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Lesson 9
A STORY OF FUNCTIONS
M4
ALGEBRA II
Problem Set Sample Solutions
1.
Periodically the U.S. Mint checks the weight of newly minted nickels. Below Is a histogram of the weights (In grams)
of a random sample of 100 new nickels.
n.211-
0.05-
0 iv
4.86 1.88 1.9C 1.92 ,.31 1.95 1.98 5.00 5.02 5.34 5.06 5.08 5.10 5.12
Weigl t (Grams)
a.
The mean and standard deviation of the distribution of nickel weights are 5.00 grams and 0.06 grams,
respectively. Mark the mean on the histogram. Mark one standard deviation above the mean and one
standard deviation below the mean.
4.94
5
5.06
0.20
0.05
0
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MATH
Lesson 9:
Cr 2014 Common Core, Inc. Al rights resenied. commoncore.org
4.86 4.88 4.90 4.92 4.94 4.96 4.98 5.00 5.02 5.04 5.06 5.08 5.10 5.12
Weigi t (Grams)
Using a Curve to Model a Data Distribution
179
Lesson 9
A STORY OF FUNCTIONS
ALGOMA 11
b.
Describe the shape of the distribution. Draw a smooth curve that comes reasonably close to passing through
the midpoints of the tops of the bars in the histogram. Is this approximately a normal curve?
The shape Is approximately normal.
4.94
5.(6
5
0.20
r. 0.15
0.05
.86 4.88 4.90 4.92 4.94 4.96 4.90 5.00 5.02 5.04 5.06 5.08 5.10 5.12
Weight (Grams)
c.
Shade the area under the curve that represents the proportion of data within one standard deviation above
and below the mean. Find the proportion of the data within one standard deviation above and below the
mean.
0.09 + 0.11 + 0.12 + 0.15 + 0.14 + 0.09 = 0.70
4.94
5
5.06
020
g 0.15
0.10
0.05
0
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MATH'
Lesson 9:
2014 Common Com, Inc. All right reser/ed. commoncoreorg
4.86 4.83 4.90 4.92 4.94 4.96 4.98 5.00 5.02 5.04 5.06 5.08 5.10 5.12
weight (grams)
Using a Curve to Model a Data Distribution
Lesson 9
A STORY OF FUNCTIONS
M4
Al CiEBRA II
Below Is a relative frequency histogram of the gross (in millions of dollars) for the all-time top-grossing American
movies (as of the end of 2012). Gross is the total amount of money made before subtracting out expenses, like
advertising costs and actors' salaries.
0.35 •
0,30.
0.25-
ft
0.20 -
1 0.15 0.10 0.35 •
0 300 350 400 450 500 550 500 650
700 750 800
tiross (clolars)
a.
Describe the shape of the distribution of all-time top-grossing movies. Would a normal curve be the best
curve to model this distribution? Explain your answer.
The shape is skewed to the riyht. A normal curve would not be the best curve to model the distribution,
b.
Which of the following is a reasonable estimate for the mean of the distribution? Explain your choice.
i.
325 million
ii.
375 million
Ili.
425 million
(iii) 425 million Is a reasonable estimate since this is a skewed distribution, and the mean will be pulled
toward the outliers,
c.
Which of the following is a reasonable estimate for the sample standard deviation? Explain your choice.
I.
50 trillion
IL
100 million
Ili.
200 million
(11) 100 million is a reasonable estimate because 50 million is too small, and 200 million is too large to be
considered a typical deviation from the mem
EUREKA
MATH
Lesson 9:
It) 2014 Conunon Cote, Inc. All rights resumed. commoncoreorg
Using a Curve to Model a Data Distribution
131
Lesson 9
A STORY OF FUNCTIONS
ALGEBRA II
.
Below is a histogram of the top speed of different types of animals.
0.35
0.30 t 0.25IX 0.20 0.15 0.10 •
035
10
a.
20
30
:0
40
Speed (mph)
00
00
70
Describe the shape of the top speed distribution.
Approximately normal
b.
Estimate the mean and standard deviation of this distribution. Describe how you made your estimate.
Mean is approximately 40 mph, and standard deviation is about 15 ni ph. Answers will vary in terms of how
the estimate was made.
Note: Students could roughly estimate the mean by locating a balance point of the distribution. The standard
deviation could be estimated by a typical deviation from the mean that was developed in Lesson 8.
C.
Draw a smooth curve that is approximately a normal curve. The actual mean and standard deviation of this
data set are 34. 1 mph and 15.3 mph, respectively. Shade the area under the curve that represents the
proportion of the data values that are within one standard deviation of the mean.
34.1
18,0
49.4
0.35 0.30 0.25
tb 0.20 •
0.15
0.10 0.05
10
EUREKA
MATH
Lesson 9:
CI 2014 Common Core, Inc. All rights reserved. commontoreorg
20
30
50
40
Speed (mph)
Using a Curve to Model a Data Distribution
60
70
80