M3PrentHall STAT

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Lesson Preview

Analyzing Data

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---What You'll learn (For help, go to Lesson 1-1.)

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OeJECTIVE

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To calculate measures ofcentral tendency

Todraw and interpret box-and-whisker plots

... And Why

To analyze a set ofwater temperatures, as in Example2

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Check Skills You'll Need

Order each set ofvalues from least to greatest. Then find the middle value. 1. 0.2 0.3 0.6 l.2 0.7 0.9 0.8 2. 11 23 15 17 21 18 21

3. 7.8 2.6 3.9 15.6 9.1 11.7 10.4 4. 76 89 80 82 86 84 86

New Vocabulary '. measures of central tendency • mean • median • mode • bimodal • quartiles • box-and-whisker plot • percentiles • outlier

OBJECTIVE

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Mea5ur~5

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of Central Tendency

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CheekU~dersta~di~g

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Find the mean, median, and mode for these values: 2.4,4.3, 3.7,3.9, 2.8,5.4,2.8. r~'

Key Co~cepts

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Read

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as"the mean ofx" or "xbar."

Statistics is the study of data analysis and interpretation. The mean, the median, and the mode are single, central values that help describe a set of data. They are called measures of central tendency.

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-Measures of Central Tendency

Measure Definition Example, using {1,2, 2, 3, 5, 5}

Mean sum of the data values 1+2+2+3+5+5 = 18=3

number of data values 6 6

Median middle value ormean of the 2; 3 =2,5 two middle values

Mode most frequently occurring value 2 and 5

A bimodal data set has two modes. If a data set has more than two modes, then the modes are probably not stat!tically useful. Ifno value occurs more frequently

than any other, then there isno mode.

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EXAMPLE Finding Measures of Central Tendency

Find the mean, median, and mode for these values: 98, 95, 99,97,89,92,97,62,90 .

- _ 98 + 95 + 99 + 97 + 89 + 92 + 97 + 62 + 90 _ 819 _ 91 Use the symbol

x

to

x - 9 - 9 - designate the mean.

62 89 90 92 95 97 97 98 99 Find the median and the mode by

M di

t

tM dt ordering the values numerically.

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• The mean is91,the median is95,and the mode is 97.

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Youcan use a graphing calculator tofind the measures of central tendency.

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EXAMPLE Real-World ~ Connection

Oceanography Find the mean, the median, andthe mode ofall the water

temperatures listed for the eastern coast ofthe Gulf ofMexico.

Gulf of Mexico Eastern Coast Water Temperatures (OF)

Use the STATfeature to enter the d~a as Ll inyour graphing calculator.

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.~: m i n (\~.!l ,ti~4~.;;t'~~

2:m a x ( ".'~ , ..•" .:' .. ~meanC .~".

4:median(', •.

5:sum( ",~.:.

6:prod( ; ;:~

7lstdDev('~_.'_{" .•~\}_>

Step 4 Use the

STATPLOT .feature to

access Plot 1. Choose the histogram, Ll, and

Frequency 1options.Then enter an appropriate viewing window.

Step 5 Graph the data. Use the TRACEfeature tomovethe cursor tothe highest

points of the graph.

On the screen, the mode appears as the minimum value for the cursor. The modes are 70 and 84.

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The modes both occur

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five times in the data .

• Themean is 73.65°F, the median is 75°F,and the modes are70°Fand 84°F.

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Check Understanding

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Find the mean, median, and mode for the water temperatures in Grand Isle,Louisiana.

Step 1

Step 2 Use the

LISTfeature toaccess the MATH menu. Find the mean.

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Lesson 12-3 Analyzing Data 649

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Step 3 Return to

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OBJECTIVE

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Box-and-Whisker Plots

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If you arrange data in increasing order, then the median divides the data set into

two equal parts. You can use the median of each of the parts to divide the set further, into four equal parts. The values separating the four parts are called

quartiles. Quartiles are shown below for the 12 water temperatures from Pensacola inExample 2.

Median of lower part (Q1) = 60.5

56 58 58

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63 65 71

t

74

Median of data set (Q2) = 72.5

Median of upper part (Q3) =83

78 82

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84 85 86

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Quartiles are sometimes called "hinges."

The values Ql, Q;,and Q3are the first, second, and third quartiles.

A box-and-whisker plot is a method of displaying data that uses quartiles to

form the center box and the minimum and maximum values to form the whiskers.

Minimum Q Q2 Q Maximum

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50 60 70 80 90

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EXAMPLE Making a Box-and-Whisker Plot

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Make a box-and-whisker plot for these values: 84,79,90,73,95,88,92,81,67.

Step 1 Find the quartile values, the minimum value, and the maximum value.

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67 73 79 81 84 88 90 92 95

t

Q2

=

median = 84

When the median is a value of the data set, it isremoved for the

calculation of Ql and Q3'

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67 73 79 81

t

Q _ 73 + 79 - 76

1 - 2

-88 90 92 95

t

Q _ 90 + 92 - 91

3 - 2

-The minimum value is

6r

and the maximum value is 95.

Step 2 Draw anumber line for the base of your box-and-whisker plot. Above

the number line, plot the three quartiles, the minimum value, and the

maximum value. I

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67 76 84 9195

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50 60 70 80 90 100 110

Step 3 Finish your box-and-whisker plot by drawing a box through Q1and Q3,a vertical line through the median, and line segments from the box outward to the minimum and maximum values.

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50 60 70 80 90 100 110

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Check

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Make abox-and-whisker plot for these values: 34,36,47,45,28,31,29,40.

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EXAMPLE Real-World

8

Connection

Oceanography Use a graphing calculator tofind the quartiles of the water

temperature data in Example 2.

Use the STAT PLOT feature toselect abox-and-whisker plot. Enter the window

values. Graph the box-and-whisker plot.

• Use the TRACE feature to find the quartiles: Ql =64.5, Q2 = 75,and Q3 = 83.5.

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Cneck Ut1derstat1dit1g

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Use thedata forjust the summer months, as shown in the graph below. Find the quartiles bygraphing the box-and-whisker plot.

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Gulf of Mexico Eastern Coast Water Temperatures (OF)

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..t-A percentile, isavalue that divides the range ofa data set into two parts such

that the part below the percentile contains agiven percent of the data. Ifanumber

x isat the 63rd percentile inadata set,then about 63% of the data are lessthan x.

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EXAMPLE Using Percentiles

Find the values at the 20th and 65th percentiles for the values below.

54-98 45 87 9864 21 61 71 82 93 65 62 98 87 24 65 97 31 47

Step 1 Order the values:

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Step 2

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21. ,24.31 4S 47 54·61 62 64 65 6S 71 8287I ' 87 93 97 98 98 98

Find the number ofvalues that fall below the 20th percentile and the number that fallbelow the 65th percentile.

Of the 20values, 20% should fallbelow the 20th percentile and 65% should fall below the 65th percentile.

20 . 20%

=

20 .0.20

=

4 20 .65%

=

20 . 0.65

=

_r:13

Since47 isgreater than 4values, 47is at the20thpercentile.

Since 87 isgreater than 13 values,

87 isat the 65th percentile .

The value at the20th percentile is47 and the value at the 65th percentile is 87.

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Find the value at eachpercentile forthe data inExample 5.

a. Oth percentile h.45th percentile c. 55th percentile

Lesson 12-3 Analyzing Data 651

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An outlier is an item of data with a value substantially different from the rest of

the items in the data set. Sometimes an outlier isan important part of the data.At other times it can represent a false reading. When you think an outlier has resulted from an error, you may remove it from the data set.

Identify an outlier for this set ofvalues:56 65 73 59 98 65 59.

Order the data.

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,EXAMPLE

Identifying

an Outlier

56 59 59 65 65 73 98

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8 25 Find the differences between adjacent values.

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a. Suppose the values in Example 6 are measurements of thewater temperature of

alake. Would you discard the outlier? Explain.

b. Suppose the data represent the number of customers ina small restaurant each night,duringone week.Would you discard the outlier? Explain.

Practice and Problem Solving

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For more practice, see Extra Practice,

Practice by Example

Examples 1 and 2

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(pages 648 and 649)

Find the mean, median, and mode of each set of values.

1>5"9 1 2 7 3 1 8 8 1 3 2. 307 309 323 304 390 398

3.'475 722 499 572 402 809 499 828 405 499 800 422 672 800

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Make a box-and-whisker plot for each set of values. 4. 12 11 15 12 19 20 19

5.'120 145 133 105 117 150

6.49 57.5 58 49.2 62 22.2 67 52.1 77 99.9 80 51.7 64

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Examples 3 and 4

(pages 650 and 651)

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7. Weather The table shows the high temperatures for one dayat different locations on the island of Maui, Hawaii. Make a box-and whisker plot of the data.

Example 5

(page 651)

Find the values at the 30th and 90th percentiles for each set of values.

8. 6283'5900 6381 6274 5700 5896 5972 6075 5993 5581

9. 7 12 3 14 17 20 5 3 17 4 13 2

15 9 15 18 16 9 1 6

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Example 6

(page 652)

Identify the outlier of each set of values.

10. 3.4 4.5 2.3 5.9 9.8 3.3 2.1 3.0 2.9

11. 17 21 19 10 15 19 14 0 11 16

652

Chapter 12 Probability and Statistics

High Temperatures on Maui

Kahului 88°F

Kihei 85°F

Lahaina 86°F

Hana 82°F

Haleakala 66°F

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12. a. What percent of the customers in the cartoon are exactly the median age?

b. Must one itemfrom adata set fallexactly atthemedian? Explain.

c. Can the company do anything about theshocking discovery? Explain.

~ 13. Meteorology On May 3,1999,59tornadoes hit Oklahoma in the largest

tornado outbreak everrecorded in the state.Sixteen of these were classified

as strong (F2 orF3) or violent (F4or F5).

"---- a. Make abox-and-whisker plot of the data.

b. Identify the outlier. Remove it horn the data set and make a revised

box-~ and-whisker plot.

~ c. Writing How does theremoval of theoutlier affect the box-and-whisker

plot? How does itaffect the median of the data set?

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Apply Your Skills

Major Tornadoes in Oklahoma, May 3, 1999

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Dilbert

IWllf1£. SHOCKING DISCOVERY WAS

lHAT 50'7. WERE

BELOW THE. MEDIAN AGE..

WHAT PERCENI ARE EXAC.TlY

THE I"\E.DIAN

AGE?

I'M PROPOSING TO STUDy THAT

IN PHASE.l"WO

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Identify the outlier of each set of values. Then describe how its value affects the

mean of the dat~.

14. 947 757 103 619 661 582 626 900 869 728 1001 596 515

15.87 104 381 215 174 199 233 186 142 228 9 53 117 129

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MARKET RESEARCH

INDICATES THAT 50%

OF YOUR CUSTOMERS AI\[ ABOVE ThE.

MEDIAN AGE..

For Exercises 16-18, use the set of values below.

1 1 11 1 1 2 3 5 8 13

21

34' 55 189 89 89 89 89 89

16. At what percentile is 1? 17. At what percentile is 34?

~8.Error Analysis A student claimsthat 89isat the 100th percentile. Explain

the student's error.

SOURCE:U.S. Geological SurveyNational Earthquake Information Center

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a. Find themean andthe median of the numbers of

annual earthquakes.

b.Do the data include an outlier that youshould discard? Explain.

c. Compare the box-and-whisker plots at the right. One shows the data above. The other shows

worldwide earthquake data from 1900 through

2000.What conclusions can you draw about

recent earthquakes? Justify your reasoning.

~. 19. Geology The table below shows thenumber ofmajor earthquakes (magnitude

7.0or greater) worldwide in the ten-year period from 1991 through 2000.

Major Earthquakes Worldwide (Magnitude 7.0 and Greater)

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

11 23 16 15 25 22 20 16 23 18

Xmin = 0

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Ymin =0

Ymax = 1

Yscl =1

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Challenge

Real-World ~. Connection

The "shot" in shot put refers

to cannonballs. For centuries,

soldiers used them in throwing contests.

Multiple Choice

20. Critical Thinking Which measure better represents a data set with several outliers-the mean or the median? Justify your answer.

~ 21. Track and Field The table shows the qualifying distances for the shot put

events for both men and women during the 1996 Olympics in Atlanta, Georgia.

1996 Olympic Qualifying Distances for Shot Put (meters)

20.43 20.42 19.81 19.61 19.57 19.40 19.12 19.07 19.05 19.01 18.98 18.48 18.39 18.37 18.21 13.02 NM NM

20.58 20.54 20.23 1.9.95 19.45· 19.39 19.39 19.37 18.96

18.69 18.67 18.53 18.29 18.23 18.22 17.98 17.29 16.51 19.93 ,19.08 19.04 18.92' 18.55 18.39 18.23 18.16 17.69 17.14_, 16.40 15.91' DNS 19.36 19.29 19.22 19.03 19.02

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18.61 18.56 18.55 17.48 16.92 16.49 15.28 13.74 DNS

NM

=

No Measure DNS

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Did Not Show

a. Identify and remove any outliers from the men's results and from the women's results.

b. Using the same number line base for both plots, make a box-and-whisker % plot for the men's results and another for the women's results.

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c. Writing Compare your box-and-whisker plots. Describe any conclusions you

can draw about Olympic-level male and female shot-putters.

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22. a. Government Make abox-and-whisker plot for the data from each of the

three types of elections shown in the table below.

Voter Turnout (percent of voting-age population)

1972 1976 1980 1984 1988 ·1992 1996 55.2 53.5 52.8 53.3 50.3 55.1- '49.0 50.7. 48.9 47.6 .47.8 44.9 ·50.8 . '45.8. 19'14 1978 1982 .1986 1990 1994 1998 35.9 34.9 38.0 . 33.5 ₃₃_.₁ _36.6 ₃₂_.₉

SOURCE: U.S. CensusBureau.Go towww.PHSchool.com foradata update.

Web Code:agg·2041

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b. Writing How does aPresidential election in the United States affect the

voter turnout rate in elections for the House of Representatives? Use your box-and-whisker plots to describe any effect that you see.

A person checked for e-mail four times each day.

The table shows

the number of new e-mails she received each time she checked during 5 days.

23. Which value is the greatest?

A. the mean number of e-mails for checks made on days 1-3 B. the mean number of e-mails for checks made on days 2-4 C.the mean number of e-mails for checks made on days 2-3

D.the median number of e-mails for checks made on days 3-4

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r~ke It to the

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Online lessonquiz at www.PHSchool.com

:.. .. .... Web Code: aga-1203

Short Response

Extended Response

24. Which statement(s) is (are) true?

'1. The mean number of e-mails for checks made over the 5 days was 5.3.

II. The mean number of e-mails for checks made on day 5 was 7.75.

III. The median number of e-mails for checks made over the 5 days was 6.5.

i=. lonly G. I and II only H. II and III only I. I, II, and III

25. Describe how you could find the scores at the 20th and 60th percentiles in a set of 80 scores.

26. Draw a box-and-whisker plot for this set of values: 123, 127, 127, 142,118,

131,137,125,131.

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Mixed Review

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Lesson 12-2 Of all the respondents to asurvey,59% are girls. Of the girls,61% read horror stories. Of the boys,49% read horror stories.

27. FindP(boy and reads horror stories). 28.Find P(reads horror stories).

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lesson 11-2 Isthe sequence arithmetic? If so,identify the common difference.

lesson 10-2

29. 16,7, ....:2,... _{30. 34}_,₅₁_,₆₈_, _... _31._2,2.2_,_{2.22, .}_._. _32. ₁_,_1,1_, _._..

Graph each equation.

33.y2 - x - 2y + 1= 0 34. x2 + 4x + 144y +4 = 0

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Instant self-check quiz online and on CD-ROM

In a poll,gymnasts were asked, "How many seconds long was your longest handstand on the balance beam?" Use the results below for Exercises 1 and 2.

Longest Handstand

0-2 '.3-5 6-10 11"':'2021-30 31-60 >60 Total

14 27 19 18 13 15 24 130

1.Graph the probability distribution. 2. FindP(6-30 seconds).

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~ 3. Writing Could the function P(x) = x 2" 2forx = 1,2,3, or 4,represent a probability distribution? Explain.

Use the table at the right to find each probability.

4. P(teacher

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teacher) 6. P(student

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no) 7. P(yes

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Did You Eat Breakfast Today?

Find the mean, median, and mode of each data set.

8.7 4 9 3 5 4 4 7 9 10 3 1 8

9. 1.2 2.1 4.6 2.5 9.7 6.2 2,6 2.4 3.1 3.8

10.Open-Ended Write a data set that includes an outlier. Make a box-and-whisker plot ofyour data setwith and without the outlier.

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What You'll learn

OBJECTIVE

V

To find the standard deviation of a set of values

To use standard deviation in rea l-world situations

OBJECTIVE

V

... And Why

To analyze energy demand. as in Example 3

OBJECTIVE

Standard Deviation

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Finding Standard Deviation

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Check Skills You

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(For help. go to Skills Handbook page 845.) Simplify each expression. If necessary, round to the nearest hundredth.

1 34.3 2 ~

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3.8.4.1.25 4.12 ~-~6_·~0~.5 _

S.~[(2- 6)2+.(7- 6)2+ (8- 6)2] .6.V~(4 - 3)2+ (5+ 3)2

New Vocabulary • measures of variation • range of a set of data • interquartile range • standard deviation • z-score

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Interactive lesson includes instant self-check, tutorials, and activities.

hwestigatiot1:

At1alyzit1g Vata Spread

1. a. Find the mean, the median, and the mode of each set of data.

I_,' Set 1 77 78 79 80 80 81 82 83

Set 2 20 60 70 80 80 90 100 140

Set 3 50 60 70 80 80 90 100 110

Set 4 20 30 40 80 80 120 130 140

b. Are the sets the same? Explain.

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2. Find the difference between the greatest and least values in ea2hs~t ~:- :-.

of data. What do these differences tell 'You about each set of data?

3. Find the quartiles of each set of data.

4. For each set, half of the data lie between 01and 03'The value

03 - 01gives you an idea of how the data are spread out. Find

03 - 01for each set of data» ,";" "

5. a. Give an example of two sets of data that are spread out differently, though the differences between their extreme values are the same. b. 0ive an example of tW9 sets of data that are spread out differently,

though theirvalues of 03 - 01are the same.

, 6. a. Summarize the similarities and differences among the four sets in

terms of central tendency and spread. Which is the most spread-out set? Which is the least? Explain.

b. Make box-and-whisker plots of the four sets of data. Do the plots.

support your conclusions from part (a)?

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Real-World ~\ Connection

By age 29,skier Paul Casey

Puckett had made the U.S.

Olympic teams of 1992, 1994,

1998,and 2002.

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Check Understanding

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The lower case Greek letter (J'corresponds to

the English letter s.

Key Concepts

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Need Help?

: 2:means summation.

Statisticians use several measures of variation todescribe how thedata inadata set are spread out.

The range of a set of data is the difference between the greatest and least values.The interquartile range isthe difference between the third and first quartiles.

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EXAMPLE Real-World ~~ Connection

Thirteen men qualified for the 2002 U.S.Men's Alpine Ski Team.Find therange and the interquartile range of their ages at the time of qualification: 27,28,29,23,

25,26,26,28,22,23,23,21,25.

greatest value - least value =29 - 21

=8

Find the range.

21 22 23 23 23

t

QJ = 23

Median

J

-25 25 26 26 27 28 28 29

t

Q3

=

27

!

28

=

27.5

Find the median.

Q3 - Ql = 27.5 - 23

= 4.5

Find the interquartile range.

•• Therange is 8 years.The inter quartile range is4.5years.

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a. Seventeen women qualified for the 2002 U.S. Women's Alpine Ski Team.Find the range and the interquartile range oftheir ages:24, 30, 29, 21, 22,22,28,21,16,

17,25,22,21,18,19,18,19.

b. Critical Thinking Can the variation, or spread, intwo setsofdata be different,

even though they have the same range? Give an example.

c. Can thevariation in two sets ofdata be different, even though they have the

same interquartile range? Give an example.

Another measure of variation isthe standard deviation, a measure ofhow each value in a data setvaries, or deviates, from the mean. The Greek letter (T (sigma)

represents standard deviation.

You can use the following procedure to calculate standard deviation.

Finding Standard Deviation

• Find themean ofthe data set:x.

• Find the difference between each value and the mean:

x

-x.

• Square each difference: (x -

xl

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L

(

x

-

x)2

• Find the average (mean) of these squares: n

• Take the square root to findthe standard deviation:

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EXAMPLE

Finding the Standard

Deviation

Find the mean and thestandard deviation for the values:48.0,53.2, 52.3,46.6,49.9.

x

= 48.0+ 53.2+ 5~.3 +46.6+ 49.9= 50.0 Find the mean.

Organize the next steps

in a table.

x X

x-x

(x - x)2.

48.0 50.0 -2.0 4.00

53.2 50.0 3.2 10.24

52.3 50.0 2.3 5.29

46.6 50.0 -3.4 11.56

49.9 50.0 -0.1 0.01

Sum:31.1 i I

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(J =

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2,

(x n- x)2

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)

31.

1=2.5

• The mean is 50.0, and the standard deviation isabout 2.5.

Find the standard deviation.

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CheekU~dersta~di~g

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Find themean and thestandard deviation for these values:50,60,70, 80,80,

90,100,110.

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Standard deviation islike acustom-made measuring stick for the variation in aset -of data. Asmall standard deviation (compared to actual data values) indicates that the data are clustered tightlyaround the mean. Asthe data become more spread out,the standard deviation increases.

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EXAMPLE

Real-World

8

Conne

c

tion

•

·

~adingMath

The watt is the metric 1.1

unit of measurement for power. One million watts of power delivered for one hour results in one

megawatt-hour (MWh).

Energy Find the mean and the standard deviation of the data for daily energy

demand in a small town during August. .

Daily Energy Demand During August (MWh)

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Frf.'.~ Sat.

.ThlJrs',

53 52 47 50 39

33 40 ·41 44 47 49 43

39 47 49 54 53 46 36

3;3 45 45 42 43 39 33

33 40 40 41 42

Step 1 Use the STAT feature to enter the data asL1.

Step 2 Use the CALC menu of STAT toaccess the

1-Var Stats option.

The mean is about 43.2 MWh; the standard deviation is about .6.0MWh.

mean

standard deviation

(12)

~

Check Ut1derstat1dit1g

e

Find the mean and standard deviation for this data set: 2 mm, 3 mm,4 mm,6mm,

7mm,9mm,10mm,12mm, 13mrn,14 mm.

OBJECTIVE

~

Using Standard Deviation

.

:0

Real-World ~j Connection

Careers Dispatchers •coordinate a utility's power

supply and demand.

<

i

Check Ut1derstat1dit1g

In adata list,every value fallswithin some number ofstandard deviations of

themean.When a value falls within one standard deviation of the mean, it isin the range of values from one standard deviation below the mean to one standard deviation above.For example, ifthe mean is50 and thestandard deviation is 10,

then a value x within one standard deviation of the mean must be in the range 40 ~ x ~ 60.

4

EXAMPLE Real-World

8

Connection

Energy Use the energy demand data from Example 3. Within how many standard deviations of the mean do all of the values fall?How might the company supplying power to the town use thisinformation?

Step 1 Draw a number line. Plot the data values and the mean.

Step 2 Mark off intervals of

6.0 on either side of the mean.

x

X X

x

·

X

XXXX

• I X

X X X

I I

I

Ie •

X X

X X X

X

'

X X

I

X

I

+

150 'I •

55:

30 :

1... ...-I I I I I I I I I

2 standard deviations

below the mean

43.2 - 2( 6.0) =31.2

: 45

---6,0

I

mean = 43,2

40

6.0

35 I

6.0 --:-

-I I

----+-1__ I

I

6.0 --..I

I I I I I I I I I

2 standard deviations

above the mean

43.2

+

2(6.0) = 55.2

1standard deviation

above the mean 43.2 +6,0 =49,2

1standard deviation

belowthe mean

43,2 - 6,0 =37,2

All of the values fall within two standard deviations of the mean.Therefore, the

: power company can expect that the dailydemand on most days in August will fall

.• within two standard deviations of the mean,

o

a. Within how many standa'rd deviations of the mean for August isademand of38.5 MWh?

b. In May,the mean daily energy demand is 35.8 MWh, with astandard deviation of

3.5 MWh. The power company prepares for any demand within three standard deviations of the mean, Are they prepared for a demand of 48MWh? Explain,

The z-score isthe number ofstandard deviations that a value isfrom the mean.

In Example 4,the value 49.2is one standard deviation above the mean, soit has

a z-score of LThe value 37.2,which is one standard deviation below themean,

has az-score of -1.

(13)

5

EXAMPLE Finding the z-score

A set of values has amean of 85 and astandard deviation of 6. Find the z-score of the value 76.

~

Check Ut1derstat1dit1g

0

Use the mean and standard deviation from Example S.Find the value that has a

z-score of 2.S.

Examples 2 and 3

(page 658)

Example 4

(page 659)

Example 5

(page 660)

660

value -. mean

z-score

=

standard deviation

=

76

(5

85 Substitute.

=

6

9 Simplify.

• = -1.5

For more practice, see Extra Practice.

P

ractice and Problem Solving

__ •••• • ~ ~ - '. •-~.., ~ W , li~ - • ~ --. .•. ,,,.. .~,., ~ ~~ •• "';:'-.'1 - -.,'~3 -on- -..-.- .•.. ~ ~•-. •...••• ""'-~

-L5 6 7 3 4 5 678

Find the range and the interquartile range of each set of values.

2. 56 78 125 34 67 91 20 3. 724 786 670 760 300 187 190 '345 456 732 891 879 324

4. 78 90 456 673 111 381 21

Find the mean and the standard deviation for each set of values.

5. 13 15 17 18 12 21 10

6. The Dow Jones Industrial average for the first 24 weeks of 1991:

2646.80 2909.90

2920.80

2913.90

2659.40

2955.20

2965.60 3027.50

\ ;,i

EXE

RCISES

I ru

(~:1,1I !:

:1

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Practice byExample Example 1

(page 657)

2501.50

2889.40

2896.80

2886.60

2730.70

2948.30 2912.40

2976.70

2830.70 2558.90 2938.90

3000.50

2934.70

2913.90 2920.20

2965.60

7. The Dow Jones Industrial average for the first 24 weeks of 2001:

10525.38

10441.90

9791.09

11301.74

10587.59 '

10466.31 10126.94

11005.37

10662.01

10799.82 '

9878.78

10821.31

10659.98

10644.62

10579.85

10990.41

10864.10

9823.41

10810.05

10977.00

10781.45

9504.78

10951.24

10623.64

Determine the whole number of standard deviations that includes all data values. 8. The mean price of the nonfiction books on abest-sellers list is $25.07; the

standard deviation is $2.62.

$26.95, $22.95, $24.00, $24.95, $29.95, $19.95, $24.95, $24.00, $27.95, $25.00

9. The mean length of Beethoven's nine symphonies is37minutes; the standard

deviation is 12 minutes.

27min, 30 min, 47min, 35 min, 30 min, 40 min, 35 min, 22min, 65 min

10.39 12.125

A data set has mean 25and standard deviation 5.Find the z-score of each value. 14.11

(14)

o

Apply Your Skills ~. Find the standard deviation for each data set. Use the standard deviations to compare each pair of data sets.

'.

'

.

15. fastest recorded speeds of various large wild cats (miles per hour): 70 50 30 40 35 30 30 40 15 ,

fastest recorded speeds ofvarious birds inflight (miles per hour): 217 106 95 56 65 37 50 31 53 25 25 25

16. the number of buttons on selected outfits: 11 5 12 8 3 12 10 10 0 5 0 2 7 10 the number of pockets in the same outfits: 5 5 5 225 3 2 0 2 0 055

Income Use the chart at the right for Exercises 17-20.

Real-World ~} Connection The fastest wild cat is the cheetah, which can run as fast as 70 milh.

17. Find the mean income for each year.

~ 18. Writing Use the range of thedata for each year to describe how farm income varied from 1998to1999.

Farm Income in Midwestern States (millions of dollars) .

"

State _,,-

-

;-<_. '1998,,;' 1.,.-. 1999

"

·,

Iowa 12,153 10,812

Kansas 8688 8565

Minnesota' 8971 8010

Missouri 5464 4902

Nebraska 9827 9429

North Dakota 3664 2921

South Dakota 4264 3974

~ 21. a. Energy Find the mean and the standard deviation for daily energy usage during ten days in June: 51.8MWh, 53.6 MWh,54.7MWh, 51.9 MWh, 49.3MWh, 52.0 MWh,53.5 MWh, 51.2MWh,60.7MWh, 59.3 MWh.

b. How many items in the data set fallwithin onestandard deviation of the mean? Within twostandard deviations? Within three standard deviations? 19. Find the standard deviation for

each year. Inwhichyear did farm income cluster more tightly'

around the mean?

20.Which state's 1998 income has a z-score ofabout 1.6?

I

Another measure of variation isvariance, which equals a2•Find the variance and the standard deviation of each data set.

22. 12 h 3 h 2 h 4 h 5 h 7h

24. $6.99 $5.50 $7.10 $9.22 $8.99

23.60 m 40 m 35ill 45 m 39 m

25. 0.7g 0.84 g 0.9 g 0.8g 0.69 g

26. Critical Thinking From your results in Exercises 22-25, which do you think isa better measure of variation-variance or staridard deviation? Explain.

27. Error Analysis Minh says that the data below fall within three standard deviations ofthe mean. Marsha disagrees, sayingthat the data fallwithin six standard deviations ofthe mean. With whom do you agree? Explain.

I I x I I I

I I x x l I I

I I x x x :1 x I I

I I x X X X ;t X X x :t I

x :*: x X x x x x

*

x x x x :*: x x

*

x

-+- _I

•

I •

•

I

•

I

, _I _I _I _I _I

70 I 75 I 80 I 85 I 90 I 95

mean =82.5

(15)

28. a. Data Collection Make a table showing the number of siblings of each

student inthe class.

b. Find the mean and standard deviation of the data.

Challenge 29. a. Use the table at the left to find the range, the mean, and the standard

deviation of the ages for each team.

b. Critical Thinking For two data sets, does the set with the larger range

necessarily have the larger standard deviation? Support your answer with

your results from part (a).

~ 30. Earnings The table at the right Worker's Median Weekly Earnings, 1999

shows the median weekly

earnings ofunion and nonunion

workers invarious occupations. Construction $778 $509

a. Find-the mean and the range Transportation

of the data for union workers and public utilities $748 $613

and for nonunion workers. Trade $499 $418

b. Find the standard deviation Manufacturing $614 $561

for each set of data. Services $554 $515

c. Within how many standard Finance, insurance,

deviations of the mean are and real estate $582 $599

earnings of $395 for union Mining $710 $735

workers? For nonunion

SOURCE:U.S.Bureau of Labor Statistics.

'~ workers? Gotowww.PHSchool.com for adata update.

~ d.Writing Compare the wages Web Code:agg-2041

of union and nonunion workers. Use your results from parts (a) through (c).

099

Ages of the Members of

the 2000 U.S. Olympic Soccer Teams

Men Women

23 32 32 29

29 23 29 22

23 23 32 28

22 23 25 23

21 18 22 22

19 ₂₂

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20 ₂₃ _'_;_~ ₃₂ 25

23 21

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26 22 24 22

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For Exercises 31-32, use the following bowling scores for sixmembers of a bowling team: 175,210,180,195,208,196.

31. What isthe mean of the scores?

32. What isthe standard deviation of the scores?

33. At a second bowling tournament, the mean of all the scores was 205, with

a standard deviation of 14. What was the z-score for a score of 2827

34. At the second tournament, a participant had az-score of -2.5, What was

the participant's bowling score?

r

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Mixed Review

.

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Lesson 12-3

Lesson 11-3

Lesson 10-3

Make abox-and-whisker plot for each set ofvalues.

35.25,25,30,35,45,45,50,55,60,60 36.20,23,25,36,37,38,39,50,52,55

Find the missing positive term in each geometric sequence.

37. 64,11I,4, ... 38. 20, 11,0.05, .. , 39. 29,

m

,

65.25, ...

Graph each circle.

40. (x - 2)2 + (y + 1)2 = 36 41. (x - 1)2 + (y - 1)2 = 4

(16)

Working With Samples

Lesson Preview

~

Check Skills You'll Need

(For help,go toLesson 7-1.)

Simplify each expression.

1

1.

V

4

1

4. -

VITI

New Vocabulary • sample • sample proportion • random sample • margin of error

What You'll Learn

OBJECTIVE

V

eV

E

To find sample proportions

To find the margin of error

... And

Why

To analyze data from a poll, asin Example 5

1

3. v'36

1

6. - v'8I

Suppose you want to know what percent of all teenagers recognize the word that means "to passthe summer in a state of slumber." Since itistoo costly and time consuming to ask every teenager, use a sample.A sample gathers information from onlypart ofapopulation.

Using any sample, you can find asample proportion. The sample proportion is theratio~, where xisthe number of times an event occurs ina sample of sizen.

1

·EXAMPLE

~b

Interactive lesson includes instant

~~ self-check, tutorials, and activities.

Finding the Sample Proportion

In a sample of 350 teenagers, 294 have never made a snow sculpture. Find the sample proportion for those whohave never made a snow sculpture. Write the answer as apercent.

!

.

sample proportion

=

~

: 294

! = 350

= 0.84

Write the formula.

Substitute 294 for x and 350 forn.

Simplify.

• Thesample proportion is 84%.

~

Check Understanding

0

In apoll of1085voters, 564 favor Candidate A.Find the sample proportion for those who favor Candidate A.

•..

R

e

ading Math

: Bias means "slant."

Samples vary in how well they reflect the entire population. In a random sample,

all members ofthepopulation ~re equally likelytobe chosen.

When apart of apopulation isoverrepresented or underrepresented in a sample,

the sample is biased. Arandom sample canhelp avoid bias ingathering data.

(17)

i '

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2

EXAMPLE Real-World ~l Connection

Public Opinion A news program reports ona proposed school dress code. The purpose of the program isto find out what percent ofthe population in its viewing

area favors thedresscode. Identify anybias ineachsampling method.

a. Viewers are invited to call the program andexpress their preferences.

The people who decide to callinmay over- orunderrepresent some views.For

example, members ofa group favoring the new dress code might encourage its

members to callin. This type of sample is called a "self-selected" sample.

b.A reporter interviews people on the street near the local highschool.

Thisis a "convenience" sample, since itisconvenient forthe reporter to

stay inone place.Because the location isnear the school,students maybe

overrepresented in the sample and the results will be biased.

c.During the program, 300 people are selected at random from the viewing area. Then each person is contacted.

• This sampling method contains the least bias. It is a random sample.

:

I

Check Understanding

0

a. Suppose the 350 teenagers in Example 1 alllive inFlorida. Isthere biasin this

sample? Explain.

b. Critical Thinking The onlywayto know atrue population proportion isto poll

every person inthe population. Such a poll is nolonger a sample, but acensus.

Describe a situation inwhich asample isunsatisfactory and a census isrequired.

The size of asample affects itsreliability.Witha small sample size,you are likely to

get a widerange of sample proportions. For example, in some samples, no one will

recognize the word that means "to pass the summer in a state of slumber." In other samples, everyone will recognize estivate. Withlarger sample sizes,you are less

likely tohave asample containing an"all ornothing" result.

The Law of Large Numbers states that the variation inadata set decreases asthe

sample size increases. Bycomparing the variation insamples,youcangetan idea

oftheir relative sizes.

3

EXAMPLE Comparing Sample Sizes

,~

,

f

Sample Size

,

, .-.... .~~...- ...•.~. ,.~ -' ~- , - ~ '" - . - -"'-"'- ~ .. OBJECTIVE

Each graph below showsthe sample proportions for20 samples.Match each graph

tothe most likely sample size.

a. 20people per sample b.5people per sample

Distributions of Sample Proportions

5r---p,-~----~

on

t~ 41---j<iI--H--,.,.,----1

"E

S

3f-m,---U--j1l--f~·1_t______

::1~ 2 i

Z"-' 1:1 :

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1

r

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0.2 0.4 0.6 0.8 1

Sample Proportion

of CorrectResponses

5r---=---~

•..•:G 4f---,,-j~I_---___1 <1)

-.0 0..3f---!:r.h".---.j

§

a

2f---..,--{;:r I,,{.ir----j

Z'O 1f---rd~;.2;ld>~'h ;,r;\:,",II."'·".r.----l

'

f

~

1

:.

'

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OL- ~"~I~~"I~-~·illll'iill~L__II~~I~

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0.2 0.4 0.6 0.8 1

Sample Proportion

of CorrectResponses

(18)

•

a. The graph on the right shows lessvariation, so it is more likely to be based on samples of a larger size,20 people per sample.

b. The graph on the left shows more variation, so itis more likely be based on 5 people per sample.

~

Check Ut1derstat1dit1g

G

A science class measured the heights of blades of grass behind the school. The class took three samples. Use the information in the table below to decide which sample most likelywas the greatest in size.Explain your reasoning.

A sample proportion should be reported with an estimate of error, called the

margin of error. The margin of error is based on the standard deviation in graphs like those in Example 3. Thelarger the sample size,the smaller the margin of error.

Margin of Error Formula

When a random sample ofsize nis taken from alarge population, the sample

proportion has a margin of error ofapproximately ±

J

-n

.

4

EXAMPLE Using the Margin of Error

A poll reports that 56% of voters favor Candidate B,with a margin of error of ±3%. Estimate the number of voters in the poll.

margin of error

=

±

J-n

Write the formula. ±

Vii =

margin10£error Rewrite the equation.

=

0

.

b

3

Substitute 0.03 for margin of error.

=33.33 Simplify.

n

=

1111 Square each side.

• The poll included about 1100voters.

"

Check Ut1derstat1dit1g

0

Estimate the sample size for each margin of error.

a. ±10% b. ±4% c. ±2%

You can use the margin of error to determine the likelyrange for the true population proportion. The graph below shows the range for the population in Example 4.

Sample Proportion

1_ -3% ---L- +3% ---I_Margin of error

1 t I

.•.I I I • I I I'

53% 56% 5~% Likely range for.the true

i"' ~i -population proportion

(19)

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Real-World

it

Connection

Some retail businesses cater to left-handers. •

,£

Check U~dersta~di~g

EXER

CISES

5

EXAMPLE Real-World

8

Conne

ction

Practice and Problem Solving

- ~ ~- .~ ~~.,., ....- "'..- - -. - =-=

o

Practice by Example Example 1

(page 663)

Example 2

(page 664)

Example 3

(pages 664-665)

Genetics A survey of 2580 students found that 9% are left-handed.

a. Find the margin oferror for the sample.

margin oferror ~

±.

I

n

Use the formula.

= ±V21580 Substitute.

= ±0.0197 Use a calculator.

The margin of error is about ±2%.

b. Use themargin of error to find an interval that islikelyto contain the true

population proportion.

The margin of error forms an interval with the sample proportion at its midpoint.

Sample P~oportion

:-- -2% ~- +2%---:

7% 9% 11%

The proportion ofstudents who are left-handed is likelyto be from7% to11%.

o

In apoll of123students, 87have never ridden aferry. Find the sample

proportion, the margin of error, and the interval likely to contain thetrue population proportion.

For more practice, see Extra Practice.

For each sample, find the sample proportion. Write it as a percent.

1.837 out of1150 insurance applicants have no citations on their driving record.

2. 27 out of60shoppers prefer generic brands when available.

3. 532 out of580 households own a color television set.

Identify any bias in each sampling method.

4. A supermarket wants to find the proportion ofshoppers who use reduced-price

-coupons.A manager interviews every shopper entering the greeting card aisle.

-5. A maintenance crew wants to estimate how many of3000 air filtersin an office

building need replacing. The crew examines five filters chosen at random on each floor of the building.

6. The student government wants to find out how many students have after-school jobs.A pollster interviews students selected at random as theyboard buses at the end ofthe school day.

7. In a survey,teenagers were asked the importance of "making your own things." The response scale ranged from 1to 5,with 5 being extremely important. Which sample most likelywas largest? Explain.

-nSample I_Score. Standard Deviation _

-

'

.

~,

A 3.6 1.2

B 3.8 1.0

C 3.8 0.5

(20)

Example 4

(page 665)

Example 5

(page 666)

8. Thetable below shows the results of a poll asking students, "How many hours a week would yousay youspend doing academic homework?" Which sample most likelywassmaller? Explain.

Find the margin of error for the sample proportion, given each sample of sizen.

9.n

=

200 10.n

=

800 11.n = 1200

Find the sample size that produces each margin of error.

12. ±8% 13. ±5% 14. ±1%

For each situation, find the margin of error for thesample. Then find an interval likelyto contain the true population proportion.

15.Of750 teenagers polled, 59% think boys and girls are portrayed asequals on television.

16. Of400teenagers surveyed, 62% do not plan tostay in their community after finishing their education.

o

Apply YourSkills

8

1Surveys For each sample, find (a) the sample proportion, (b) the margin of error,

and (c) the interval likely to contain the true population proportion.

17. Inarand omsample of 408 grocery shoppers, 258 prefer one large tripper week toseveral smaller ones.

18. Of 500teenagers surveyed, 460 would like to see adults in their community do more tosolve drug problems.

19.In'a survey of 32people, 30 return amilkcarton tothe refrigerator immediately after usingit.

20. In a survey of 16people, one person never locks hiscar.

~ 21. Writing Write a news article describing the

. sample proportion and margin oferror for

'the poll results shown atthe right.

22. Reasoning How is the margin oferror affected if you double the sample size?Explain.

.

Do you save

'.

more

than

5%

' .

ofvour lncome?

- Jr. ~

yes

·no

,.

23. a. Data Collection Write a survey question to find out thenumber ofstudents atyour school

who plan tocontinue their education after highschool b. Describe the sampling method youwould use. c. Conduct your survey.

Results:

yes

370

.no 583

24. Critical Thinking A sample proportion provides an estimate for the percent of an entire population that favors anevent. Is a sample proportion an experimental oratheoretical probability? Explain.

Lesson 12-5 Working With Samples

667

(21)

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~' 25. computer_T_h_{e s}_ur_{vey as}Use An onli_k_s _h_ow _mune_c_{h tim}adverti_{e yo}sement_u _s_p_e_n_{d onli}asks y_nou t_e _eaco parti_h _weecip_{k. I}ate_denin_ti_fya sur_avey ._n_y _b_i_as

inthis method. If appropriate, suggest a method more likely to produce a

random sample ..

For help with reading

: and solving Exercise 25, : see p. 670.

An event occurs xtimes in a sample of size n.Find its sample proportion and margin of error.

26.x = 96

n

=

900

28.x

=

100

n = 250

29.x

=

273

n = 435

27.x

=

20

n = 64

C

3

Challenge 30. a. It costs $20 to interview each person for a survey. Find the cost to obtain a

±3% margin of error.

b. Critical Thinking Find the cost to obtain a ±2% margin of error. Why do

you think polls 'with smaller margins of error are rare?

~. 31. a. Elections A poll 0050 voters shows that a candidate ispreferred by 56%

of the voters while 44% prefer the opponent. Should the candidate be concerned? Explain.

b. A later poll of 600 voters shows the candidate is preferred by55% of the

voters. Should this candidate feel more or less confident, given the results of

the second poll? Explain. '

~. 32. Wildlife Wild animal populations are often estimated through the use of the

capture-tag-recapture method. Several animals are captured, tagged, and

released back into the wild. The an.imals continue to roam freely. Then, some

time later at the same site, several more animals are captured, and the number

of tagged animals isrecorded. An estimate of the population can then be

calculated. This method of estimation assumes that the fraction of tagged animals in the second sample isequivalent to the fraction of tagged animals in the entire population.

tagged animals insecond sample tagged animals in population

animals insecond sample = population (P)

Use the formula above to predict the black bear population of the northern

coastal plain of South Carolina. Researchers tagged fourteen black bears in the

fall and captured eleven bears the following summer. Of the eleven bears, three

were tagged.

Exercise 32

Multiple Choice 33. In a sample of 625 airline travelers, 485 collected "airline miles" toward

free trips. What does the number

~i~

represent?

A. the probability that a passenger collects airline miles

B. the sample proportion of the travelers who do not collect airline miles C. the sample proportion of the travelers who collect airline miles D. the margin of error for the sample

34. A random sample of people answered the question "Do you collect airline miles?" The margin of error for the sample was ±2%. The sample

proportion of people who answered no was 130' How many people in the sample answered no?

F. 15 G. 225 H. 750 l. 2500

(22)

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Web Code: aga-1205

Short Response

Extended Response

lesson 12-4

lesson 11-4

lesson 10-4

35. A research group had a stack of survey responses. The number of respondents was more than 5000 and fewer than 5500. When the researchers divided the respondents into 13 equal groups, there were no extra respondents. Similarly, there were no extra respondents when they

divided the responses into 7 equal groups or 11 equal groups. How many respondents were there?

A.l00l B. 5005 C. 5031 D. 500,500

36.What is the margin of error for a random sample of size 3600? Show

your work.

37.Ina poll of 2750 airline travelers, 138 said they never check their luggage when they fly. Find the sample proportion, the margin of error, and the interval likely to contain the true population proportion.

Find the mean and the standard deviation for each data set.

38. 0 km, 1km, 1 km, 1km, 2km, 2km, 2km, 3km, 3km, 4 km, 5 km, 10 km

39. 1OZ,1 oZ,2 oz,2 OZ,3OZ,4OZ,5OZ,6OZ,8OZ,9OZ,10 OZ,10 OZ,12 OZ,20 oz

Use summation notation to write each arithmetic series for the specified number ofterms.

40. 3 + 8 + 13 + ... ;n = 5

42. -14 + (-8) + (-2) + ... ;n = 6

41. 41+ 33 + 25 + ... ;n = 8

43.-27 + (-21) + (-15) + ;11 = 10

Find the equation of each ellipse centered at the origin.

44. height: 20 units

width: 6units

45. height: 12units width: 10units

46. height: 24 units

width: 36 units

Algebra at Work

When questions arise about consumer products or services, a market researcher gathers statistical information to help answer the questions. The

information a market researcher collects and analyzes helps companies improve their products and make decisions about their customer base. Quantitative

research allows a'market researcher to analyze data from alarge population of

potential customers. Market research strategies for gathering information

include the following.

• mail surveys • telephone surveys

• focus groups • .in-person interviews

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research, go to www.PHSchool.com.

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