Lesson 1.2.1
1-27. Mark’s scores on his first nine assignments are: 10, 10, 9, 9, 10, 8, 9, 10, and 8.
a. What are the median, mode, and range of his scores? [ median = 9, mode =
10,
range = 2 ]
a. What is his mean (average) score so far? [ About 9.2 ]
b. Mark did not do the tenth assignment, so he got a zero on it. Zero is an outlier
for these assignments. What is his new mean? [ About 8.3 ]
1-28. There are five students of different
ages whose median age is 13 years.
a. What are two possibilities for the
ages of the three oldest students?
[ 13 and any two greater ages ]
b. If each student is a different age,
how many students must be
younger than 13? [ 2 students ]
c. In complete sentences, describe what a median is to another student in the
class.
Pretend that the student was absent the day you learned about medians. [ If the
data is arranged in order from least to greatest, the median is the number
in
the middle. If two numbers share the middle, then the median is the mean
of those two numbers. ]
1-29. Inspect each data set. Without making a calculation, decide if each statement
about
the data set is true or false. Explain how you decided. [ a: False; The mean cannot
be greater than any piece of data. b: True. c: False; If the data were arranged
in numerical order, it would be easy to see that the median is 14. d: False; 2
occurs more often than any other value. ]
a. True or False: “The set of 24, 25 26, and 28 has a mean of 30.”
b. True or False: “The set of 0, 1, 2, 2, 3, 3, and 4 has two modes.”
c. True or False: “15 is the median of this data set: 12, 14, 15, 13, and 16.”
d. True or False: “5 is the mode of 0, 2, 2, 3, 4, 5 because it has the greatest
value.”
Lesson 1.2.2
1-37. DO COLLEGE ATHLETES EARN MORE?
A 2005 study at the Sate University of New York looked at college athlete’s earnings
six years after graduation to determine if college athletes tended to earn more in their
full-time jobs than non-athletes.
a. Create two histograms, one for nonathletes,
and one for athletes. The
horizontal axis should have bins labeled
$0-27, $27-54, etc. The vertical axis
should be labeled from 0 to 30. The
height of the bars will be the number of
athletes or non-athletes.
[ See histograms at right. ]
b. A salary over $216,000 could be
considered to be wealthy. How many
non-athletes make over $216,000?
How many athletes make over
$216.000? [ 1% for both. ]
c. By looking at your histogram, estimate
the median salary for athletes and nonathletes.
[ About $95,000 for both. ]
d. Between what salaries do “typical” athletes earn? “Typical” non-athletes?
[ Between $54,000 and $135,000 for both. ]
e. Do college athletes earn more after graduation than non-athletes? Justify your
answer by comparing the distributions as completely as you can (center, shape,
typical range, outliers). [ There seems to be very little difference in the
distributions of salaries. ]
1-38. Mrs. Sakata is correcting math tests. Here are the scores for the first fourteen
tests
she has corrected: 62, 65, 93, 51, 55, 12, 79, 85, 55, 72, 78, 83, 91, and 76. Which
score does not seem to fit in this set of data? How will the outlier score affect the
mean and median of the data? Explain. [ 12 is much lower than the rest of the
data. It will make the class mean lower, but will not affect the median much. ]
1-39. Considering the histograms of salaries you drew in problem 1-37, which would be
a
better measure of center, the mean, or the median? [ Since the distributions are
fairly symmetric with no apparent outliers, either the mean or the median would
be appropriate. ]
1-40. The weights of 19 hummingbirds are given below in
ounces. Justify whether the mean or median is a
better choice for a “typical” hummingbird’s weight
and calculate it.
11, 8, 10, 10, 10, 9, 4, 9, 7, 5, 11, 8, 9, 11, 10, 10,
10, 9, 10 [ The histogram at right shows that the
data is not symmetrical. median = 10 ounces ]
Lesson 1.2.3
1-50. Assume that the histograms below represent the amount of time it took two
different
groups of 100 people to run a 5K race. Assume that the mean of each histogram is
the same. Which group has a greater mean absolute deviation? Why? [ Group A;
If the mean is the “center” of the data, then Group B has more data points near
the center and Group A has more data points farther from the center. ]
1-51. Describe the shape of the distributions in problem 1-50 above. [ Group A:
doublepeaked
and symmetric, Group B: single-peaked and symmetric ]
1-52. Given the 2 sets of data below, without doing any calculation, which one has a
smaller mean deviation? Why? [ Set 2; The data values are closer to the center. ]
Set 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Set 2: 3, 3, 4, 4, 5, 5, 6, 6, 7, 7
1-53. The ice cream consumption in several countries is given in the table below.
a. How many countries are represented? [ 14 ]
b. Display the data in a histogram. Make the bins 4 liters per person wide
(0-4, 4-8, etc.). [ See histogram above right. ]
c. Describe the shape of the distribution. [ Close to single-peaked and
symmetric. ]
d. Why is it appropriate to calculate a mean for this data? Calculate the mean.
[ The data is nearly symmetric; 9.7 liters per person. ]
e. Measure the spread (variability) of the data by finding the mean absolute
deviation. [ 5.17 liters per person ]
f. In complete sentences, completely describe the distribution of ice cream
consumption by discussing the center, shape, spread, and outliers. [ The mean
of the distribution is a good choice for the center of the data. The mean is
9.7 liters per person. The data is single-peaked and symmetric. The mean
deviation is 5.17 liters per person. Japan in an outlier. ]
1-54. The Tigers and the Panthers are in the same league and each play 12 games
during a
basketball season. The points they scored each game are shown in the table below.
a. Make a histogram for each team. Each
bin should have an interval of 10 points
(20-30, 30-40, etc.). [ See histograms
at right. ]
b. Calculate the mean and mean absolute
deviation for each team.
[ Tigers: mean = 54, mean absolute
deviation = 8.83; Panthers: mean = 54,
mean absolute deviation = 11.5 ]
c. Why was finding the mean absolute
deviation for the Tigers not a good choice?
(In the next lesson, you will find an
alternative.) [ The distribution is skewed
and not symmetrical. ]
d. Write a complete explanation for which team
you think is “better” and why. Be sure you compare the center, shape, spread,
and outliers. [ Both teams have the same average, but the Tigers have a
lower mean absolute deviation. They are more consistent. The Panthers
have some very good scores, but they also have some very bad scores. ]\
Lesson 1.2.4
1-62. ALDEN’S SODA
Alden created a box plot for the calories in 11 different brands of sodas as shown
below.
a. How do you think Alden collected the data for his box plot?
[ He probably went to the store and read labels for
different brands of soda. ]
b. According to this graph, give as complete a description
about the calories of the 11 brands of soda as you can?
Consider the center, shape, spread, and outliers. [ Sample responses: There is
at least one diet soda (with zero calories) that is an outlier, the maximum
calories are 160, half the sodas have between 100 and 140 calories, the IQR
is 40. The median soda is 120 calories. The distribution is not symmetric. ]
1-63. How do box plots help compare data? Think about this question as you compare
the
data below that shows the ages of the students at three schools.
a. Which school is a K-8 school and how do you know? Does that school have
more students in K-2, grades 3-5, or 6.8? Why? [ School A. We know because
of the age range (from 5 to 14). School A has more students in K-2 because
half of the students are less than 8 years old according to the median. ]
b. What does the box plot for School C tell you about 11 year-olds at that school?
[ 11 year-olds make up about 50% of the school. ]
c. How many students attend School C? [ It cannot be determined from the
box
plot. ]
d. Make a conjecture about how the data for these plots was collected. [ All
three
schools would have to be visited. The registration records at the schools
give
students birth dates. From these, the ages of students could be determined
and recorded. ]
e. What statements can you make about the grades at School B based on its box
plot? Consider the center, shape, spread, and outliers. [ School B has about
the
same number of students in each grade. Median is 11 years, the
distribution
is symmetric, IQR is two years, and there are no apparent outliers. ]
1-64. Levi used the box plot below to say, “Half of the
class walked more than 30 laps at the walk-athon.”
Levi also knows that his class has more
than 20 students.
a. Do you agree with him? Explain your reasoning. If you do not
agree with him, what statement could he say about those who
walked more than 30 laps? [ No, because 30 is the third
quartile of the data. This means that only about 25% of the
students walked more than 30 laps. The actual percentage of
students varies with the number of data points. ]
b. Levi wants to describe the portion of students who walked between 20 and 30
laps (the box). What statement could he say? [ This portion represents
roughly 50% of the students. It could slightly vary depending on the
number of data points. ]
c. How could you alter a single data point and not change the graph? How could
you change one data value and only move the median to the right? [ You could
change one of the data points between 30 and the maximum, but keep in
the
same range of values. You could add data greater than the current median.]
1-65. Lucy and Marissa each designed a box plot to represent this data set:
16 18 19 19 25 26 27 32 35
Their plots are shown below. Which plot is scaled correctly and why?
Explain the mistakes in the incorrect plot.
[ (a) is done correctly since it
has both the equally spaced number line and the box plot. (b) has an
equally spaced number line that is numbered incorrectly. ]
Lesson 1.2.5
1-70. Jerome is keeping track of how many books he and his friends have read during
the
first 100 days of school. Make a stem-and-leaf plot of how many books each person
has read to help Jerome present the data to his teacher. The numbers of books are:
12, 17, 10, 24, 18, 31, 17, 21, 20, 14, 30, 9, 25.
a. Make a stem-and-leaf plot of the data.
[ See graphic at right. ]
b. Jerome wants to present the data with a plot
that makes it possible to calculate the mean,
median, and mode. Can he do this with a
stem-and-leaf plot? He is not asking you to
calculate them, but he wants you to tell him if
it is possible and why. [ Because the stemandleaf shows all of the data points, Jerome can calculate all of the
measures of central tendency. ]
c. Use the stem-and-leaf plot to describe how the data is spread. That is, is it
spread out, or is it concentrated mostly in a narrow range? [ It appears to be
concentrated at lower values and not spread too widely. ]
1-71. Would it be helpful for Jerome to create a dot plot to display and analyze the data
from problem 1-70? Why or why not?
[ No, there are too many data points that
do not repeat. ]
1-72. Create a histogram of the data from problem 1-71.
What are the advantages and disadvantages of
displaying the data using a histogram? [ See
histogram at right and answers to part (a) of
problem 1-69. ]
1-73. Create a box plot of the data from problem 1-71.
What are the advantages and disadvantages of
displaying the data using a histogram? [ See box
plot at right and answers to part (c) of problem
1-69. ]