GOALS Level

ITEM #7309

C

Math 4 MEASUREMENT AND RATES, STATISTICS, AND PROBABILITY

for LEVEL C MATH 1 Rational Numbers, Number Sense, Ratios, Proportions, and Percents MATH 2 Patterns and Expressions, Equations and Inequalities, and Functions MATH 3 Lines, Angles, Triangles, Transformations, and Formulas MATH 4 Measurement and Rates, Statistics, and Probability

ISBN 978-0-88336-137-5

Syracuse, New York 800.448.8878 www.newreaderspress.com

9 780883 361375

CASAS

®

TA B L E O F C ON T E N TS UNIT 1: MEASUREMENT AND RATES Strategy 1

Convert Measurements Within a System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Strategy 2

Convert Metric Units with D ­ ecimal Placement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Strategy 3

Understand Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Strategy 4

Convert Rates to Solve Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Strategy 5

Graph Proportional Relationships. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Unit 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

UNIT 2: STATISTICS Strategy 6

Describe Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Strategy 7

Use Measures of Center to Compare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Strategy 8

Use Scatter Plots and Pictographs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Strategy 9

Create and Interpret Bar Graphs, Line Graphs, and Circle Graphs. . . . . . . . . . . . . . . . . 39

Strategy 10 Draw Inferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Unit 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

UNIT 3: PROBABILITY Strategy 11 Understand Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Strategy 12 Develop Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Strategy 13 Determine Probability of Compound Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Unit 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

CASAS Practice Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3

STRATEGY 6

Describe Data The mean, median, and mode are values that describe the center of a set of data and are sometimes called the measures of central tendency or measures of center. The mean is the average of the numbers. The median is the middle number in a sorted list of numbers. To find the median, place the numbers in order from least to greatest and find the middle number. The mode is the value that appears most often in a set. Characteristics of a data set help determine which measure of central tendency best describes the center of that data set. An outlier is a value that “lies outside” (is much less than or greater than) most of the other values in a data set. The mean is a good measure of center when a data set does not include outliers. When a data set includes outliers, the mean may be raised or lowered, so the median is a better choice. Though mode may be used with numerical data, it is more often used to describe non-numerical data, such as favorite color. Variability is a measure of the spread of a data set. The spread is how far away a typical value is from the center. The range and interquartile range of a data set are measures of variability. The range is the difference between the lowest and highest values. The interquartile range (IQR) is the difference between the first (lower) and third (upper) quartiles. Quartiles divide a rank-ordered data set into four equal parts. When a dot plot has most of the data on the left side with one or more outliers, or a “tail,” on the right, it is said to be skewed right. In a data set that is skewed right, the mean is typically greater than the median. When a dot plot has most of the data on the right side with a “tail” on the left, it is said to be skewed left. In a data plot that is skewed left, the mean is typically less than the median. Because the mean of a skewed data set is affected by the much smaller or greater values in the “tail,” the median is often a better measure of central tendency. The range may also be affected by the outliers in the tail, which means that the IQR is often a better measure of spread or variability than range.

E xa m p l e 1 The data in the dot plot show the number of hours that each member in a club spends walking in one week. Identify the outlier of the data set and determine how the outlier affects the mean and median. What is the best measure of central tendency, and why? X

X X X

X X

X

0

1

2

3

X X 4

5

X 6

7

8

© New Readers Press. All rights reserved.

With Outlier

Without Outlier

+ 2 + 3 + 5 + 5 + 8 28 = 10 = 2.8 mean = 0 + 1 + 1 + 1 + 210 median = 2 mode = 1 range = 8 − 0 = 8

mean = 0 + 1 + 1 + 1 + 29 + 2 + 3 + 5 + 5 = 20 = 2.2 9 median = 2 mode = 1 range = 5 − 0 = 5

The outlier increases the mean but has no effect on the median or the mode. The best measure of central tendency is the median because it more accurately reflects the center of the data set.

Unit 2: Statistics

25

Ex a m p l e 2 The data in the dot plot show the number of answers students got correct on a 20-item test. What is the best measure of center, and why? Correct Answers

15

16

17

18

19

20

+ 2(18) + 3(19) + 2(20) mean = 2(15) + 2(16) + 4(17)15 = 263 = 17.53 15 median = 17 mode = 17 range = 20 − 15 = 5

There is no outlier in this data set. The best measure of center is the mean because it accurately reflects the center of the data.

Ex a m p l e 3 The data in the dot plot show the hours that each employee spends on a project in one week. What are the range and the interquartile range of the data? Which is the better measure of variability of the data, and why? X

X X

X X X

X

X X

X

10 11 12 13 14 15 16 17 18

Range Interquartile Range Subtract the lowest number from the highest number. Use the median to divide the data into two equal groups. 10, 11, 11, 12, 12 | 12, 13, 15, 15, 18 range = 18 − 10 = 8

Find the median of the lower quartile of data. 10, 11, (11), 12, 12



Find the median of the upper quartile of data. 12, 13, (15), 15, 18

Subtract the medians of the upper and lower quartiles of data to find the IQR. IQR = 15 − 11 = 4 So, the range of the data is 8 and the IQR is 4.

In this case, the IQR is a better measure of variability because it is less affected by the outliers or skewed data. The IQR tells you the range for the middle 50% of the data.

26 Unit 2: Statistics

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In the data set, 18 is an outlier. If you remove the outlier from the data, the range would be 5, not 8, and would be much closer to the IQR of 4.

GU ID E D PR ACT I C E 1. The data in the dot plot show the numbers of hours that part-time employees work in a week. What is the better measure of variability, and why?

30

31

32

33

2. The dot plot shows the number of days in a week that members of a soccer team spend working out. What is the best measure of center?

34

X

X X X

X X X X X

X X X

X X

X

0

1

2

3

4

5

A. mean B. mode C. median D. range

 THINK Measures of variability are range and interquartile range (IQR). Does the data have outliers? Solution: range = 34 – 30 = 4 IQR = 33 – 31 = 2 There are no outliers in the data, so the range, which represents 100% of the data, is exactly double the IQR, which represents 50% of the data. Thus, the range is a better measure of variability because it gives you a complete and accurate measure of the variability in the data set.

 THINK Does the data have an outlier? Solution: There are no outliers in the data; therefore, the mean would be the best measure of center. (A) mean

IN D E PEND E N T P R ACT I C E 3. The data in the dot plot show the hours students in a class spend studying in one week. Which is the best measure of variability?

4. The data in the dot plot show the height (in inches) of plants in a greenhouse. Which is the best measure of center? X X X X X X X X

6

7

8

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Time (hours)

A. range B. mean C. median D. interquartile range

9

10

X X X X X X X X X

X

9 10 11 12 13 14 15 16 17 18 19 20

A. range B. mean C. median D. mode

Unit 2: Statistics

27

C ASA S PRA CT I C E 5. The data in the dot plot show how many pages each student read last night.

6

7

8

8. The data in the dot plot show the scores on a recent quiz in Mrs. Riley’s class.

9 10 11 12 13 14 15

Which is the best measure of variability? A. interquartile range

X

B. mean

3

4

5

X X X X X

X X X X X X X

X X X X X X X X

X X X X

X X

6

7

8

9

10

Quiz Scores

C. median

Which is the best measure of central tendency?

D. range

A. range

6. The data in the dot plot show the travel time (in minutes) of each employee’s commute to work.

B. mean C. median D. mode

5

6

7

8

9

10 11 12 13

Travel Time (minutes)

9. The data in the dot plot show how many tools different mechanics have in their toolboxes.

Which is the best measure of central tendency? A. range B. mean C. median D. mode 7. The data set below shows the amount Travis spent for lunch on 11 days this month. $5, $5 $6.50, $7, $7, $8, $8, $8, $8.50, $10, $18.50 Which is the most reasonable measure of central tendency for the data set? A. range B. mean

Which is the best measure of variability? A. interquartile range B. mean C. median D. range 10. The data in the dot plot show how much gasoline Benny uses traveling to and from work each day over the course of four weeks.

C. median D. mode X X

X X X X X X X

X X X X X X

X X X X X

Gasoline Consumed (gallons)

TEST TIP: When determining the best measure

Which is the best measure of central tendency?

of center or variability, first determine if there are outliers in the data. Then choose the measure that best represents the data.

B. mean

A. range C. median D. mode

28 Unit 2: Statistics

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1 .3 1 .4 1 .5 1 .6 1 .7 1 .8 1 .9 2 .0

STRATEGY 7

Use Measures of Center to Compare Mean, median, and mode are measures of center. Range, interquartile range (IQR), and mean absolute deviation (MAD) are measures of variability. You can use measures of center and variability to compare two sets of data. If the data in a data set are spread out, the IQR is big. If the IQR is small, the data are closer together. The mean absolute deviation (MAD) of a data set is the average distance between each data value and the mean. Similar to the IQR, if the MAD is big, the data are spread out, and if the MAD is small, the data are closer together. To find the MAD of a data set, first find the mean of the data set. Then subtract the mean from each data point and find the average of the absolute value of each difference.

E xa m p l e 1 The data in the dot plot show the number of miles each runner ran last week. Determine the mean absolute deviation of the data set.

X X

X

X X X

21

22

23

X X X X X

X X

24

25

X X X 26

27

28

First, find the mean of the data set. 21 + 21 + 22 + 23 + 23 + 23 + 24 + 24 + 24 + 24 + 24 + 25 + 25 + 27 + 27 + 27 16

= 384 = 24 16

Then, subtract the mean from each data point and find the average of the absolute value of each difference. |21 – 24| + |21 – 24| + |22 – 24| + |23 – 24| + |23 – 24| + |23 – 24| + |24 – 24| + |24 – 24| + |24 – 24| + |24 – 24| + |24 – 24| + |25 – 24| + |25 – 24| + |27 – 24| + |27 – 24| + |27 – 24| 16

+ 0 + 0 + 0 + 1 + 1 + 3 + 3 + 3 22 = 3 + 3 + 2 + 1 + 1 + 1 + 0 + 016 = 16 = 1.375

The mean absolute deviation of the data is 1.375. This means that the average distance of each data value from the mean is 1.375 miles. The MAD is small, which indicates that the data are clustered together.

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Unit 2: Statistics

29

Ex a m p l e 2 The data in the dot plots show the heights of soccer players and basketball players. Compare the two sets of data using the measures of center and the measures of variability.

X 65

X 66

67

68

X

X X

X X

X X X

X X X X X

69

70

71

72

73

X X X 74

X 75

76

X 77

78

79

80

81

82

X X X

X X

X X

X X

X X

78

79

80

81

82

83

84

Height of Soccer Players (in.)

65

66

67

68

69

70

71

72

X

X

X

X X X

73

74

75

76

77

X X X 83

84

Height of Basketball Players (in.)

Measures of Center Soccer Players Basketball Players Mean = 72 Mean = 79 Median = 72.5 Median = 79 Mode = 73 Mode = 76, 78, 84 Each measure of center for the height of basketball players is greater than each measure of center for the height of soccer players. This tells you that the average basketball player is taller than the average soccer player. Measures of Variability Soccer Players Basketball Players Range = 13 Range = 11 IQR = 3 IQR = 5.5 MAD = 2.1 MAD = 2.7

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The variability of the two data sets is about the same. This tells you that the spread of data within each data set is about the same.

30 Unit 2: Statistics

GU ID E D PR ACT I C E 2. Which data set has more variability?

Look at the following information. Then answer questions 1 and 2. The data in the dot plots compare electricity costs for customers using two different electric companies.

X 162

X 166

XX 170

X X X XXX

X X

174

178

X X X X

X

182

X

X XX

186

X

190

194

Company A (dollars)

X 162

166

X 170

X 174

X X X XXXX 178

XX XX XX 182

X X

Measures of variability include the range, IQR, and MAD.

X X X

186

 THINK

X 190

194

Company B (dollars)

1. What do the measures of center tell you about these two data sets?

Solution: Find the measures of variability for each data set: Company A: range = 30 IQR = 10 MAD ≈ 6.45 Company B: range = 24 IQR = 7 MAD ≈ 4.27 The data set showing the electric costs for Company A has greater variability.

 THINK What are the three measures of center? Solution: Find the measures of center for each dot plot: Company A: mean ≈ 178 median = 178 mode = 173 and 183 Company B: mean ≈ 181 median = 182 mode = 180, 182, 183, and 188

WORKPLACE CONNECTION: Retailers use measures of center and measures of variability to compare the sales of similar products to determine which products to sell.

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The measures of center tell you that customers using Company B generally pay a little more than customers using Company A.

Unit 2: Statistics

31

INDEPEND E N T P R A CT I C E Look at the following information. Then answer questions 3 and 4.

Look at the following information. Then answer questions 5 and 6.

The data in the dot plots show the grade point averages of students in two different classes.

The data in the dot plots show the number of hours volunteers work each week for two different organizations.

X 0 .0

0 .5

1 .0

1 .5

2 .0

X X

X X X X X X X

2 .5

3 .0

X X X X X X 3 .5

X X X X

X 0

X X X

X X X 2

X

X 4

X

X X

6

X X 8

10

12

14

X

X

12

14

Organization A (h)

4 .0

Class A

X X

0 .0

0 .5

1 .0

1 .5

X X X

X X X

X X

X

2 .0

2 .5

3 .0

3 .5

4 .0

Class B

3. Find the measures of center and measures of variability of the two sets of data.

4. What conclusions can you draw from the two sets of data? Select all that apply. A. Students in Class A have higher grade point averages than students in class B. B. A student in Class A would most likely have a grade point average of 3.0. C. Class B’s data have less variability than Class A’s data. D. The mean and median of Class A’s data are the same. E. The mean and median of Class B’s data are the same.

32 Unit 2: Statistics

X X

X

X X X

0

2

4

6

X X

X 8

X X

X X 10

X X

Organization B (h)

5. Find the measures of center and measures of variability of the two sets of data.

6. What conclusions can you draw from the two sets of data? Select all that apply. A. Volunteers in Organization B spend more time volunteering than volunteers in Organization A. B. The median of Organization A’s data and Organization B’s data is the same. C. The range of Organization B’s data is greater than the range of Organization A’s data. D. The mean and median of Organization A’s data are the same. E. The mean and median of Organization B’s data are the same.

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X

X X

X X X

C A SA S PR ACT I C E 11. What conclusions can you draw from the two sets of data?

Look at the following information. Then answer questions 7 through 11.

A. Students in grade 6 appear to have heavier backpacks than students in grade 9.

The data in the dot plots show the weights of the backpacks of students in grade 6 and grade 9. X X X X X X X X X X X

X

B. Grade 6’s data and grade 9’s data appear to have about the same amount of variability. C. The mean and median of grade 6’s data are the same.

X X X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

D. The mean and median of grade 9’s data are the same.

Grade 6

X X X X X X X X X X X

X X

X

X

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Grade 9

7. What is the IQR for grade 6? A. 5

C. 11

B. 6

D. 12

8. What is the IQR for grade 9? A. 5

C. 11

B. 6

D. 12

12. The data in the dot plots show the number of hours employees at a department store worked during a non-holiday week and the number of hours they worked during a holiday week. What conclusions can you draw from the two sets of data? X X X

X

X X X X X

18

22

24

20

26

X X X X X X X X X X 28

C. 11

B. 6

D. 14

10. What is the median value for grade 9? A. 5

C. 11

B. 6

D. 14

32

34

36

38

X 40

42

44

Number of hours Worked During a Non-Holiday Week

X X X X X X X X X X X X X X X X X X X X X X X X X

9. What is the median value for grade 6? A. 5

30

X X X X X

18

20

22

24

26

28

30

32

34

36

38

40

42

44

Number of Hours Worked During a Holiday Week

A. Employees appear to work more hours during a non-holiday week. B. An employee will likely work at least 30 hours during a holiday week. C. The mean and median of the hours worked during a holiday week are the same. D. The mean and median of the hours worked during a non-holiday week are the same.

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TEST TIP: When comparing two sets of data, first analyze the shape of the plot. The shape of the plot tells if the data are skewed right or skewed left. Skewed data will impact the best measure of center or variability.

Unit 2: Statistics

33

GOALS Level

ITEM #7309

C

Math 4 MEASUREMENT AND RATES, STATISTICS, AND PROBABILITY

for LEVEL C MATH 1 Rational Numbers, Number Sense, Ratios, Proportions, and Percents MATH 2 Patterns and Expressions, Equations and Inequalities, and Functions MATH 3 Lines, Angles, Triangles, Transformations, and Formulas MATH 4 Measurement and Rates, Statistics, and Probability

ISBN 978-0-88336-137-5

Syracuse, New York 800.448.8878 www.newreaderspress.com

9 780883 361375

CASAS

®