2-5 Statistics & Analyzing Data

8/14/2019 2-5 Statistics & Analyzing Data

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2-5 Statistics &

Analyzing Data


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Lesson 2-1 Rational Numbers on the Number Line

Lesson 2-2 Adding and Subtracting Rational Numbers

Lesson 2-3 Multiplying Rational Numbers

Lesson 2-4 Dividing Rational Numbers

Lesson 2-5 Statistics: Displaying and Analyzing Data

Lesson 2-6 Probability: Simple Probability and Odds

Lesson 2-7 Square Roots and Real Numbers
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Example 1 Identify Coordinates on a Number LineExample 2 Graph Numbers on a Number Line

Example 3 Absolute Value of Rational Numbers

Example 4 Expressions with Absolute Value


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Name the coordinates of the points graphed on thenumber line.

The dots indicate each point on the graph.

Answer: The coordinates are {9,7,6,3}.


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Name the coordinates of the points graphed on thenumber line.

The bold arrow on the graph indicates that the graphcontinues infinitely in that direction.

Answer: The coordinates are {11, 12, 13, 14, }.


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Graph .

Answer:


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Graph {1.5, 0, 1.5, }.

Answer:


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Graph {integers less than6 or greater than orequal to 1}.

Answer:


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Graph each set of numbers.

a. {5, 2, 3, 5}

b.

Answer:

Answer:


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c. {integers less than or equal to2 or greater than 4}

Answer:


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Find .

unit from 0 in the negative direction.

Answer:


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0.25 is 0.25 unit from 0 in the positive direction.

Answer:

Find .


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Find each absolute value.

a.

b.

Answer:

Answer: 6.3


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Replace y with 12.

Simplify.Answer:


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Answer: 10


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Example 1Use a Number Line to Add Rational Numb

Example 2 Add Rational Numbers

Example 3

Subtract Rational Numbers to Solve a Pro


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Use a number line to find .

+8

5

Step 1 Draw an arrow from 0 to 8.

Step 2 Then draw a second arrow 5 units to the left torepresent adding5.

Step 3 The second arrow ends at the sum 3.

Answer:


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Use a number line to find .

14

Step 1 Draw an arrow from 0 to1.

Step 2 Draw a second arrow 4 units to the left.

Step 3 The second arrow ends at the sum5.

Answer:


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Use a number line to find each sum.

a.

b.

Answer:3

Answer:9


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Find .

Answer:

Subtract the lesserabsolute value from the

greater absolute value.

Since the number withthegreater absolute valueis

14,the sum is negative.


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Find .

Both numbers are

negative, so the sumis negative.

Answer:


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Find each sum.

a.

b.

Answer:17

Answer:


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Stocks In the past year, a publishing companys stockwent from $52.08 per share to $70.87 per share. Findthe change in the price of the stock.

Explore The stock price began at $52.08 per share andended at $70.87. You need to determine the

change in price for the year.

Plan Subtract to find the change in price.

ending price minus beginning price

52.0870.87


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SolveTo subtract 52.08,add its inverse.

Subtract theabsolute values.

Theabsolute valueof70.87 is greater,

so the resultispositive.


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Answer:The price of the stock changed by $18.79.

Examine The problem asks for the change in a stocksprice in the past year. Since the change in price

was positive, the price increased. This makessense since the ending price is more than thebeginning price.


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Stocks The stock in a company went from $46.98 to$35.09 over a one-month period. Find the change inprice for the stock.

Answer:$11.89


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Example 1 Multiply IntegersExample 2 Simplify Expressions

Example 3 Multiply Rational Numbers

Example 4 Multiply Rational Numbers to Solve a Probl

Example 5 Evaluate Expressions


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Find (10)(11).

different signs negative product

Answer:110


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Find each product.

a. (4)(8)

b. (6)(12)

Answer:32

Answer: 72


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Simplify the expression

Associative Property ()

Substitution

Distributive Property

Simplify.Answer:


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Answer: 3x

Simplify the expression


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Find

same signs positive product

Answer:


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Find

Answer:


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Stocks The value of a companys stock dropped by$1.25 per share. By what amount did the total value ofthe companys stock change if the company hasissued 500,000 shares of stock?

To find the change in the total value of the companysstock, multiply the price lost per share by the numberof shares.

different signs negative product

Answer: The total value of the companys stockchanged by $625,000.


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Substitution

Answer: same signs positive product


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Answer:


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Example 1 Divide IntegersExample 2 Simplify Before Dividing

Example 3 Divide Rational Numbers

Example 4 Divide Rational Numbers to Solve a Proble

Example 5 Simplify Algebraic Expressions

Example 6 Evaluate Algebraic Expressions


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Find .

Answer: positive quotient


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Find .

Divide.

negative quotientAnswer:


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Find each quotient.

a.

b.

Answer: 20

Answer:15


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Simplify

Simplify the numerator first.

Multiply.

Answer: different signs negative quotient


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Simplify

Answer: 3


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Find .

Answer: Use a calculator.

different signs

negative quotient


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Find .

Multiply by the reciprocal

of

Answer: same signs positive quotient


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Find each quotient.

a.

b.

Answer: 15.3

Answer:


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Baseball The perimeter of a square baseball diamondis 360 feet. Find the length of one side of the diamond.

To find the length of one side, divide the perimeter by thenumber of sides.

same signs positive quotient

Answer: The length of one side is 90 feet.


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The perimeter of a triangular building is 450 feet. Findthe length of each side.

Answer: 150 feet


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Simplify

The fraction bar

indicates division.

Multiply by

the reciprocal of13.


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Answer: Simplify.

Distributive Property.


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Answer:

Simplify


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Answer: Use a calculator.different signs negative quotient

Evaluate if w = 2, x = 9.1 and y = 4.

Replace w with 2, x with9.1and y with 4.

Find the numerator anddenominator separately.


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Evaluate if s = 2.3, t= 5 and u = 4.

Answer:14.375


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Example 1 Create a Line PlotExample 2 Use a Line Plot to Solve a Problem

Example 3 Create a Stem-and-Leaf Plot

Example 4 Back-to-Back Stem-and-Leaf Plot

Example 5 Analyze Data

Example 6Determine the Best Measure of Central Te


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Draw a line plot for the data.

11 2 10 2 7 2 7 4 9 0 6 9 7 2 0 4 10 7 6 9

Step 1 The values of the data range from2 to 11, soconstruct a number line containing these values.

Step 2 Then place an a number for eachtime it occurs.


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Draw a line plot for the data.

3 5 7 6 0 4 6 4 7 0 0 2 3 7

Answer:


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Traffic The highway patrol did a radar survey of thespeeds of cars along a stretch of highway for 1minute. The speeds (in miles per hour) of the 20cars that passed are listed below.

72 70 72 74 68 69 70 72 74 7579 75 74 72 70 64 69 66 68 67

Make a line plot of the data.

The lowest value is 64 and the highest value is 79, so use

a scale that includes those values. Place an above eachvalue for each occurrence.


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Answer:


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Which speed occurs the most frequently?

Answer: Looking at the line plot, we can easily see that72 miles per hour occurs most frequently.


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Family Size Students in Mrs. Barretts class listed the

number of family members in their households below.

6 4 8 3 3 5 4 4 3 5 5 2 5 6 3 5 6 2 4 4 4

a. Make a line plot of the data.

b. Which family size occurs the most frequently?

Answer:

Answer: 4


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Use the data below to make a stem-and-leaf plot.

85 115 126 92 104 107 78 131 114 9285 116 100 121 123 131 88 97 99 11679 90 110 129 108 93 84 75 70 132

The greatest common place value is tens, so the digits inthe tens place are the stems.


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Stem7

8

9

1011

12

13

Leaf0 5 8 9

4 5 5 8

0 2 2 3 7 9

0 4 7 80 4 5 6 6

1 3 6 9

1 1 2

Answer:


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Use the data below to make a stem-and-leaf plot.

3 5 7 11 10 15 21 1113 25 32 37 21 10 12

Stem0

1

2

3

Leaf3 5 7

0 0 1 1 2 3 5

1 1 5

2 7

Answer:


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Weather Monique wants to compare the monthly

average high temperatures of Dallas and Atlantabefore she decides to which city she wants to move.The table shows the monthly high temperatures (F)for both cities.

50 55 64 72

75 85 88 87

81 72 63 54

54 59 68 77

83 91 95 95

87 78 66 57

AtlantaDallas

Monthly AverageHigh Temperature


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Make a stem-and-leaf plot to compare the data.

To compare the data we can use a back-to-back stem-and-leaf plot. Since the data represent similarmeasurements, the plot will share a common stem.

Answer: Dallas Stem Atlanta9 7 4 5 0 4 5

8 6 6 3 4

8 7 7 2 2 5

7 3 8 1 5 7 85 5 1 9


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What is the difference between the highest average

temperatures in each city?

Answer: 95 88 or7


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Which city has higher average temperatures?

Answer: Looking at the temperatures of 80 and above,we can see that Dallas has a higher number of averagetemperatures above 80.


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Ms. Smith wants to compare the final grades for

two of her classes. The table shows the scores forboth classes.

Class BClass A

91696471

85727576

83779892

5762518176999687


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a. Make a back-to-back stem-and-leaf plot to compare

the data.

Answer: Class A Stem Class B

1 5 7

4 6 2 96 5 1 7 2 6 7

7 1 8 3 5

8 6 2 9 1 9


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Which measure of central tendency best

represents the data?

Stem

4

56

7

8

Leaf

1 1 2 4 4 4 5 8

02 5 7

3 9

1


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Determine the mean, median, and mode.

The mean is about 5.5. Add the data and divide by 15.

The median is 4.8. The middle value is 4.8.

The mode is 4.4. The most frequent value is 4.4.

Answer: Either the median or the mode best representthe set of data since both measures are located in thecenter of the majority of the data. In this instance, the

mean is too high.


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Which measure of central tendency best

represents the data?

Answer: The mean is about 2.9. The median is 2.5. Themode is 1.1. Either the mean or median can be used torepresent the data. The mode is too low.

Stem

1

23

4

5

Leaf

0 1 1 5 6 8

3 7 82

6

4 5 9


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Politics The number of electoral college votes for the

12 most populous states in the 2000 Presidentialelection are listed below. Which measure of centraltendency best represents the data?

21 22 18 23 15 2514 32 13 33 13 54

The mean is about 23.6. Add the data and divide by 12.

The median is 21.5. The middle value is 21.5.

The mode is 13. The most frequent value is 13.

Answer: Either the mean or median can be used to bestrepresent the data. The mode is too low.


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The number of points scored by the basketball team

during each game in the season is listed below. Whichmeasure of central tendency best represents the data?

48 45 52 63 59 64 67 72 58

51 81 62 73 68 82 73 70 65

Answer: Either the mean or the median can be used tobest represent the data. The mode is too high.


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Example 1 Find Probabilities of Simple Events

Example 2 Odds of an Event

Example 3 Odds Against an Event

Example 4 Probability and Odds


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Find the probability of rolling a number greater

than 2 on a die.

There are six possible outcomes. Four of the outcomesare favorable. That is, four of the six outcomes arenumbers greater than two.

Sample space: 1,2,3,4,5,6

4 numbersgreater than 2

6 possible

outcomes

Answer:


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A class contains 6 students with black hair, 8 with

brown hair, 4 with blonde hair, and 2 with red hair.Find P(black).

Thereare 6 students with black hair and 20 total students.

number of favorable outcomes

number of possible outcomes

Answer: The probability of selecting a student with black

hair is

Simplify.


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brown hair, 4 with blonde hair, and 2 with red hair.Find P(red or brown).

Thereare 2 students with red hair and 8 students with

brown hair. So there are 2 + 8 or10 students with redor brownhair.

number of favorable outcomes

number of possible outcomes

Simplify.


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Answer: The probability of selecting a student with red

or brown hair is


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brown hair, 4 with blonde hair, and 2 with red hair.Find P(not blonde).

There are 6 + 8 + 2 or16 students who do not haveblonde hair.

number of favorable outcomesnumber of possible outcomes

Simplify.

Answer: The probability of selecting a student who does

not have blonde hair is


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a. Find the probability of rolling a

number less than 3 on a die.

b. A gumball machine contains 40 red gumballs, 30green gumballs, 50 yellows gumballs, and 40 bluegumballs. Find P(red).

Answer:

Answer:


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c. A gumball machine contains 40 red gumballs, 30

green gumballs, 50 yellows gumballs, and 40 bluegumballs. Find P(green or yellow).

d. A gumball machine contains 40 red gumballs, 30green gumballs, 50 yellows gumballs, and 40 bluegumballs. Find P(not blue).

Answer:

Answer:


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Find the odds of rolling a number greater than 2.

There are six possible outcomes, 4 are successes and2 are failures.

Sample space: 1,2,3,4,5,6

4 numbers

greater than 2

2 numbersless than orequal to 2

Answer: The odds of rolling a number greater than


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Find the odds of rolling a number less than 4.

Answer:


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A card is selected at random from a standard

deck of 52 cards. What are the odds againstselecting a 2 or 3?

There are four 2s and four 3s in a deck of cards, andthere are 52 4 4 or44 cards that are not a 2 or a 3.

number of ways notto picka 2 or3

number of ways to picka 2 or3

Answer: The odds against selecting a 2 or 3 are 11:2.


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A card is selected at random from a standard

deck of 52 cards. What are the odds againstselecting a 5, 6, or 7?

Answer: 10:3


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Travel Melvin is waiting to

board a flight to Washington,D.C. According to the airline, the flight he is waiting foris on time 80% of the times it flies. What are the oddsthat the plane will be on time?

The probability that the plane will be on time is 80%, sothe probability that it will not be on time is 20%.

odds of the plane being on time

Answer: The odds that the plane will be on time are 4:1.


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If the probability that it will snow this weekend is 70%,

what are the odds that it will snow?

Answer: 7:3


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Example 1 Find Square Roots

Example 2 Classify Real Numbers

Example 3 Graph Real Numbers

Example 4 Compare Real Numbers

Example 5 Order Real Numbers

Example 6 Rational Approximation


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Find .

represents the positive and negative square

roots of


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Answer:


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Find .

represents the positive square root of0.0144.

Answer:


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Find each square root.

a.

b. Answer: 0.6

Answer:


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Name the set or sets of numbers to which

belongs.

Answer: Because , which is

neither a repeating nor terminating decimal, this numberis irrational.


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belongs.

Answer: Because 1 and 6 are integers and

, whichis a repeating decimal,the number is a rationalnumber.


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belongs.

Answer: Because this number is a

natural number, a whole number, an integer anda rational number.


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Name the set or sets of numbers to which327

belongs.

Answer: This number is an integer and a

rational number.


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Name the set or sets of numbers to which each real

number belongs.

a.

b.

c.

d.

Answer: rationals

Answer: naturals, whole, integers,rationals

Answer: irrationals

Answer: integers, rationals


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Graph .

The heavy arrow indicates that all numbers to the left of 8are included in the graph. The dot at 8 indicates that 8 isincluded in the graph.


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The heavy arrow indicates that all the points to the right of5 are included in the graph. The circle at5 indicatesthat5 is notincluded in the graph.

Graph .


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Graph each solution set.

a.

b.

Answer:

Answer:


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Replace the with , or= to make thesentence true.

Since the numbers are equal.

Answer:


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Replace the with , or= to make thesentence true.

Answer:


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Replace each with , or= to make eachsentence true.

a.

b.

Answer:

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