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Chapter 3Descriptive Statistics: Numerical MeasuresLearning Objectives Understand the purpose of measures of location. Be able to compute the mean, weighted mean, geometric mean, median, mode, quartiles, and various percentiles. Understand the purpose of measures of variability. Be able to compute t...

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Cha pt e r3 De s c r i pt i v eSt a t i s t i c s :Nume r i c a lMe a s ur e s

Le a r ni ngObj e c t i v e s 1 .

Unde r s t a ndt h ep ur p os eofme a s u r e sofl oc a t i on .

2 .

Bea bl et oc omp ut et h eme a n,we i g ht e dme a n,g e ome t r i cme a n ,me d i a n,mode ,qua r t i l e s , a n dv a r i ou s pe r c e n t i l e s .

3 .

Unde r s t a ndt h ep ur p os eofme a s u r e sofva r i a bi l i t y .

4 .

Bea bl et oc omp ut et h er a n g e ,i nt e r qu a r t i l er a n g e ,v a r i a nc e , s t a n da r dde v i a t i on , a ndc o e ffic i e n tof v a r i a t i o n.

5 .

Unde r s t a nds ke wne s sa same a s u r eoft h es ha p eofada t ad i s t r i bu t i o n. Le a r nhowt or e c o gn i z ewh e na da t adi s t r i but i oni sne ga t i v e l ys k e we d,r oug hl ys y mme t r i c , a n dp os i t i v e l ys ke we d .

6 .

Unde r s t a ndho wzs c or e sa r ec o mpu t e da ndh o wt he ya r eu s e da same a s ur eofr e l a t i v el oc a t i onofa da t ava l ue .

7 .

Knowh o wChe b y s h e v’ st h e or e ma ndt hee mpi r i c a lr ul ec a nb eus e dt ode t e r mi n et hepe r c e nt a g eo ft he da t awi t h i nas p e c i fie dn umb e rofs t a nd a r dd e vi a t i onsf r om t h eme a n .

8 .

Le a r nho wt oc o ns t r u c ta5 –nu mbe rs umma r ya n dab oxpl o t .

9 .

Bea bl et oc omp ut ea n di nt e r pr e tc o v a r i a nc ea ndc or r e l a t i o na sme a s ur e so fa s s o c i a t i onbe t we e nt wo v a r i a b l e s .

1 0.

Unde r s t a ndt h er ol eofs umma r yme a s ur e si nda t ada s hb oa r ds .

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Cha pt e r3

Sol ut i ons : x

1 .

xi 75  15 5 n

1 0, 12 , 16,1 7, 20 Me di a n=16( mi ddl eva l ue ) x

2 .

xi 96  16 n 6

1 0,12,16,17,20 ,21

Median = x 3 . a .

16  17 16.5 2

wi xi 6(3.2)  3( 2)  2( 2.5)  8( 5) 70.2   3.69 6 3  2 8 19 wi

3.2  2  2.5  5 12.7 .   3175 4 4 b. 4. Period 1 2 3 4 5

Rate of Return (%) -6.0 -8.0 -4.0 2.0 5.4

The mean growth factor over the five periods is: xg n  x1   x2   x5  5  0.940  0.920  0.960  1.020  1.054 5 0.8925 0.9775 So the mean growth rate (0.9775 – 1)100% = –2.25%. 5 .

1 5, 20 , 25,2 5, 27 , 28, 3 0,3 4 20 (8) 1.6 i 100 25 (8) 2 i 100 65 (8) 5.2 i 100

2n dp os i t i o n=2 0 20  25  22.5 2

6t hp os i t i o n=28 3-2

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De s c r i pt i v eSt a t i s t i c s :Nume r i c a lMe a s u r e s

28  30

75 (8) 6 i 100 Mean  6 .

7.

 29

xi 657   59.73 n 11

Me di a n=57

6 t hi t e m

Mo de=53

I ta pp e a r s3t i me s

a.

The mean commute time is 26.9 minutes.

b.

The median commute time is 25.95 minutes.

c.

The data are bimodal. The modes are 23.4 and 24.8.

d.

8.

2

75 i 48 36 100 , so the third quartile is the mean of the values of The index for the third quartile is 28.5  28.5  28.5. 2 the 36th and 37th observations in the sorted data, or x

 x i 350  18.42 n 19

x

 x i 120   6.32 n 19

a.

b.

c.

120 (100)  34.3% 350 of3 p oi nts hot swe r ema d ef r o mt he2 0f e e t , 9i n c hl i n edur i ngt h e1 9ga me s .

d. Mo vi n gt he3 po i ntl i n eba c kt o20f e e t , 9i nc h e sha sr e duc e dt henu mbe ro f3 po i nts hot st a ke np e r g a mef r om 19 . 07t o18. 42 , or19 . 07–1 8. 4 2=. 6 5s hot spe rg a me .Thep e r c e nt a g eo f3p oi nt sma d e p e rga meha sbe e nr e duc e df r om 35 . 2 %t o34 . 3%, o ron l y. 9 %.Themov eh a sr e duc e dbo t ht he n umb e ro fs ho t st a k e npe rga mea ndt h epe r c e n t a geofs h ot sma depe rga me , butt hedi ffe r e nc e sa r e s ma l l . Theda t as up por tt heAs s oc i a t e dPr e s sSp or t sc onc l us i ont ha tt h emo v eha sn otc ha ng e dt he g a medr a ma t i c a l l y . The20 0809s a mpl eda t as ho ws12 03p oi ntba s k e t si nt h e19ga me s .Thu s , t heme a nnu mbe ro f p oi nt ss c o r e df r om t he3po i ntl i n ei s1 20 ( 3) / 19=1 8. 95poi nt sp e rga me .Wi t ht hepr e vi ou s3 po i nt l i n ea t19f e e t ,9i n c he s , 19. 07s h ot sp e rga mea nda35 . 2 %s u c c e s sr a t ei n di c a t et ha tt h eme a n n umb e ro fpoi nt ss c or e df r omt he3poi ntl i newa s19. 07 ( . 3 52 ) ( 3 )=20. 14po i n t sp e rga me .The r ei s o nl yame a nof2 0. 1 4–1 8. 95=1 . 19p oi n t spe rg a mel e s sbe i n gs c or e df r omt he20f e e t , 9i nc h3p oi ntl i n e . x

9 . a .

 x i 148  14.8 10 n

b. Or de rt hed a t af r om l o w6. 7t oh i gh3 6. 6

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Cha pt e r3

Me di a n

 50  i  10 5  100 

t h t h Us e5 a n d6 pos i t i ons .

10.1 16.1 13.1 Median  2 c . Mode=7. 2( oc c ur s2t i me s )

 25  i  10 2.5  100  d.  75  i   10 7.5  100 

r d Us e3 p os i t i on .Q1 = 7 . 2

t h Us e8 p os i t i on .Q3= 17 . 2

e . Σxi=$ 14 8b i l l i o n Thepe r c e nt a geoft o t a le n do wme n t she l db yt h e s e2. 3% ofc o l l e ge sa n du ni v e r s i t i e si s( 14 8/ 413 ) ( 10 0)=3 5. 8 %. f . Ade c l i n eof23 % wo ul dbeade c l i n eo f. 23 ( 1 48)=$34b i l l i o nf ort h e s e10c ol l e ge sa n dun i v e r s i t i e s . Wi t ht hi sd e c l i ne , a dmi n i s t r a t o r smi g htc ons i de rbudg e tc ut t i n gs t r a t e gi e ss u c ha s    

1 0.a.

Hi r i n gf r e e z e sf orf a c ul t ya n ds t a ff De l a y i n gore l i mi n a t i n gc on s t r uc t i o np r o j e c t s Ra i s i n gt u i t i o n I n c r e a s i n ge nr ol l me nt s

x 3200 x  i  160 20 n

Order the data from low 100 to high 360

Median

 50  i   20 10  100  Use 10th and 11th positions

 130  140    135 2  Median = 

Mode = 120 (occurs 3 times)

b.

 25  i   20 5  100  Use 5th and 6th positions  115 115  115 Q1   2    75  i   20 15  100  Use 15th and 16th positions

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De s c r i pt i v eSt a t i s t i c s :Nu me r i c a lMe a s u r e s

 180 195  Q3   187.5 2  

c.

 90  i   20 18  100  Use 18th and 19th positions  235  255    245 2  90 percentile  th

90% of the tax returns cost $245 or less. 10% of the tax returns cost $245 or more. 11. a.

The median number of hours worked per week for high school science teachers is 54.

b.

The median number of hours worked per week for high school English teachers is 47.

c.

The median number of hours worked per week for high school science teachers is greater than the median number of hours worked per week for high school English teachers; the difference is 54 – 47 = 7 hours.

12. a. b.

c.

d.

The minimum number of viewers that watched a new episode is 13.3 million, and the maximum number is 16.5 million. The mean number of viewers that watched a new episode is 15.04 million or approximately 15.0 million; the median is also 15.0 million. The data is multimodal (13.6, 14.0, 16.1, and 16.2 million); in such cases the mode is usually not reported.  25  i   21 5.25  100  The data are first arranged in ascending order. The index for the first quartile is , so the first quartile is the value of the 6th observation in the sorted data, or 14.1. The index for the 75 21 15.75 i 100 , so the third quartile is the value of the 16th observation in the third quartile is sorted data, or 16.0. A graph showing the viewership data over the air dates follows. Period 1 corresponds to the first episode of the season, period 2 corresponds to the second episode, and so on.

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Cha pt e r3

18.0 16.0 Viewers (millions)

14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 0

5

10

15

20

25

Period This graph shows that viewership of The Big Bang Theory has been relatively stable over the 2011– 2012 television season. 1 3.

Us i n gt h eme a nweg e t

xcity

=15. 5 8 ,

xhighway

=1 8 . 9 2

Fo rt h es a mpl e swes e et h a tt heme a nmi l e a g ei sbe t t e ro nt h eh i g hwa yt ha ni nt h ec i t y . Ci t y 13 . 21 4. 415 . 215. 315 . 315. 315 . 9161 6. 116 . 21 6. 216 . 71 6. 8  Me di a n Mo de :15. 3 Hi g hwa y 17 . 21 7. 418 . 318. 51 8. 618. 61 8. 719. 01 9. 219 . 41 9. 420 . 621. 1  Me di a n Mo de :18. 6 ,1 9. 4 Th eme di a na n dmo da lmi l e a g e sa r ea l s ob e t t e ront hehi g hwa yt ha ni nt h ec i t y . 14.

For March 2011:  25  i   50 12.50 100   , so the first quartile is the value of the 13th The index for the first quartile is observation in the sorted data, or 6.8. 50 i 50  25.0 100 The index for the median is , so the median (or second quartile) is the average of the values of the 25th and 26th observations in the sorted data, or 8.0.

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De s c r i pt i v eSt a t i s t i c s :Nu me r i c a lMe a s u r e s

75 50 37.50 i 100 , so the third quartile is the value of the 38th The index for the third quartile is observation in the sorted data, or 9.4.

For March 2012: The minimum is 3.0  25 50 12.50 i  100   , so the first quartile is the value of the 13th The index for the first quartile is observation in the sorted data, or 6.8. 50 i 50  25.0 100 , so the median (or second quartile) is the average of The index for the median is the values of the 25th and 26th observations in the sorted data, or 7.35. 75 i 50 37.50 100 , so the third quartile is the value of the 38th The index for the third quartile is observation in the sorted data, or 8.6.

It may be easier to compare these results if we place them in a table.

First Quartile Median Third Quartile

March 2011 6.8 8.0 9.4

March 2012 6.8 7.35 8.6

The results show that in March 2012 approximately 25% of the states had an unemployment rate of 6.8% or less, the same as in March 2011. However, the median of 7.35% and the third quartile of 8.6% in March 2012 are both less than the corresponding values in March 2011, indicating that unemployment rates across the states are decreasing. 15.

To calculate the average sales price we must compute a weighted mean. The weighted mean is

501 34.99  1425 38.99  294 36.00  882 33.59  715 40.99  1088 38.59  1644 39.59  819 37.99 501 1425  294  882  715  1088  1644  819

= 38.11 Thus, the average sales price per case is $38.11. 16. a . Gr a d ex i 4( A) 3( B) 2( C) 1( D) 0( F)

We i g htWi 9 15 33 3 0 60Cr e d i tHour s

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Cha pt e r3

x

wi xi 9(4)  15( 3)  33( 2)  3(1) 150    2.50 9  15  33  3 60 wi

b . Ye s ;s a t i s fie st h e2 . 5g r a d epo i n ta v e r a g er e q u i r e me n t x

1 7.a .

 fi M i 9191(4.65) 2621(18.15) 1419(11.36) 2900(6.75)  9191 2621 1419 2900 N

126,004.14  7.81 16,131

Th ewe i gh t e da v e r a g et o t a lr e t ur nf o rt h eMor ni n g s t a rf u n dsi s7. 8 1%. b .I ft h ea mou nti n v e s t e di ne a c hf undwa sa v a i l a bl e ,i two ul db ebe t t e rt ou s et h os ea mou nt sa swe i g ht s . Th ewe i gh t e dr e t u r nc omp ut e di np a r t( a )wi l lbeag o oda p pr o xi ma t i o n, i ft hea moun ti n v e s t e di nt he v a r i ou sf u nd si sa p pr o xi ma t e l ye qua l . 2000(4.65)  4000(18.15)  3000(11.36) 1000(6.75) 2000  4000  3000  1000 c . Po r t f o l i oRe t u r n= 122,730  12.27 10, 000

Th ep or t f ol i or e t u r nwoul db e12. 27 %. 1 8. Assessment 5 4 3 2 1 Total

19.

Deans 44 66 60 10 0 180

fiMi 220 264 180 20 0 684

x

De a n s :

 fi M i 684  3.8 180 n

x Re c r u i t e r s :

 fi M i 444   3.7 n 120

Recruiters 31 34 43 12 0 120

fiMi 155 136 129 24 0 444

To calculate the mean growth rate we must first compute the geometric mean of the five growth factors: Year 2007

% Growth 5.5

Growth Factor xi

2008

1.1

1.011

2009

-3.5

0.965

2010

-1.1

0.989

2011

1.8

1.018

1.055

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De s c r i pt i v eSt a t i s t i c s :Nu me r i c a lMe a s u r e s

x g  n  x1   x2   x5   5  1.055  1.011  0.965  0.989  1.018  5 1.036275 1.007152 The mean annual growth rate is (1.007152 – 1)100 = 0.7152%. 20.

Year 2004

Stivers End of Year Growth Value Factor $11,000 1.100

Trippi End of Year Growth Value Factor $5,600 1.120

2005

$12,000

1.091

$6,300

1.125

2006

$13,000

1.083

$6,900

1.095

2007

$14,000

1.077

$7,600

1.101

2008

$15,000

1.071

$8,500

1.118

2009

$16,000

1.067

$9,200

1.082

2010

$17,000

1.063

$9,900

1.076

2011

$18,000

1.059

$10,600

1.071

For the Stivers mutual fund we have:  x 18000=10000  1

  x2    x8   , so   x1   x2    x8   =1.8 and

xg n  x1   x2   x8  8 1.80 1.07624 So the mean annual return for the Stivers mutual fund is (1.07624 – 1)100 = 7.624% For the Trippi mutual fund we have:  x 10600=5000  1

  x2    x8   , so   x1   x2   x8   =2.12 and

xg n  x1   x2   x8  8 2.12 1.09848 So the mean annual return for the Trippi mutual fund is (1.09848 – 1)100 = 9.848%. While the Stivers mutual fund has generated a nice annual return of 7.6%, the annual return of 9.8% earned by the Trippi mutual fund is far superior. 21.

  x   x    x9    x   x    x9  , so  1 2 =1.428571, and so 5000=3500  1 2

xg n  x1   x2   x9  9 1.428571 1.040426 So the mean annual growth rate is (1.040426 – 1)100 = 4.0404% 22.

  x   x    x6    x   x    x6   25,000,000=10,000,000  1 2 , so  1 2 =2.50, and so

xg n  x1   x2   x6  6 2.50 1.165 3-9 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Cha pt e r3

So the mean annual growth rate is (1.165 – 1)100 = 16.5% 2 3.

Ra n g e2 0-1 0=10 1 0, 12 , 16,1 7,2 0 25 (5) 1.25 i 100 2 ndp os i t i o n )=1 2 Q1 ( 75 (5) 3.75 i 100 4 t hp o s i t i on )=1 7 Q3 ( I QR=Q3–Q1=17–12=5 x

xi n

2 4. s2 



75 15 5

( xi  x) 2 64  16 n1 4

s  16  4

2 5.

15 , 2 0,2 5,25 , 27, 2 8,3 0,3 4

Ra n g e=34–15=1 9 20 25 22.5 2

25 (8) 2 i 100

Q1 

75 (8) 6 i 100

Q3 

28 30 2

 29

I QR=Q3 –Q1=29–22 . 5=6. 5 x

xi

s2 

n



204 25.5 8

( xi  x) 2 242  34.57 7 n1

s  34.57 588 . 2 6.a . Ra n g e=190–16 8=22

( ) 2 376 b .  xi  x  3-1 0 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

De s c r i pt i v eSt a t i s t i c s :Nu me r i c a lMe a s u r e s

2 3 6=7 5 . 2 =7 s 5

. 8.67 c . s  752

 8.67  Coefficient of Variation  100% 4.87%  178  d . 27. a.

The mean price for a round–trip flight into Atlanta is $356.73, and the mean price for a round–trip flight into Salt Lake City is $400.95. Flights into Atlanta are less expensive than flights into Salt Lake City. This possibly could be explained by the locations of these two cities relative to the 14 departure cities; Atlanta is generally closer than Salt Lake City to the departure cities.

b.

For flights into Atlanta, the range is $290.0, the variance is 5517.41, and the standard Deviation is $74.28. For flights into Salt Lake City, the range is $458.8, the variance is 18933.32, and the standard deviation is $137.60. The prices for round–trip flights into Atlanta are less variable than prices for round–trip flights into Salt Lake City. This could also be explained by Atlanta’s relative nearness to the 14 departure cities.

28. a. b.

The mean serve speed is 180.95, the variance is 21.42, and the standard deviation is 4.63. Although the mean serve speed for the twenty Women's Singles serve speed leaders for the 2011 Wimbledon tournament is slightly higher, the difference is very small. Furthermore, given the variation in the twenty Women's Singles serve speed leaders from the 2012 Australian Open and the twenty Women's Singles serve speed leaders from the 2011 Wimbledon tournament, the difference in the mean serve speeds is most likely due to random variation in the players’ performances.

2 9.a . Ra n g e=6 0–28=3 2 I QR=Q3 –Q1=55–45=1 0

x b .

435 48.33 9

( xi  x) 2 742 s2 

( xi  x ) 2 742   92.75 8 n 1

s  92.75 9.63 c . Th ea v e r a g ea i rq ua l i t yi sa bou tt hes a me .But ,t hev a r i a b i l i t yi sgr e a t e ri nAna he i m. 3 0.

Da ws onSupp l y :Ra n g e=1 1–9=2

s

4.1 0.67 9 3-1 1

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Cha pt e r3