Frequency Tables, Line Plots, and Histograms

Pre-AlgebraPre-Algebra

A survey asked 22 students how many hours of TV

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

in a frequency table. Then make a line plot.

Lesson 12-1

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

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

Additional Examples

Number Tally Frequency

List the numbers of hours in order.

1234

Count the tally marks andrecord the frequency.

3766

Use a tally mark for each result.

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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.

Lesson 12-1

“How many cases are you trying?”

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


Additional Examples


(continued)

Lesson 12-1

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


Additional Examples


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

Lesson 12-2

Additional Examples

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


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.


Lesson 12-2

(continued)

Additional Examples


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.

Lesson 12-2

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

Additional Examples



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

Lesson 12-2

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.

Additional Examples



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?

Lesson 12-2

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.

Additional Examples


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.

Lesson 12-3

Additional Examples



(continued)

Lesson 12-3

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.

Additional Examples



Study the graphs below. Which graph gives the impression of

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

Lesson 12-3

Additional Examples



(continued)

Lesson 12-3

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.

Additional Examples



What makes the graph misleading? Explain.

Lesson 12-3

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

Additional Examples


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.

Lesson 12-4

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

Additional Examples



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

digit is odd and the second digit is even?

Lesson 12-4

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

Additional Examples



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.

Lesson 12-4

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

Additional Examples



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?

Lesson 12-4

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

Additional Examples


Independent and Dependent EventsIndependent and Dependent Events

Lesson 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.

Additional Examples



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?

Lesson 12-5

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.

Additional Examples



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?

Lesson 12-5

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.

Additional Examples


Permutations and CombinationsPermutations and Combinations

Find the number of permutations possible for the

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

Lesson 12-6

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.

Additional Examples



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

students for a photograph?

Lesson 12-6

7 students Choose 3.

7P3= 7 • 6 • 5 = 210 Simplify

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

Additional Examples



In how many ways can you choose two states from the

table when you write reports about the areas of states?

Lesson 12-6

State Area (mi2)Alabama ColoradoMaine Oregon Texas

50,750103,72930,86596,003

261,914

Make an organized list of all the combinations.

Additional Examples



(continued)

Lesson 12-6

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.

Additional Examples



How many different pizzas can you make if you can

choose exactly 5 toppings from 9 that are available?

Lesson 12-6

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

Additional Examples



Tell which type of arrangement—permutations or

combinations—each problem involves. Explain.

Lesson 12-6

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.

Additional Examples


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.

Lesson 12-7

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

Additional Examples


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).

Lesson 12-7

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

Additional Examples


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.

Lesson 12-8

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.

Additional Examples


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.

Lesson 12-8

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.

Additional Examples


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?

Lesson 12-9

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.

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

Additional Examples


(continued)

Lesson 12-9

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


Additional Examples


(continued)

Lesson 12-9

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

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Additional Examples

Documents

Frequency Tables, Line Plots, and Histograms