12-1 frequency tables, line plots, and histograms pre-algebra lesson 12-1 formulate a set of five...

12-1

Frequency Tables, Line Plots, and HistogramsFrequency Tables, Line Plots, and HistogramsPRE-ALGEBRA LESSON 12-1PRE-ALGEBRA LESSON 12-1

Formulate a set of five different numbers whose median is 95 and whose mean is 100.

90, 92, 95, 110, 113

Sample answer

Frequency Tables, Line Plots, and HistogramsFrequency Tables, Line Plots, and HistogramsPRE-ALGEBRA LESSON 12-1PRE-ALGEBRA LESSON 12-1

(For help, go to Lesson 3-3.)

Find the median and mode of each data set.

1. 6, 9, 9, 5, 9 2. 73, 78, 77, 73, 79

3. 300, 100, 200, 150, 300 4. 3, 5, 7, 9, 3, 4, 6, 3, 7

Check Skills You’ll Need

12-1

PRE-ALGEBRA LESSON 12-1PRE-ALGEBRA LESSON 12-1

Solutions

1. 5, 6, 9, 9, 9 2. 73, 73, 77, 78, 79median = 9 median = 77mode = 9 mode = 73

3. 100, 150, 200, 300, 300 4. 3, 3, 3, 4, 5, 6, 7, 7, 9median = 200 median = 5mode = 300 mode = 3

Frequency Tables, Line Plots, and HistogramsFrequency Tables, Line Plots, and Histograms

12-1

A survey asked 22 students how many hours of TV

they watched daily. The results are shown. Display the data

in a frequency table.


1 3 4 3 1 1 2 3 4 1 32 2 1 3 2 1 2 3 2 4 3

Number Tally Frequency

List the numbers of hours in order.

1

2

3

4

Count the tally marks andrecord the frequency.

3

7

6

6

Use a tally mark for each result.

|||

|||| |

|||| |

|||| ||


Quick Check

12-1

Twenty-one judges were asked how many cases they were trying

on Monday. The frequency table below shows their responses. Display the

data in a line plot. Then find the range.


“How many cases are you trying?”

Number Frequency 0 3 1 5 2 4 3 5 4 4


12-1

(continued)


The greatest value in the data set is 4 and the least value is 0. So the range is 4 – 0, or 4.

For a line plot, follow these steps 1 , 2 , and 3 .

3 Write a title that describes the data.Cases Tried by Judges

2 Mark an x for each response.

x xx x x x

x x x x xx x x x xx x x x x

1 Draw a number line with the choices below it.

0 1 2 3 4


Quick Check

12-1


Solve.

1. Maria surveyed friends to find out how many pets each one has. The responses are below. Display the data in a frequency table.0, 1, 1, 2, 3, 0, 1, 1, 3, 2, 1, 2, 1, 0, 1, 0, 0

2. Here are the numbers of books students read in the last month: 5, 4, 0, 12, 4, 5, 4, 3, 10, 5, 12, 3, 5, 7, 3, 10, 5, 0, 6. Display the data in a line plot. Then find the range.

Pets Tally Frequency0 IIII 51 IIII II 72 III 33 II 2

range: 12


12-1

Box-and-Whisker PlotsBox-and-Whisker PlotsPRE-ALGEBRA LESSON 12-2PRE-ALGEBRA LESSON 12-2

12-2

If the population of the United States is approximately 250 million and the median age is 33 yr, roughly how many people in the United States are younger than 33 years old?

125 million



Find each median.

1. 12, 10, 11, 7, 9, 8, 10, 5

2. 4.5, 3.2, 6.3, 5.2, 5, 4.8, 6, 3.9

3. 55, 53, 67, 52, 50, 49, 51, 52, 52, 52

4. 101, 100, 100, 105, 102, 101


12-2


Solutions

1. 5, 7, 8, 9, 10, 10, 11, 12

median = = 9.5

2. 3.2, 3.9, 4.5, 4.8, 5, 5.2, 6, 6.3

median = = 4.9

3. 49, 50, 51, 52, 52, 52, 52, 53, 55, 67

median = = 52

4. 100, 100, 101, 101, 102, 105

median = = 101

9 + 102

4.8 + 52

52 + 522

101 + 1012

12-2

Box-and-Whisker PlotsBox-and-Whisker Plots

The data below represent the wingspans in

centimeters of captured birds. Make a box-and-whisker plot.

61 35 61 22 33 29 40 62 21 49 72 75 28 21 54


Step 1 Arrange the data in order from least to greatest. Find the median.21 21 22 28 29 33 35 40 49 54 61 61 62 72 75

Step 2 Find the lower quartile and upper quartile, which are the medians of the lower and upper halves.21 21 22 28 29 33 35 40 49 54 61 61 62 72 75lower quartile = 28upper quartile = 61

12-2

Step 3 Draw a number line.

Mark the least and greatest values, the median, and the quartiles.

Draw a box from the first to the third quartiles.

Mark the median with a vertical segment.

Draw whiskers from the box to the least and greatest values.


(continued)

Quick Check

12-2

Draw a number line for both sets of data. Use the range of data points to choose a scale.

Draw the second box-and-whisker plot below the first one.


Use box-and-whisker plots to compare test scores from

two math classes.


Class A: 92, 84, 76, 68, 90, 67, 82, 71, 79, 85, 79

Class B: 78, 93, 81, 98, 69, 95, 74, 87, 81, 75, 83

Quick Check

12-2


Describe the data in the box-and-whisker plot.


The lowest score is 55 and the highest is 85.

Half of the scores are at or between 66 and 80 and thus within 10 points of the median, 76.

One fourth of the scores are at or below 66 and one fourth of the scores are at or above 80.

Quick Check

12-2


The plots below compare the percents of students who

were eligible to those who participated in extracurricular activities

in one school from 1992 to 2002. What conclusions can you draw?


About 95% of the students were eligible to participate in extracurricular activities. Around 60% of the students did participate.A little less than two thirds of the eligible students participated in extracurricular activities. Quick Check

12-2


Solve.

1. Use the data to make a box-and-whisker plot. Student heights (in.) are: 60, 66, 59, 67, 68, 63, 62, 61, 69, 64, 61.

a. What is the median height?

b. Between what heights do 50% of the students fall?

63 in.

between 61 in. and 67 in.

12-2


2. The box-and-whisker plots below compare prices for the same items at Mary’s Discount Store and Ed’s Clothing. What conclusions can you draw?

Prices at the discount store are more tightly grouped around the median price of $25. Half the items cost from $18 to $45. For less expensive items, there is not much difference in the prices at the two stores. For more expensive items, the discount store offers lower prices.

12-2

Using Graphs to PersuadeUsing Graphs to PersuadePRE-ALGEBRA LESSON 12-3PRE-ALGEBRA LESSON 12-3

12-3

Use estimation to create a circle graph. Twenty percent of the students surveyed always prefer a salad. Thirty percent always prefer a hamburger. The rest prefer a hamburger sometimes and a salad sometimes.

Check students’ graphs. The graphs should show 20% (72°) salad, 30% (108°) hamburger, and 50% (180°) salad or hamburger.



Find the slope of AB in each graph.

1. 2.


12-3


Solutions

1. 2.12

12

12-3

Using Graphs to PersuadeUsing Graphs to Persuade

Which title would be more appropriate for the graph

below: “Texas Overwhelms California” or “Areas of California and

Texas”? Explain.


12-3


(continued)


Because of the break in the vertical axis, the bar for Texas appears to be more than six times the height of the bar for California.

Actually, the area of Texas is about 267,000 mi2, which is not even two times the area of California, which is about 159,000 mi2.

The title “Texas Overwhelms California” could be misleading. “Areas of Texas and California” better describes the information in the graph.

Quick Check

12-3


Study the graphs below. Which graph gives the impression of

a sharper increase in rainfall from March to April? Explain.


12-3


(continued)


In the second graph, the months are closer together and the rainfall amounts are farther apart than in the first graph.

Thus the line appears to climb more rapidly from March to April in the second graph.

Quick Check

12-3


What makes the graph misleading? Explain.


The “cake” on the right has much more than two times the area of the cake on the left.

Quick Check

12-3


Solve.

1. Name two ways you can use graphs to give different impressions of the same data.

2. What makes this graph misleading? Explain.

Answers may vary. Sample: Use breaks in scales; vary the size of scale intervals.

Both dimensions of the “photo” on the right were changed. This gives the impression that the difference in costs is much greater than it really is.

12-3

Counting Outcomes and Theoretical ProbabilityCounting Outcomes and Theoretical ProbabilityPRE-ALGEBRA LESSON 12-4PRE-ALGEBRA LESSON 12-4

12-4

If you roll two standard number cubes, what is the probability that the sum will be 1? 11? Less than 4?

118

0; 112;



A bag has 5 blue (B) chips, 4 red (R) chips, and 3 tan (T) chips. Find each probability for choosing a chip at random from the bag.

1. P(R) 2. P(not R) 3. P(B)

4. P(R or B) 5. P(T) 6. P(B or T)


12-4


Solutions

1.

2.

3.

4.

5.

6.

favorable outcomesall possible outcomes =

drawing a blue chip12 =

512

13

412


drawing a red chip12 = =


drawing a chip that is not red12 =

812 =

23


drawing a red or blue chip12 =

912 =

34


drawing a tan chip12 =

312 =

14


drawing a blue or tan chip12 =

812 =

23

12-4

Counting Outcomes and Theoretical ProbabilityCounting Outcomes and Theoretical Probability

The school cafeteria sells sandwiches for which you can choose

one item from each of the following categories: two breads (wheat or

white), two meats (ham or turkey), and two condiments (mayonnaise or

mustard). Draw a tree diagram to find the number of sandwich choices.


There are 8 possible sandwich choices.

mayonnaise

Each branch of the “tree” represents one choice—for example, wheat-ham-mayonnaise.

wheat

white

ham

turkey

ham

turkey

mayonnaisemustardmayonnaisemustardmayonnaisemustard

mustard

Quick Check

12-4


How many two-digit numbers can be formed for which the first

digit is odd and the second digit is even?


There are 25 possible two-digit numbers in which the first digit is odd and the second digit is even.

5 • 5 = 25

first digit,possible choices

second digit,possible choices

numbers,possible choices

Quick Check

12-4


Use a tree diagram to show the sample space for guessing

right or wrong on two true-false questions. Then find the probability of

guessing correctly on both questions.


The tree diagram shows there are four possible outcomes, one of which is guessing correctly on both questions.

P(event) = Use the probability formula.number of favorable outcomesnumber of possible outcomes

The probability of guessing correctly on two true/false questions is .14

rightright

wrong

right

wrongwrong

=14

Quick Check

12-4


In some state lotteries, the winning number is made up of five

digits chosen at random. Suppose a player buys 5 tickets with different

numbers. What is the probability that the player has a winning number?


First find the number of possible outcomes. For each digit, there are 10 possible outcomes, 0 through 9.

1st digitoutcomes

10

2nd digitoutcomes

10

3rd digitoutcomes

10

5th digitoutcomes

10

4th digitoutcomes

10

totaloutcomes= 100,000• • • •

Then find the probability when there are five favorable outcomes.

P(winning number) = =number of favorable outcomesnumber of possible outcomes

5100,000

5100,000

The probability is , or .120,000 Quick Check

12-4


Use the following information for Questions 1 and 2. In a game, a numbercube is tossed to determine the number of spaces to move, and a coin istossed to determine forward or backward movement.

1. How many possible outcomes are there?

2. What is the theoretical probability you will move four spaces?

3. How many different three-digit whole numbers are possible using the digits 1, 2, 3, 4, and 5?

125

12

16

12-4

Independent and Dependent EventsIndependent and Dependent EventsPRE-ALGEBRA LESSON 12-5PRE-ALGEBRA LESSON 12-5

12-5

3 4

Multiply. Express the answers in lowest terms.

3 b. 5 6

a. 12 15

c. 710

2 21

5 2 or 2 1

2 3 5

115



Multiply.

1. • 2. • 3. •

4. • 5. • 6. •

35

15

14

24

47

36

59

48

410

210

910

89


12-5


Solutions

1. 2. 3.

4. 5. 6. 518

325

2072 =

18

216 =

27

1242 =

45

7290 =

8100

225=

12-5


You roll a number cube once. Then you roll it again. What is the

probability that you get 5 on the first roll and a number less than 4 on the

second roll?

The probability of rolling 5 and then a number less than 4 is .112

P(5, then less than 4) = P(5) • P(less than 4)

= •16

36

336

112

= , or

P(5) =16 There is one 5 among 6 numbers on a number cube.

P(less than 4) =36 There are three numbers less than 4 on a number cube.

Quick Check

12-5

Independent and Dependent EventsIndependent and Dependent Events

Bluebonnets grow wild in the southwestern United States. Under the best conditions in the wild, each bluebonnet seed has a 20% probability of growing. Suppose you plant bluebonnet seeds in your garden and use a fertilizer that increases to 50% the probability that a seed will grow. If you select two seeds at random, what is the probability that both will grow in your garden?


P(two seeds grow) = P(a seed grows) • P(a seed grows)

The probability that two seeds will grow is 25%.

P(a seed grows) = 50%, or 0.50 Write the percent as a decimal.

= 0.50 • 0.50 Substitute.

= 0.25 Multiply.

= 25% Write 0.25 as a percent.

Quick Check

12-5

Independent and Dependent EventsIndependent and Dependent Events

Three girls and two boys volunteer to represent their class at a

school assembly. The teacher selects one name and then another from a

bag containing the five students’ names. What is the probability that both

representatives will be boys?


P(boy, then boy) = P(boy) • P(boy after boy)

The probability that both representatives will be boys is .110

220

110

= , or Simplify.

= •25

14

Substitute.

P(boy after boy) =14

If a boy’s name is drawn, one of the four remaining students is a boy.

P(boy) =25 Two of five students are boys.

Quick Check

12-5


Solve.

1. You roll a number cube once. Then you roll it again. What is the probability that you get 6 on the first roll and a number greater than

3 on the second roll?

2. Suppose there are three white marbles and three black marbles in a bag and you want to remove two marbles. What is the probability that you will select a white marble and then a black marble? Express your answer as a percent.

30%

112

12-5


Solve.

722

533

;

12-5

3. Each of five girls and seven boys wants to be one of the two announcers for a variety show. To be fair, a teacher puts the names of the twelve students in a hat and draws two. What is the probability that the teacher will draw the names of two boys? Of two girls?

Permutations and CombinationsPermutations and CombinationsPRE-ALGEBRA LESSON 12-6PRE-ALGEBRA LESSON 12-6

12-6

Bianca’s family needs to choose exterior paint for their new house. The wall colors are white, green, and beige. The trim colors are white, green, blue, and cocoa. How many combinations of wall color and trim are possible?

12



Use the Counting Principle to find the number of outcomes.

1. Roll 2 number cubes.

2. Choose three different letters.

3. Select a month and a day of the week.

4. Toss a coin 4 times.


12-6


Solutions

1. first number second number6 outcomes 6 outcomes

6 6 = 36

2. first letter second letter third letter(different from 1st) (different from 1st & 2nd)

26 choices 25 choices 24 choices

26 25 24

= 15,600

12-6


Solutions (continued)

3. month day12 choices 7 choices

12 7 = 84

4. first toss second toss third toss fourth toss2 outcomes 2 outcomes 2 outcomes 2 outcomes

2 2 2 2

= 16

12-6

Permutations and CombinationsPermutations and Combinations

Find the number of permutations possible for the

letters H, O, M, E, and S.


1st letter5 choices

5

2nd letter4 choices

4

3rd letter3 choices

3

5th letter1 choice

1

4th letter2 choices

2 = 120• • • •

There are 120 permutations of the letters H, O, M, E, and S.

Quick Check

12-6


In how many ways can you line up 3 students chosen from 7

students for a photograph?


7 students Choose 3.

7P3= 7 • 6 • 5 = 210 Simplify

You can line up 3 students from 7 in 210 ways.

Quick Check

12-6


In how many ways can you choose two states from the

table when you write reports about the areas of states?


State Area (mi2)Alabama ColoradoMaine Oregon Texas

50,750103,72930,86596,003

261,914

Make an organized list of all the combinations.

12-6


(continued)


AL, CO AL, ME AL, OR AL, TX Use abbreviations of each CO, ME CO, OR CO, TX state’s name. First, list all

ME, OR ME, TX pairs containing Alabama. OR, TX Continue until every pair

of states is listed.

There are ten ways to choose two states from a list of five.

Quick Check

12-6


How many different pizzas can you make if you can

choose exactly 5 toppings from 9 that are available?


9 toppings Choose 5.

9C5= 9P5

5P5

You can make 126 different pizzas.

= = 126 Simplify.9 • 81 • 7 • 62 • 51

51 • 41 • 31 • 21 • 1

Quick Check

12-6


Tell which type of arrangement—permutations or

combinations—each problem involves. Explain.


a. How many different groups of three vegetables could you choose from six different vegetables?

b. In how many different orders can you play 4 DVDs?

Combinations; the order of the vegetables selected does not matter.

Permutations; the order in which you play the DVDs matters.

Quick Check

12-6


Find the number of permutations or combinations.

1. Find the number of permutations of the last four digits in the phone number 555–1234.

2. In how many ways can you arrange six out of eight books on a shelf?

3. In how many ways can you choose three different items from a menu containing seven items?

Does each problem involve permutations or combinations?

4. How many different groups of four ice cream toppings can you choose from sixteen toppings?

5. In how many different ways can five students line up?

35

24

20,610

combinations

permutations

12-6

Experimental ProbabilityExperimental ProbabilityPRE-ALGEBRA LESSON 12-7PRE-ALGEBRA LESSON 12-7

12-7

There are 15 girls in a class of 27 students. Using lowest terms, whatfraction of the students are girls? What fraction are boys?

59

49

;



Write each decimal or fraction as a percent.

1. 0.8 2. 0.53 3. 0.625

4. 5. 6.35

58

712


12-7


Solutions

1. 0.8 = 80% 2. 0.53 = 53% 3. 0.625 = 62.5%

4. = 0.6 = 60% 5. = 0.583 = 58 % 6. = 0.625 = 62.5%35

58

712

13

12-7

Experimental ProbabilityExperimental Probability

A medical study tests a new medicine on 3,500 people. It

produces side effects for 1,715 people. Find the experimental

probability that the medicine will cause side effects.


The experimental probability that the medicine will cause side effects is 0.49, or 49%.

P(event) = number of times an event occurs

number of times an experiment is done

= = 0.491,7153,500

Quick Check

12-7

Experimental ProbabilityExperimental Probability

Simulate the correct guessing of answers on a multiple-choice test

where each problem has four answer choices (A, B, C, and D).


Use a 4-section spinner to simulate each guess. Mark the sections as 1, 2, 3, and 4. Let “1” represent a correct choice.

P(event) = = = number of times an event occurs

number of times an experiment is done15

1050

Here are the results of 50 trials.

22431 13431 43121 21243 3343432134 12224 42213 34424 32412

The experimental probability of guessing correctly is .15

Quick Check

12-7


Solve.

1. Jenny made 42 out of 70 free throws. What is the experimental probability that she will make her next free throw?

2. If 152 of 722 students would choose fish for lunch, what is the experimental probability that a student chosen at random will

choose fish?

about 21%

60%

12-7

Random Samples and SurveysRandom Samples and SurveysPRE-ALGEBRA LESSON 12-8PRE-ALGEBRA LESSON 12-8

12-8

Write and solve the proportion: The ratio of scouts to leaders is 25 to 1. If there are 125 scouts, how many leaders are there?

251

1255

= ; 5



Solve each proportion.

1. = 2. =

3. = 4. =

n450

832

n24,000

725

4100

n144,000

2250

n50,000


12-8


Solutions

1. = 2. =

32n = 3600 25n = 168,000

= =

n = 112.5 n = 6,720

3. = 4. =

250n = 100,000 100n = 576,000

= =

n = 400 n = 5,760

n450

832

n24,000

725

4100

n144,000

2250

n50,000

32n32

360032

25n25

168,00025

250n250

100,000250

100n100

576,000100

12-8

Random Samples and SurveysRandom Samples and Surveys

You want to find out how many people in the community

use computers on a daily basis. Tell whether each survey plan

describes a good sample. Explain.


a. Interview every tenth person leaving a computer store.

b. Interview people at random at the shopping center.

c. Interview every tenth student who arrives at school on a school bus.

This is not a good sample. People leaving a computer store are more likely to own computers.

This is a good sample. It is selected at random from the population you want to study.

This is not a good sample. This sample will be composed primarily of students, but the population you are investigating is the whole community.

Quick Check

12-8

Random Samples and SurveysRandom Samples and Surveys

From 20,000 calculators produced, a manufacturer takes a

random sample of 300 calculators. The sample has 2 defective calculators.

Estimate the total number of defective calculators.


Estimate: About 133 calculators are defective.

defective sample calculatorssample calculators

defective calculatorscalculators= Write a proportion.

2300

n20,000= Substitute.

2(20,000) = 300n Write cross products.

2(20,000)300

300n300= Divide each side by 300.

133 n Simplify.

Quick Check

12-8


Solve.

1. To find out the type of music the general population prefers, you survey people at random at a local art museum. Does the survey

plan describe a good sample?

2. Of 80 sweaters, 6 have flaws. Estimate how many of 38,000 sweaters have flaws.

2,850 sweaters

Not a good sample; it includes only people who visit the art museum.

12-8

Problem Solving Strategy: Simulate the ProblemProblem Solving Strategy: Simulate the ProblemPRE-ALGEBRA LESSON 12-9PRE-ALGEBRA LESSON 12-9

12-9

In a certain part of a state, only the letters J, K, L, M, and N can be used to form a 2-letter beginning for a license plate. How many different 2-letter beginnings are possible?

25 beginnings; 20 using two different letters and 5 using the same letter twice

Problem Solving Strategy: Simulate the ProblemProblem Solving Strategy: Simulate the ProblemPRE-ALGEBRA LESSON 12-9PRE-ALGEBRA LESSON 12-9


Create a frequency table showing the number of times each letter of the alphabet appears in the sentence below.

A simulation is a model of a real experience.


12-9


Solutions

Problem Solving Strategy: Simulate the ProblemProblem Solving Strategy: Simulate the Problem

12-9

A softball player has an average of getting a base hit 2

times in every 7 times at bat. What is an experimental probability

that she will get a base hit the next time she is at bat?


You can use a spinner to simulate the problem. Construct a spinner with seven congruent sections. Make five of the sections blue and two of them red. The blue sections represent not getting a base hit and the red sections represent getting a base hit. Each spin represents one time at bat.


12-9

(continued)


Use the results given in the table below. “B” stands for blue and “R” stands for red.

B B B B R R B B R BB R B B R B B B B RR B B R B B B B R BB B B B B B B B B BB B R B B B R B B BB B B R R B B B B RR B B B B B B B B BB B B B B B R R B BB R R B B R B B B BB B B B B R B B B R


12-9

(continued)


Make a frequency table.

An experimental probability that she gets a base hit the next time she is at bat is 0.22, or 22%.

Makes a Base Hit Doesn’t Make a Base Hit |||| |||| |||| |||| || |||| |||| |||| |||| |||| |||| |||| ||||

|||| |||| |||| |||| |||| |||| |||| |||


Quick Check

12-9


1. You and a friend play a game in which you each toss a coin. You score a point for each heads and your friend scores a point for each tails. The first person to score ten points wins. The score is 8 to 6 in your favor. Describe a simulation that completes the game and use it to find an experimental probability that your friend will win.

Check students’ work.


12-9

12-1 frequency tables, line plots, and histograms pre-algebra lesson 12-1 formulate a set of five...

Documents