complete sentence in context. The average annual
chocolate sales was 6 billion dollars.
∑(𝑥−𝑥̅ )2
b. Calculate the standard deviation: s = √
𝑛−1
. Write the answer in a complete sentence in context.
S = 5.6 billion dollars. Typical chocolate sales are 6 billion dollars ±5.6 billion dollars.
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c. Using this standard deviation, one could then expect typical annual chocolate sales to be between
Checkpoint
2.1 countries were around 0.4 to 11.6 billion
which two values? Typical annual chocolate
sales Topic
for these
dollars.
Question
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16.
Points: 10 out of 10
Answer the following questions
with a letter I, II, III, or IV.
Explain your choice in complete
sentences for each question.
Histograms can be used more
than once and some answers
might have more than one
answer.
Which of the histograms could represent a distribution of weights of babies
for a large random sample of male newborns at a local hospital?
A.
A. WhichI graph would represent a distribution of the ages of math 075 students where there is a high
percentage of students who recently graduated high school and very few students who over 50?
B. Explain.
II II - Most of the data will be clustered on the lower end (left) and very little data will be on
the higher end (right).
C.
III
B. Name all graphs where the mean would be chosen as the best measure of center. Explain. III – The
D. meanIVis a good measure of center for symmetric graphs only.
C. Name all graphs where the IQR (interquartile range) would be chosen as the best measure of spread.
Explain. I, II, & IV – The IQR is a good measure of spread for non-symmetric graphs since it is not
Feedback
sensitive to outliers.
Good job! We expect the distribution of weights to have a central peak
D. Which
would
represent a distribution for the heights of koala bears? Explain. III –
around
an graph
average
weight.
Measurements of species tend to be symmetric. Most fall within a typical range with fewer high
and low values.
Please answer the question below. Your response will not be graded, but will be available for your
17. The ten top grossing Pixar Animated movies for the US box office up to June 2010 are shown below,
instructor to read.
in millions of dollars.
a. Find the median
Question
6 A typical Pixar movie made about 246 million dollars.
b. Find the interquartile range (IQR). 261-163 = 98 million dollars
Points: 0 out of 0
Here are data on 77 cereals. The data describes the grams of
carbohydrates (carbs) in a serving of cereal. Compare
the distribution of
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carbohydrates in adult and child cereals.
c. Interpret the meaning of the IQR in context. Examples: The typical spread of revenue for Pixar movies
was 98 million dollars. This means that the spread between the middle 50% of the revenues was 98
million dollars. Pixar typically made from 163 to 261 million dollars.
Movie
$Millions
Toy Story
A Bug’s Life
Toy Story 2
Monsters, Inc.
Finding Nemo
The Incredibles
Cars
Ratatouille
WALL-E
Up
192
163
246
256
340
261
244
206
224
293
18. The following graphs show the distributions of the ages in years of Math 075 students in the Fall of
2014.
Histogram of Ages
350
300
Frequency
250
200
1 50
1 00
50
0
15
30
45
60
75
Ages
8
90
Dotplot of Ages
24
36
48
60
72
84
96
Ages
Each symbol represents up to 5 observations.
Boxplot of Ages
1 00
90
80
Ages
70
60
50
40
30
20
10
Note: Age 26 is the first outlier.
Descriptive Statistics: Ages
Variable
Mean StDev
Ages
21.128 6.812
Minimum
15.0
Q1
18.0
Median Q3
19.0
21.0
Maximum
98.0
IQR
3.0
a. Was this a categorical or quantitative study? Quantitative (ages)
b. What is the variable (variables)? ages
c. What is the shape of the distribution in the ages? Right skewed
d. Which measure of typical center is best to use? Mean or Median? Explain. The median would be a
better representation of the typical center since the graph is skewed.
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e. Which measure of typical spread is best to use? Standard Deviation or IQR? Explain. The IQR would be
a better representation of the typical spread since the graph is skewed.
f. What is the typical center? Complete sentence in context. Using the provided descriptive statistics, the
median is 19.0. This means that a typical age for a Math 075 student was 19 years old.
g. What is the typical spread? Complete sentence in context. The IQR is given a 3. This means that the
spread between the middle 50% of the ages was only 3 years.
h. What ages are considered unusual for this group? Were there any students that were unusually
younger or older for this sample? It was given that 26 was the first outlier so any student 26 and older is
considered unusual for this group. There were many outliers in this group (too many to count). The
oldest being close to 100.
19. According to the data above for the ages, the mean was 21.1 years with a standard deviation of 6.8
years. The following question is to practice standard deviation. In reality, since the graph was skewed,
these values are not a good representation of what was typical for this group. The median and IQR
would be used instead.
But for practice:
a. What is the range of ages from one standard deviation below the mean to one standard deviation
above the mean? 14.3 to 27.9 years old. (Typical ages)
b. What is the range of ages from two standard deviations below the mean to two standard deviations
above the mean? 7.5 to 34.7 years old. (Anyone over 34.7 years old is unusual for this group)
c. What is the range of ages from three standard deviations below the mean to three standard
deviations above the mean? 0.7 to 41.5 years old. (Anyone over 41.5 years old was extremely unusual
for this group)
d. Is the age of 25 years more than one standard deviation above the mean? Show by converting to a z
score using the formula z 
xx
25  21.1
 0.6 No it is not more than one standard deviation
. z
s
6.8
above the mean. The z score is 0.6. This means a 25 year old was typical for this group.
e. There was a 98 old student which is unusual. How unusual is she, highly unusual (z-score above 2) or
extremely unusual (z-score above 3)? z 
98  21.1
 11.3 She was extremely unusually with a z score
6.8
of 11.3! Much higher than a z score of 3 meaning this is extremely rare and highly unlikely to happen
again.
20. A dietitian is interested in comparing the sodium content of real cheese with the sodium content of
a cheese substitute in milligrams and asks you (the statistician) to provide data that supports her belief
that cheese substitutes typically contain more sodium. You collect the sodium content of several real
cheeses and chees substitutes. Using computer technology, you provide the following box plots and
sample statistics.
Using the following statistics and graphs, decide whether the dietitian’s belief is correct. Support your
decision with the statistics provided. (Include discussion of the shapes, any outliers and the best
measures of center and spread to support your decision). Answers will vary but the conclusion should
be that: The typical sodium content for real cheese was between 56.3 and 292.5 mg (Half of the samples
fell within this range. The typical sodium content for cheese substitute was between 197.5 and 305 mg.
(Half of the samples fell within this range. Although there was more variability in the real cheese, a
typical sample had lower sodium in general. Note, however that the maximum typical value of sodium
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was about the same for both. (Even though real cheese is the better choice we can note that the upper
25% of the samples were higher in sodium content compared to the substitute cheese due to real
cheese having more variability.)
real cheese
N
Mean
SD
Minimum
8
193.1mg
133.2mg
40mg
cheese substitute
N
Mean
8
253.8mg
SD
68.6mg
Q1
56.3mg
Median
Q3
200mg
292.5mg
Minimum
Q1
130mg
197.5mg
Median
265mg
Maximum
420mg
Q3
Maximum
305mg
340mg
Boxplot of real cheese and cheese substitute
400
Data
300
200
100
0
real cheese
cheese substitute
21. In the real cheese/cheese substitute boxplots, which type had more variability?
(Using the descriptive statistics) The typical spread for real cheese was (Q3 – Q1) 236.2 mg. The typical
spread for cheese substitute was 107.5 mg. Thus the real cheese had more variability. The cheese
substitute was more consistent.
22. The mean for each pair of graphs is given just above each histogram. For each pair of graphs
presented below indicate whether one of the graphs has a larger standard deviation than the other or if
the two graphs have the same standard deviation. Try to identify the characteristics of the graphs
that make the standard deviation larger or smaller.
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1. B since it is skewed
2. B since there is more
variability
3. Both have the same. The
distributions in the graphs are
identical, the mean is just
higher for the second one. The
data is spread out the same for
both graphs.
23. Which would have a larger standard deviation? The mile times of the male high school track teams in
the U.S. or the mile times of the male participants in the last Olympics? High school teams since their
times would be more spread out (more variability).
24. In 2007, the mean property crime (per 100,000 people) for the 26 states east of the Mississippi River
was 409 with a standard deviation of 193. Assume the distribution was roughly symmetric and
unimodal.
a. Between which two values would you expect to find about 68% of the rates? 68% is one standard
deviation so between 216 and 602 crimes per 100000 people.
b. Between which two values would you expect to find about 95% of the rates? 95% is two standard
deviations so between 23 and 795 crimes per 100000 people.
c. If an eastern state had a violent crime rate of 503 crimes per 100,000 people, would you consider this
unusual? Explain. No, 503 crimes falls within one standard deviation. 503 crimes is within the typical
values.
25. When would you choose the median as the best measure of center? Median is appropriate for nonsymmetrical graphs. (Mean would be appropriate for symmetrical graphs.)
26. When would you choose the standard deviation as the best measure of spread? Standard deviation
is appropriate for symmetrical graphs. (IQR would be appropriate for non-symmetrical graphs.)
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