Methods for Describing Sets of Data

Transcription

1 6 Chapter Methods for Describig Sets of Data Chapter. I a bar graph, a bar or rectagle is draw above each class of the qualitative variable correspodig to the class frequecy or class relative frequecy. I a pie chart, each slice of the pie correspods to the relative frequecy of a class of the qualitative variable..4 First, we fid the frequecy of the grade A. The sum of the frequecies for all 5 grades must be. Therefore, subtract the sum of the frequecies of the other 4 grades from. The frequecy for grade A is: ( ) = 184 = 16 To fid the relative frequecy for each grade, divide the frequecy by the total sample size,. The relative frequecy for the grade B is 36/ =.18. The rest of the relative frequecies are foud i a similar maer ad appear i the table: Grade o Statistics Exam Frequecy Relative Frequecy A: B: C: D: F: Below Total 1..6 a. The graph show is a pie chart. b. The qualitative variable described i the graph is opiio o library importace. c. The most commo opiio is more importat, with 46.% of the respoders idicatig that they thik libraries have become more importat. d. Usig MINITAB, the Pareto diagram is: 5 Chart of Percet 4 Percet 3 1 More Same Importace Less Of those who respoded to the questio, almost half (46%) believe that libraries have become more importat to their commuity. Oly 18% believe that libraries have become less importat. Copyright 17 Pearso Educatio, Ic.

2 Methods for Describig Sets of Data 7.8 a. From the pie chart, 5.4% or.54 of adults livig i the U.S. use the iteret ad pay to dowload music. From the data, 56 out of 1,3 adults or 56/1,3 =.54 of adults i the U.S. use the iteret ad pay to dowload music. These two results agree. b. Usig MINITAB, a pie chart of the data is: Pie Chart of Dowload-Music Category Pay No Pay No Pay 33.% Pay 67.%.1 a. Data were collected o 3 questios. For questios 1 ad, the resposes were either yes or o. Sice these are ot umbers, the data are qualitative. For questio 3, the resposes iclude character couts, roots of empathy, teacher desiged, other, ad oe. Sice these resposes are ot umbers, the data are qualitative. b. Usig MINITAB, bar charts for the 3 questios are: Chart of Classroom Pets Cout 3 1 Yes Classroom Pets No Copyright 17 Pearso Educatio, Ic.

3 8 Chapter Chart of Pet Visits 4 3 Cout 1 Yes Pet Visits No Chart of Educatio 3 5 Cout Character Couts Roots of Empathy Teacher desiged Educatio Other Noe c. May differet thigs ca be writte. Possible aswers might be: Most of the classroom teachers surveyed (61/ ) keep classroom pets. A little less tha half of the surveyed classroom teachers (35 / ) allow visits by pets..1 Usig MINITAB, the pie chart is: Pie Chart of Loc Rural 5.7% Category Urba Suburba Rural Suburba 3.8% Urba 61.5% Copyright 17 Pearso Educatio, Ic.

4 Methods for Describig Sets of Data 9.14 a. The two qualitative variables graphed i the bar charts are the occupatioal titles of cla idividuals i the cotiued lie ad the occupatioal titles of cla idividuals i the dropout lie. b. I the Cotiued Lie, about 63% were i either the high or the middle grade. Oly about % were i the oofficial category. I the Dropout Lie, oly about % were i either the high or middle grade while about 64% were i the oofficial category. The percetages i the low grade ad provicial official categories were about the same for the two lies..16 Usig MINITAB, the Pareto chart is:.14 Chart of Allocatio.1 Relative frequecy #5 #8 #6 #7 #1 #11 # #3 #4 #9 #1 Track Proportio withi all data. From the graph, it appears that tracks #5 ad #8 were over-utilized ad track #1 is uderutilized..18 a. Usig MINITAB, the Pareto chart of the total aual shootigs ivolvig the Bosto street gag is:.5 Chart of Total Shootigs Proportio of Total Shootigs Year Proportio withi all data. Copyright 17 Pearso Educatio, Ic.

5 1 Chapter b. Usig MINITAB, the Pareto chart of the aual shootigs of the Bosto gag members is:.5 Chart of Gag Member Shootigs Proportio of Gag Members Year Proportio withi all data. c. Because the proportio of shootigs per year dropped drastically after 7 for both the total aual shootigs ad aual shootigs of the Bosto street gag members, it appears that Operatio Ceasefire was effective.. Usig MINITAB, the side-by-side bar graphs showig the distributio of dives for the three match situatios are: Chart of Team behid, Tied, Team ahead Left Middle Right Team behid Tied Proportio.8 Team ahead Left Middle Right Dive Proportio withi all data. From the graphs, it appears that whe a team is tied or ahead, there is o differece i the proportio of times the goal-keeper dives right or left. However, if the team is behid, the goal-keeper teds to dive right much more frequetly tha left. Copyright 17 Pearso Educatio, Ic.

6 Methods for Describig Sets of Data 11. Usig MINITAB, a bar graph of the data is: 9 Chart of Measure Freq Total Visitors Payig visitors Big shows Measure Fuds raised Members The researcher cocluded that there is a large amout of variatio withi the museum commuity with regards to... performace measuremet ad evaluatio. From the data, there are oly 5 differet performace measures. I would ot say that this is a large amout. Withi these 5 categories, the umber of times each is used does ot vary that much. I would disagree with the researcher. There is ot much variatio..4 Usig MINITAB a bar chart for the Extict status versus flight capability is: 8 Chart of Extict, Flight Cout Flight Extict No Abset Yes No No Yes No Yes Yes It appears that extict status is related to flight capability. For birds that do have flight capability, most of them are preset. For those birds that do ot have flight capability, most are extict. Copyright 17 Pearso Educatio, Ic.

7 1 Chapter The bar chart for Extict status versus Nest Desity is: 6 Chart of Extict, Nest Desity 5 4 Cout 3 1 Nest Desity Extict H Abset L H No L H Yes L It appears that extict status is ot related to est desity. The proportio of birds preset, abset, ad extict appears to be very similar for est desity high ad est desity low. The bar chart for Extict status versus Habitat is: Chart of Extict, Habitat 4 3 Cout 1 Habitat Extict A TA Abset TG A TA No TG A TA Yes TG It appears that the extict status is related to habitat. For those i aerial terrestrial (TA), most species are preset. For those i groud terrestrial (TG), most species are extict. For those i aquatic, most species are preset..6 The differece betwee a bar chart ad a histogram is that a bar chart is used for qualitative data ad a histogram is used for quatitative data. For a bar chart, the categories of the qualitative variable usually appear o the horizotal axis. The frequecy or relative frequecy for each category usually appears o the vertical axis. For a histogram, values of the quatitative variable usually appear o the horizotal axis ad either frequecy or relative frequecy usually appears o the vertical axis. The quatitative data are grouped ito itervals which appear o the horizotal axis. The umber of observatios appearig i each iterval is the graphed. Bar charts usually leave spaces betwee the bars while histograms do ot..8 I a histogram, a class iterval is a rage of umbers above which the frequecy of the measuremets or relative frequecy of the measuremets is plotted. Copyright 17 Pearso Educatio, Ic.

8 Methods for Describig Sets of Data 13.3 a. This is a frequecy histogram because the umber of observatios is displayed rather tha the relative frequecies. b. There are 14 class itervals used i this histogram. c. The total umber of measuremets i the data set is Usig MINITAB, the relative frequecy histogram is:.5. Relative frequecy Class Iterval a. Usig MINITAB, the relative frequecy histogram is: Histogram of RDER.5. Relative Frequecy RDER Value b. From the graph, the proportio of subjects with RDER values betwee 75 ad 15 is about.18. The exact proportio is13 / b. From the graph, the proportio of subjects with RDER values below 15 is about / The exact proportio is.36 a. Because the label o the vertical axis is Percet, this is a relative frequecy histogram. Copyright 17 Pearso Educatio, Ic.

9 14 Chapter b. From the graph, the percetage of the 99 seior maagers who reported a high level of support for corporate sustaiability is about %..38 Usig MINITAB, the stem-ad-leaf display is: Stem-ad-Leaf Display: Depth Stem-ad-leaf of Depth N = 18 Leaf Uit = (3) The data i the stem-ad-leaf display are displayed to 1 decimal place while the actual data is displayed to decimal places. To 1 decimal place, there are 3 umbers that appear twice 14., 15.7, ad However, to decimal places, oe of these umbers are the same. Thus, o molar depth occurs more frequetly i the data..4 a. Usig MINITAB, the dot plot of the hoey dosage data is: Dotplot of Hoey Dosage Group Improvemet Score b. Both 1 ad 1 occurred 6 times i the hoey dosage group. c. From the graph i part c, 8 of the top 11 scores (7.7%) are from the hoey dosage group. Of the top 3 scores, 18 (6%) are from the hoey dosage group. This supports the coclusios of the researchers that hoey may be a preferable treatmet for the cough ad sleep difficulty associated with childhood upper respiratory tract ifectio. Copyright 17 Pearso Educatio, Ic.

10 Methods for Describig Sets of Data 15.4 a. Usig MINITAB, the stem-ad-leaf display is: Stem-ad-Leaf Display: Spider Stem-ad-leaf of Spider N = 1 Leaf Uit = (3) b. The spiders with a cotrast value of 7 or higher are i bold type i the stem-ad-leaf display i part a. There are 3 spiders i this group. c. The sample proportio of spiders that a bird could detect is 3 /1.3. Thus, we could ifer that a bird could detect a crab-spider sittig o the yellow cetral part of a daisy about 3% of the time..44 a. A stem-ad-leaf display of the data usig MINITAB is: Stem-ad-Leaf Display: FNE Stem-ad-leaf of FNE N = 5 Leaf Uit = () b. The umbers i bold i the stem-ad-leaf display represet the bulimic studets. Those umbers ted to be the larger umbers. The larger umbers idicate a greater fear of egative evaluatio. Thus, the bulimic studets ted to have a greater fear of egative evaluatio. c. A measure of reliability idicates how certai oe is that the coclusio draw is correct. Without a measure of reliability, ayoe could just guess at a coclusio. Copyright 17 Pearso Educatio, Ic.

11 16 Chapter.46 a. Usig MINITAB, histograms of the two sets of SAT scores are: Histogram of TOT11, TOT14 TOT TOT Frequecy It appears that the distributios of both sets of scores are somewhat skewed to the right. Although the distributios are ot idetical for the two years, they are similar. b. Usig MINITAB, a histogram of the differeces of the 14 ad 11 SAT scores is: 35 Histogram of Diff 3 5 Frequecy Diff -5 5 c. It appears that there are more differeces above tha below. Thus, it appears that i geeral, the 14 SAT scores are higher tha the 11 SAT scores. However, there are may differeces that are very close to. d. Wyomig had the largest improvemet i SAT scores from 11 to 14, with a icrease of 65 poits. Copyright 17 Pearso Educatio, Ic.

12 Methods for Describig Sets of Data Usig MINITAB, the side-by-side histograms are: Histogram of ZETA without, ZETA with GYPSUM ZETA without ZETA with GYPSUM Frequecy The additio of calcium/gypsum icreases the values of the zeta potetial of silica. All of the values of zeta potetial for the specimes cotaiig calcium/gypsum are greater tha all of the values of zeta potetial for the specimes without calcium/gypsum..5 A measure of cetral tedecy measures the ceter of the distributio while measures of variability measure how spread out the data are..5 A skewed distributio is a distributio that is ot symmetric ad ot cetered aroud the mea. Oe tail of the distributio is loger tha the other. If the mea is greater tha the media, the the distributio is skewed to the right. If the mea is less tha the media, the distributio is skewed to the left..54 a. For a distributio that is skewed to the left, the mea is less tha the media. b. For a distributio that is skewed to the right, the mea is greater tha the media. c. For a symmetric distributio, the mea ad media are equal..56 Assume the data are a sample. The mode is the observatio that occurs most frequetly. For this sample, the mode is 15, which occurs 3 times. The sample mea is: x x The media is the middle umber whe the data are arraged i order. The data arraged i order are: 1, 11, 1, 13, 15, 15, 15, 16, 17, 18, 18. The middle umber is the 6th umber, which is The media is the middle umber oce the data have bee arraged i order. If is eve, there is ot a sigle middle umber. Thus, to compute the media, we take the average of the middle two umbers. If is odd, there is a sigle middle umber. The media is this middle umber. Copyright 17 Pearso Educatio, Ic.

13 18 Chapter A data set with 5 measuremets arraged i order is 1, 3, 5, 6, 8. The media is the middle umber, which is 5. A data set with 6 measuremets arraged i order is 1, 3, 5, 5, 6, 8. The media is the average of the middle two umbers which is 5..6 a. xi i The mea is x 3.3. This is the average umber of sword shafts buried at each grave site. b. To fid the media, the data must be arraged i order. The data arraged i order are: There are a total of 13 observatios, which is a odd umber. The media is the middle umber which is. Half of the grave sites had or fewer sword shafts buried ad half had or more. c. The mode is the umber that occurs most frequetly. I this case, the mode is..6 a. xi i The mea is x This is the Performace 8 8 Axiety Ivetory scale for participats i 8 differet studies. b. To fid the media, the data must be arraged i order. The data arraged i order are: There are a total of 8 observatios, which is a eve umber. The media is the average of the middle umbers which are 4 ad 43. The media is Half of the studies had a PAI scale less tha 4.5 ad half had a value greater tha 4.5. c. If 39 were elimiated, the mea becomes xi i x The data arraged i order are ow: The media is the middle umber which is 43. The mea icreased by.946 while the media oly icreased by a. There are 35 observatios i the hoey dosage group. Thus, the media is the middle umber, oce the data have bee arraged i order from the smallest to the largest. The middle umber is the 18 th observatio which is 11. Copyright 17 Pearso Educatio, Ic.

14 Methods for Describig Sets of Data 19 b. There are 33 observatios i the DM dosage group. Thus, the media is the middle umber, oce the data have bee arraged i order from the smallest to the largest. The middle umber is the 17 th observatio which is 9. c. There are 37 observatios i the cotrol group. Thus, the media is the middle umber, oce the data have bee arraged i order from the smallest to the largest. The middle umber is the 19 th observatio which is 7. d. Sice the media of the hoey dosage group is the highest, the media of the DM groups is the ext highest, ad the media of the cotrol group is the smallest, we ca coclude that the hoey dosage is the most effective, the DM dosage is the ext most effective, ad othig (cotrol) is the least effective..66 a. x 77.7 The mea of the drivig performace idex values is: x The media is the average of the middle two umbers oce the data have bee arraged i order. After arragig the umbers i order, the th ad 1 st umbers are 1.75 ad The media is: The mode is the umber that occurs the most frequetly ad is 1.4. b. The average drivig performace idex is The media is Half of the players have drivig performace idex values less tha ad half have values greater tha Three of the players have the same idex value of 1.4. c. Sice the mea is greater tha the media, the data are skewed to the right. Usig MINITAB, a histogram of the data is: Histogram of Performace 1 8 Frequecy Performace Copyright 17 Pearso Educatio, Ic.

15 Chapter.68 a. The mea umber of at species discovered is: x x The media is the middle umber oce the data have bee arraged i order: 3, 3, 4, 4, 4, 5, 5, 5, 7, 49, 5. The media is 5. The mode is the value with the highest frequecy. Sice both 4 ad 5 occur 3 times, both 4 ad 5 are modes. b. For this case, we would recommed that the media is a better measure of cetral tedecy tha the mea. There are very large umbers compared to the rest. The mea is greatly affected by these umbers, while the media is ot. c. The mea total plat cover percetage for the Dry Steppe regio is: x x The media is the middle umber oce the data have bee arraged i order: 7, 4, 4, 43, 5. The media is 4. The mode is the value with the highest frequecy. Sice 4 occurs times, 4 is the mode. d. The mea total plat cover percetage for the Gobi Desert regio is: x x The media is the mea of the middle umbers oce the data have bee arraged i order: 14, 16,, 3, 3, 56. The media is The mode is the value with the highest frequecy. Sice 3 occurs times, 3 is the mode. e. Yes, the total plat cover percetage distributios appear to be differet for the regios. The percetage of plat coverage i the Dry Steppe regio is much greater tha that i the Gobi Desert regio. Copyright 17 Pearso Educatio, Ic.

16 Methods for Describig Sets of Data 1.7 a. Usig MINITAB, the simple statistics are: Descriptive Statistics: ZETA without, ZETA with GYPSUM N for Variable N Mea Media Mode Mode ZETA without ZETA with GYPSUM For the liquid solutios prepared without calcium/gypsum, the mea zeta potetial measuremet is -5.7, the media is -5.5 ad the mode is -5.. The average zeta potetial measuremet for liquid solutios prepared without calcium/gypsum is Half of the zeta potetial measuremets for liquid solutios prepared without calcium/gypsum are less tha or equal to -5.5 ad half are greater tha The most commo zeta potetial measuremet for liquid solutios prepared without calcium/gypsum is -5.. b. For the liquid solutios prepared with calcium/gypsum, the mea zeta potetial measuremet is , the media is ad the mode is The average zeta potetial measuremet for liquid solutios prepared with calcium/gypsum is Half of the zeta potetial measuremets for liquid solutios prepared with calcium/gypsum are less tha or equal to ad half are greater tha The most commo zeta potetial measuremet for liquid solutios prepared with calcium/gypsum is c. The iterpretatio remais the same as i Exercise.48. The additio of calcium/gypsum icreases the values of the zeta potetial of silica. The mea, media ad mode of the values of zeta potetial for the specimes cotaiig calcium/gypsum are greater tha the mea, media ad mode of the values of zeta potetial for the specimes without calcium/gypsum..7 a. The mea umber of power plats is: xi i x 3.95 The media is the mea of the middle umbers oce the data have bee arraged i order: 1, 1, 1, 1, 1,,, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 7, 9, 11 The media is The umber 1 occurs 5 times. The mode is 1. b. Deletig the largest umber, 11, the ew mea is: xi i x Copyright 17 Pearso Educatio, Ic.

17 Chapter The media is the middle umber oce the data have bee arraged i order: 1, 1, 1, 1, 1,,, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 7, 9 The media is 3. The umber 1 occurs 5 times. The mode is 1. By droppig the largest measuremet from the data set, the mea drops from 3.95 to The media drops from 3.5 to 3 ad the mode stays the same. c. Deletig the lowest ad highest measuremets leaves the followig: 1, 1, 1,, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 7 The ew mea is: xi i x The trimmed mea has the advatage that some possible outliers have bee elimiated..74 The primary disadvatage of usig the rage to compare variability of data sets is that the two data sets ca have the same rage ad be vastly differet with respect to data variatio. Also, the rage is greatly affected by extreme measures..76 The variace of a data set ca ever be egative. The variace of a sample is the sum of the squared deviatios from the mea divided by 1. The square of ay umber, positive or egative, is always positive. Thus, the variace will be positive. The variace is usually greater tha the stadard deviatio. However, it is possible for the variace to be smaller tha the stadard deviatio. If the data are betwee ad 1, the variace will be smaller tha the stadard deviatio. For example, suppose the data set is.8,.7,.9,.5, ad.3. The sample mea is: x x The sample variace is: s x 3. x The stadard deviatio is s Copyright 17 Pearso Educatio, Ic.

18 Methods for Describig Sets of Data 3.78 a. s x x s b. c. s s x 1 x x 17 x s s a. Rage 4 4 s x 8 x b. Rage 6 6 s x 17 x c. Rage 8 ( ) 1 s x 7 x d. Rage ( 3) 5 s x ( 5) x s s s 8.88 s This is oe possibility for the two data sets. Data Set 1:, 1,, 3, 4, 5, 6, 7, 8, 9 Data Set :,, 1, 1,,, 3, 3, 9, 9 The two sets of data above have the same rage = largest measuremet smallest measuremet 9 9. The meas for the two data sets are: x x Copyright 17 Pearso Educatio, Ic.

19 4 Chapter x x The dot diagrams for the two data sets are show below. Dotplot of Data x-bar1 Group 1 x-bar 4 Data a. s x 8 x s b. c. d. x 55 s x square feet s feet s x ( 15) x x 4 s s x square ouces s ouce.86 a. The rage is the differece betwee the largest ad smallest umbers. For this data set, the rage is b. The sample variace is s x 4 x c. The sample stadard deviatio is s Copyright 17 Pearso Educatio, Ic.

20 Methods for Describig Sets of Data 5 d. Both the rage ad the stadard deviatio have the same uits of measure as the origial variable..88 a. The rage is the differece betwee the largest ad smallest observatios ad is meters. b. The variace is: s x 16.3 x square meters c. The stadard deviatio is s meters..9 a. For Group A, the rage is 67.. From the pritout, the rage is b. For Group A, the stadard deviatio is From the pritout, s s c. Group B has the more variable permeability data. Group B has the largest rage, the largest variace ad the largest stadard deviatio..9 a. Rage s x 79 x b. Droppig the largest measuremet: Rage 91 8 x 68 x 334 s s s s s By droppig the largest observatio from the data set, the rage decreased from 1 to 8, the variace decreased from 7.54 to 5.35 ad the stadard deviatio decreased from.743 to.44. c. Droppig the largest ad smallest measuremets: Rage 91 8 x 67 x 333 s s s By droppig the largest ad smallest observatios from the data set, the rage decreased from 1 to 8, the variace decreased from 7.54 to ad the stadard deviatio decreased from.743 to.18. Copyright 17 Pearso Educatio, Ic.

21 6 Chapter.94 The Empirical Rule applies oly to data sets that are moud-shaped that are approximately symmetric, with a clusterig of measuremets about the midpoit of the distributio ad that tail off as oe moves away from the ceter of the distributio..96 Sice o iformatio is give about the data set, we ca oly use Chebyshev's rule. a. At least of the measuremets will fall betwee x s ad x s. b. At least 3/4 or 75% of the measuremets will fall betwee x s ad x s. c. At least 8/9 or 89% of the measuremets will fall betwee x 3s ad x 3s..98 a. b. x 6 x s x 6 x Iterval x Number of Measuremets i Iterval s s 1.83 Percetage s, or (6.41, 1.17) / 5.7 or 7% x s, or (4.58, 11.9) 4 4 / 5.96 or 96% x 3s, or (.75, 13.73) 5 5 / 5 1 or 1% c. The percetages i part b are i agreemet with Chebyshev's rule ad agree fairly well with the percetages give by the Empirical Rule. d. Rage s rage / 4 7 / The rage approximatio provides a satisfactory estimate of s..1 a. Usig MINITAB, the histogram is: 1 Histogram of Wheels 1 8 Frequecy Wheels Copyright 17 Pearso Educatio, Ic.

22 Methods for Describig Sets of Data 7 Although the distributio is somewhat moud-shaped, the distributio is skewed to the right. b. Usig MINITAB, the mea ad stadard deviatio are: Descriptive Statistics: Wheels Variable N Mea StDev Wheels c. x s 3.14 (1.371) (.47,5.956) d. Accordig to Chebyshev s rule, at least 3 of the measuremets will fall withi 4 stadard deviatios of the mea. e. Accordig to the Empirical Rule, approximately 95% of the measuremets will fall withi stadard deviatios of the mea. f. Twety-six of the twety-eight observatios fall withi the iterval. The proportio is The Empirical Rule does provide a good estimate of the proportio. The 8 actual percetage is 9.9% which is close to 95%..1 a. If the data are symmetric ad moud shaped, the the Empirical Rule will describe the data. About 95% of the observatios will fall withi stadard deviatio of the mea. The iterval two stadard deviatios below ad above the mea is x s 39 (6) 391 (7,51). This rage would be 7 to 51. b. To fid the umber of stadard deviatios above the mea a score of 51 would be, we subtract the mea from 51 ad divide by the stadard deviatio. Thus, a score of 51 is stadard deviatios above the mea. From the Empirical Rule, about.5 of 6 the drug dealers will have WR scores above 51. c. By the Empirical Rule, about 99.7% of the observatios will fall withi 3 stadard deviatios of the mea. Thus, early all the scores will fall withi 3 stadard deviatios of the mea. The iterval three stadard deviatios below ad above the mea is x 3s 39 3(6) 3918 (1,57). This rage would be 1 to a. Because the histogram i Exercise.34 is skewed to the right, Chebyshev s rule is more appropriate for describig the distributio of the RDER values. b. The iterval x s is (63.4) ( 48.9, 4.67). At least ¾ or 75% of the observatios will be betwee ad a. By Chebyshev s rule, at least 75% of the observatios will fall withi stadard deviatios of the mea. This iterval is x s.9 (1.1).9. ( 1.3, 3.1). Copyright 17 Pearso Educatio, Ic.

23 8 Chapter b. No. A value of 7 would be (7.9) / stadard deviatios above the mea. This would be very uusual..18 a. There are observatios with missig values for egg legth, so there are oly 13 useable observatios. x 7,885 x s x (7,885) x 77, , , s s 1, b. The data are ot symmetrical or moud-shaped. Thus, we will use Chebyshev s Rule. We kow that there are at least 8/9 or 88.9% of the observatios withi 3 stadard deviatios of the mea. Thus, at least 88.9% of the observatios will fall i the iterval: x 3s (43.99) ( 71.3, 19.69) Sice it is impossible to have egative egg legths, at least 88.9% of the egg legths will be betwee ad a. The mea ad stadard deviatio for Group A are 73.6 ad The histogram of the data from Group A is skewed to the right so Chebyshev s rule is more appropriate. We kow at least 8/9 or 88.9% of the observatios will fall withi 3 stadard deviatios of the mea. This iterval is x 3s (14.48) (3.18, 117.6). b. The mea ad stadard deviatio for Group B are ad The histogram of the data from Group B is skewed to the left so Chebyshev s rule is more appropriate. We kow at least 8/9 or 88.9% of the observatios will fall withi 3 stadard deviatios of the mea. This iterval is x 3s (1.97) (6.63, ). c. The mea ad stadard deviatio for Group C are 83.7 ad.5. The histogram of the data from Group C is skewed to the right so Chebyshev s rule is more appropriate. We kow at least 8/9 or 88.9% of the observatios will fall withi 3 stadard deviatios of the mea. This iterval is x 3s (.5) (.9, 143.). d. It appears that weatherig type B results i faster decay..11 To decide which group the patiet is most likely to come from, we will compute the z-score for each group. Copyright 17 Pearso Educatio, Ic.

24 Methods for Describig Sets of Data 9 x Group T: z x Group V: z x Group C: z The patiet is most likely to have come from Group T. The z-score for Group T is z This would ot be a uusual z-score if the patiet was i Group T. The z-scores for the other groups are both greater tha. We kow that z-scores greater tha are rather uusual..114 a. The 5 th percetile is also called the media. b. The Q L is the lower quartile. This is also the 5 th percetile or the score which has 5% of the observatios less tha it. c. The Q U is the upper quartile. This is also the 75 th percetile or the score which has 75% of the observatios less tha it..116 For moud-shaped distributios, we ca use the Empirical Rule. About 95% of the observatios will fall withi stadard deviatios of the mea. Thus, about 95% of the measuremets will have z-scores betwee - ad..118 We first compute z-scores for each x value. a. b. c. d. x 1 5 z 5 x 14 z 3 1 x z 1 x 1 5 z The above z-scores idicate that the x value i part a lies the greatest distace above the mea ad the x value of part b lies the greatest distace below the mea..1 The mea score is 153. This is the arithmetic average score of U.S. twelfth graders o the mathematics assessmet test. The 5 th percetile score is 111. This idicates that 5% of the U.S. twelfth graders scored 111 or lower o the assessmet test. The 75 th percetile score is 177. This idicates that 75% of the U.S. twelfth graders scored 177 or lower o the Copyright 17 Pearso Educatio, Ic.

25 3 Chapter assessmet test. The 9 th percetile score is 197. This idicates that 9% of the U.S. twelfth graders scored 197 or lower o the assessmet test..1 a. From Exercise.35, the proportio of fup/fumic ratios that fall above 1 is.34. The percetile rak of 1 is (1.34)1% 96.6 percetile. b. From Exercise.35, the proportio of fup/fumic ratios that fall below.4 is.695. The percetile rak of.4 is (.695)1% 69.5 th percetile..14 a. xx 3 39 The z-score associated with a score of 3 is z 1.5. This meas s 6 that a score of 3 is 1.5 stadard deviatios below the mea. b. xx The z-score associated with a score of 39 is z. Half or.5 of the s 6 observatios are below a score of Sice the 9th percetile of the study sample i the subdivisio was.37 mg/l, which is less tha the USEPA level of.15 mg/l, the water customers i the subdivisio are ot at risk of drikig water with uhealthy lead levels..18 a. If the distributio is moud-shaped ad symmetric, the the Empirical Rule ca be used. Approximately 68% of the scores will fall withi 1 stadard deviatio of the mea or betwee 53% 15% or betwee 38% ad 68%. Approximately 95% of the scores will fall withi stadard deviatios of the mea or betwee 53% (15%) or betwee 3% ad 83%. Approximately all of the scores will fall withi 3 stadard deviatios of the mea or betwee 53% 3(15%) or betwee 8% ad 98%. b. If the distributio is moud-shaped ad symmetric, the the Empirical Rule ca be used. Approximately 68% of the scores will fall withi 1 stadard deviatio of the mea or betwee 39% 1% or betwee 7% ad 51%. Approximately 95% of the scores will fall withi stadard deviatios of the mea or betwee 39% (1%) or betwee 15% ad 63%. Approximately all of the scores will fall withi 3 stadard deviatios of the mea or betwee 39% 3(1%) or betwee 3% ad 75%. c. Sice the scores o the red exam are shifted to the left of those o the blue exam, a score of % is more likely to occur o the red exam tha o the blue exam..13 Yes. From the graph i Exercise.19 c, we ca see that there are 4 observatios with z- scores greater tha 3. There is the a gap dow to.18. Those 4 observatios are quite differet from the rest of the data. After those 4 observatios, the data are fairly similar. We kow that by rakig the data, we ca reduce the ifluece of outliers. But, by doig this, we lose valuable iformatio..13 The iterquartile rage is the distace betwee the upper ad lower quartiles..134 For a moud-shaped distributio, the Empirical Rule ca be used. Almost all of the observatios will fall withi 3 stadard deviatios of the mea. Thus, almost all of the observatios will have z-scores betwee -3 ad 3. Copyright 17 Pearso Educatio, Ic.

26 Methods for Describig Sets of Data The iterquartile rage is IQR Q Q The lower ier fece Q L 1.5(IQR) 6 1.5(5).5. The upper ier fece Q U 1.5(IQR) 6 1.5(5) 1.5. The lower outer fece Q L 3(IQR) 6 3(5) 15. The upper outer fece Q U 3(IQR) 6 3(5) 16. U L With oly this iformatio, the box plot would look somethig like the followig: The whiskers exted to the ier feces uless o data poits are that small or that large. The upper ier fece is 1.5. However, the largest data poit is 1, so the whisker stops at 1. The lower ier fece is.5. The smallest data poit is 18, so the whisker exteds to.5. Sice 18 is betwee the ier ad outer feces, it is desigated with a *. We do ot kow if there is ay more tha oe data poit below.5, so we caot be sure that the box plot is etirely correct..138 To determie if the measuremets are outliers, compute the z-score. a. b. c. d. xx z.77 s 11 Sice this z-score is less tha 3 i magitude, 65 is ot a outlier. xx 157 z 3.73 s 11 Sice this z-score is more tha 3 i magitude, 1 is a outlier. xx 7 57 z s 11 Sice this z-score is less tha 3 i magitude, 7 is ot a outlier. xx z 3.77 s 11 Sice this z-score is more tha 3 i magitude, 98 is a outlier. Copyright 17 Pearso Educatio, Ic.

27 3 Chapter.14 a. Usig MINITAB, the box plot for data is give below. 1 Boxplot of Data Data b. I this data set, there is oe outlier. It correspods to the value 14. xx a. The z-score is z s 3 b. Yes, we would cosider this measuremet a outlier. Ay observatio with a z-score that has a absolute value greater tha 3 is cosidered a highly suspect outlier. xx The z-score correspodig to 155 is: z 3.5. Because this s observatio is more tha 3 stadard deviatios from the mea, it is cosidered a highly suspect outlier. It would ot be cosidered typical of the study sample..146 a. The approximate 5 th percetile PASI score before treatmet is 1. The approximate media before treatmet is 15. The approximate 75 th percetile PASI score before treatmet is 7.5. b. The approximate 5 th percetile PASI score after treatmet is 3.5. The approximate media after treatmet is 5. The approximate 75 th percetile PASI score after treatmet is 7.5. c. Sice the 75 th percetile after treatmet is lower tha the 5 th percetile before treatmet, it appears that the ichthyotherapy is effective i treatig psoriasis..148 Usig MINITAB, a boxplot of the data is: Boxplot of Rockfall Rockfall Copyright 17 Pearso Educatio, Ic.

28 Methods for Describig Sets of Data 33 From the boxplot, there is o idicatio that there are ay outliers. We will ow use the z-score criterio for determiig outliers. From Exercises.61 ad.88, x 9.7 ad s The z-score associated with the miimum value is xx z 1.18 ad the z-score associated with the maximum value is s 4.95 xx z Neither of these idicates there are ay outliers. s a. Usig MINITAB, the boxplots of the three groups are: 18 Boxplot of Hoey, DM, Cotrol Data Hoey DM Cotrol b. The media improvemet score for the hoey dosage group is larger tha the media improvemet scores for the other two groups. The media improvemet score for the DM dosage group is higher tha the media improvemet score for the cotrol group. c. Because the iterquartile rage for the DM dosage group is larger tha the iterquartile rages of the other groups, the variability of the DM group is largest. The variability of the hoey dosage group ad the cotrol group appear to be about the same. d. There appears to be oe outlier i the hoey dosage group ad oe outlier i the cotrol group..15 a. x 47 z 3 1 b. The z-score is low eough to suspect that the libraria's claim is icorrect. Eve without ay kowledge of the shape of the distributio, Chebyshev's rule states that at least 8/9 of the measuremets will fall withi 3 stadard deviatios of the mea (ad, cosequetly, at most 1/9 will be above z 3 or below z 3 ). c. The Empirical Rule states that almost oe of the measuremets should be above z 3 or below z 3. Hece, the libraria's claim is eve more ulikely. d. Whe, x 47 z 1.5 Copyright 17 Pearso Educatio, Ic.

29 34 Chapter This is ot a ulikely occurrece, whether or ot the data are moud-shaped. Hece, we would ot have reaso to doubt the libraria's claim..154 A bivariate relatioship is a relatioship betwee quatitative variables..156 A positive associatio betwee two variables meas that as oe variable icreases, the other variable teds to also icrease. A egative associatio betwee two variables meas that as oe variable icreases, the other variable teds to decrease..158 Usig MINITAB, the scatterplot is as follows: 18 Scatterplot of Variable vs Variable Variable Variable It appears that as variable 1 icreases, variable also icreases..16 Usig MINITAB, a scatter plot of the data is: Scatterplot of SLUGPCT vs ELEVATION SLUGPCT ELEVATION If oe uses the oe obvious outlier (Dever), the there does appear to be a tred i the data. As the elevatio icreases, the sluggig percetage teds to icrease. However, if the outlier is removed, the it does ot look like there is a tred to the data. Copyright 17 Pearso Educatio, Ic.

30 Methods for Describig Sets of Data a. A scattergram of the data is: 9 Scatterplot of Strikes vs Age Strikes Age b. There appears to be a tred. As the age icreases, the umber of strikes teds to decrease..164 Usig MINITAB, a scatterplot of the data is: 7 Scatterplot of Freq vs Resoace 6 5 Freq Resoace 15 5 There is a icreasig tred ad there is very little variatio i the plot. This supports the researcher s theory..166 a. Usig MINITAB, a graph of the Athropogeic Idex agaist the Natural Origi Idex is: 9 Scatterplot of F-Athro vs F-Natural F-Athro F-Natural Copyright 17 Pearso Educatio, Ic.

31 36 Chapter This graph does ot support the theory that there is a straight-lie relatioship betwee the Athropogeic Idex agaist the Natural Origi Idex. There are several poits that do ot lie o a straight lie. b. After deletig the three forests with the largest athropogeic idices, the graph of the data is: Scatterplot of F-Athro vs F-Natural F-Athro F-Natural After deletig the 3 data poits, the relatioship betwee the Athropogeic Idex agaist the Natural Origi Idex is much closer to a straight lie..168 Usig MINITAB, a scattergram of the data is: 7 Scatterplot of Mass vs Time Mass Time Yes, there appears to be a egative tred i this data. As time icreases, the mass teds to decrease. There appears to be a curviliear relatioship. As time icreases, mass decreases at a decreasig rate..17 Oe way the bar graph ca mislead the viewer is that the vertical axis has bee cut off. Istead of startig at, the vertical axis starts at 1. Aother way the bar graph ca mislead the viewer is that as the bars get taller, the widths of the bars also icrease..17 a. This graph is misleadig because it looks like as the days are icreasig, the umber of barrels collected per day is also icreasig. However, the bars are the cumulative umber of barrels collected. The cumulative value ca ever decrease. Copyright 17 Pearso Educatio, Ic.

32 Methods for Describig Sets of Data 37 b. Usig MINITAB, the graph of the daily collectio of oil is: Chart of Barrells 5 Barrells May-16 May-17 May-18 May-19 May- Day May-1 May- May-3 From this graph, it shows that there has ot bee a steady improvemet i the suctioig process. There was a icrease for 3 days, the a levelig off for 3 days, the a decrease..174 The rage ca be greatly affected by extreme measures, while the stadard deviatio is ot as affected..176 The z-score approach for detectig outliers is based o the distributio beig fairly moudshaped. If the data are ot moud-shaped, the the box plot would be preferred over the z- score method for detectig outliers..178 Oe techique for distortig iformatio o a graph is by stretchig the vertical axis by startig the vertical axis somewhere above..18 From part a of Exercise.179, the 3 z-scores are 1, 1 ad. Sice oe of these z-scores are greater tha i absolute value, oe of them are outliers. From part b of Exercise.179, the 3 z-scores are, ad 4. There is oly oe z-score greater tha i absolute value. The score of 8 (associated with the z-score of 4) would be a outlier. Very few observatios are as far away from the mea as 4 stadard deviatios. From part c of Exercise.179, the 3 z-scores are 1, 3, ad 4. Two of these z-scores are greater tha i absolute value. The scores associated with the two z-scores 3 ad 4 (7 ad 8) would be cosidered outliers. From part d of Exercise.179, the 3 z-scores are.1,.3, ad.4. Sice oe of these z-scores are greater tha i absolute value, oe of them are outliers..18 rage / 4 / a. x x x x 6 5 Copyright 17 Pearso Educatio, Ic.

33 38 Chapter s x 3 x b. x x x 5 x s x 5 x c. x x x 49 x 7 7 s x 49 x s s s d. x x x 1 x 3 4 s x 1 x s e. a) x s 6 (5.) ( 4.4, 16.4) All or 1% of the observatios are i this iterval. b) x s 6.5 (5.3) ( 4.39, 16.89) All or 1% of the observatios are i this iterval. c) x s 7 (6.14) ( 5.8, 19.8) All or 1% of the observatios are i this iterval. d) x s 3 () 3 (3,3) All or 1% of the observatios are i this iterval. Copyright 17 Pearso Educatio, Ic.

34 Methods for Describig Sets of Data Suppose we costruct a relative frequecy bar chart for this data. This will allow the archaeologists to compare the differet categories easier. First, we must compute the relative frequecies for the categories. These are foud by dividig the frequecies i each category by the total 837. For the burished category, the relative frequecy is133 / The rest of the relative frequecies are foud i a similar fashio ad are listed i the table. Pot Category Number Foud Computatio Relative Frequecy Burished / Moochrome / Slipped / Curviliear Decoratio / Geometric Decoratio / Naturalistic Decoratio 4 4 / Cycladic White clay 4 4 / Coical cup clay / 837. Total A relative frequecy bar chart is:.6 Chart of Pot Category.5 Relative Frequecy Burished Moochrome Slipped Curviliear Geometric Pot Category Naturalistic Cycladic Coical Proportio withi all data. The most frequetly foud type of pot was the Moochrome. Of all the pots foud, 55% were Moochrome. The ext most frequetly foud type of pot was the Paited i Geometric Decoratio. Of all the pots foud, 19.7% were of this type. Very few pots of the types Paited i aturalistic decoratio, Cycladic white clay, ad Coical cup clay were foud. Copyright 17 Pearso Educatio, Ic.

35 4 Chapter.188 a. Usig MINITAB, the stem-ad-leaf display is as follows. Character Stem-ad-Leaf Display Stem-ad-leaf of Books N = 14 Leaf Uit = (3) b. The leaves that correspod to studets who eared a A grade are highlighted i the graph above. Those studets who eared A s teded to read the most books. xi i c. The mea is x This is the average umber of books read per studet. To fid the media, the data must be arraged i order. I this problem, the data are already arraged i order. There are a total of 14 observatios, which is a eve umber. The media is the average of the middle umbers which are 3 ad 34. The media is Half of the studets read more tha 3 books ad half read fewer. The mode is the observatio appearig the most. I this data set, there are two modes - 34 ad 4 because each appears times i the data set. The most frequet umber of books read is either 34 or 4. d. Sice the mea ad the media are almost the same, the distributio of the data set is approximately symmetric. This ca be verified by the stem-ad-leaf display i part a. e. For those studets who eared A, the mea is xi i x The variace is s x 96 x 11, The stadard deviatio is s s Copyright 17 Pearso Educatio, Ic.

36 Methods for Describig Sets of Data 41 f. For those studets who eared a B or C, the mea is xi i x The variace is s x 147 x 3, The stadard deviatio is s s g. The studets who received A s have a more variable distributio of the umber of books read. The variace ad stadard deviatio for this group are greater tha the correspodig values for the B-C group. xx 4 37 h. The z-score for a score of 4 books is z.345. Thus, someoe who s 8.71 read 4 books read more tha the average umber of books, but that umber is ot very uusual. xx i. The z-score for a score of 4 books is z 1.8. Thus, someoe who s 8.56 read 4 books read may more tha the average umber of books. Very few studets who received a B or a C read more tha 4 books. j. The group of studets who eared A s is more likely to have read 4 books. For this group, the z-score correspodig to 4 books is.34. This is ot uusual. For the B-C group, the z-score correspodig to 4 books is 1.8. This is close to stadard deviatios from the mea. This would be fairly uusual..19 A pie chart of the data is: Pie Chart of Drive Star % 4.1% % Category % More tha half of the cars received 4 star ratigs (6.%). A little less tha a quarter of the cars tested received ratigs of 3 stars or less. Copyright 17 Pearso Educatio, Ic.

37 4 Chapter.19 a. Usig MINITAB, the descriptive statistics are: Descriptive Statistics: Ratio Variable N Mea StDev Miimum Maximum Ratio xx The z-score associated with the largest ratio is z.45 s.634 xx The z-score associated with the smallest ratio is z 1.98 s.634 xx The z-score associated with the mea ratio is z s.634 b. Yes, I would cosider the z-score associated with the largest ratio to be uusually large. We kow if the data are approximately moud-shaped that approximately 95% of the observatios will be withi stadard deviatios of the mea. A z-score of.45 would idicate that less tha.5% of all the measuremets will be larger tha this value. c. Usig MINITAB, the box plot is: Boxplot of Ratio Ratio From this box plot, there are o observatios marked as outliers..194 a. Usig MINITAB, a histogram of the data is: 1 Histogram of ph 1 8 Percet ph Copyright 17 Pearso Educatio, Ic.

38 Methods for Describig Sets of Data 43 From the graph, it looks like the proportio of wells with ph levels less tha 7. is: b. Usig MINITAB, a histogram of the MTBE levels for those wells with detectible levels is: 8 Histogram of MTBE-Level MTBE-Detect = Detect 7 6 Percet MTBE-Level 4 5 From the graph, it looks like the proportio of wells with MTBE levels greater tha 5 is: c. The sample mea is: x i1 x The variace is: i x i , , x 1, s The stadard deviatio is: s s x s 7.47 (.8169) (5.793, 9.68). From the histogram i part a, the data look approximately moud-shaped. From the Empirical Rule, we would expect about 95% of the wells to fall i this rage. I fact, 1 of 3 or 95.1% of the wells have ph levels betwee ad d. The sample mea of the wells with detectible levels of MTBE is: xi i x Copyright 17 Pearso Educatio, Ic.

39 44 Chapter The variace is: s x xi The stadard deviatio is: s s x s (8.756) ( 14.6,.94). From the histogram i part b, the data do ot look moud-shaped. From Chebyshev s Rule, we would expect at least ¾ or 75% of the wells to fall i this rage. I fact, 67 of 7 or 95.7% of the wells have MTBE levels betwee ad a. Usig MINITAB, the dot plot for the 9 measuremets is: Dotplot of Cesium Cesium b. Usig MINITAB, the stem-ad-leaf display is: Character Stem-ad-Leaf Display Stem-ad-leaf of Cesium N = 9 Leaf Uit = (3) Copyright 17 Pearso Educatio, Ic.

40 Methods for Describig Sets of Data 45 c. Usig MINITAB, the histogram is: Histogram of Cesium. 1.5 Frequecy Cesium d. The stem-ad-leaf display appears to be more iformative tha the other graphs. Sice there are oly 9 observatios, the histogram ad dot plot have very few observatios per category. e. There are 4 observatios with radioactivity level of -5. or lower. The proportio of measuremets with a radioactivity level of -5. or lower is 4/ a. From the histogram, the data do ot follow the true moud-shape very well. The itervals i the middle are much higher tha they should be. I additio, there are some extremely large velocities ad some extremely small velocities. Because the data do ot follow a moud-shaped distributio, the Empirical Rule would ot be appropriate. b. Usig Chebyshev's rule, at least11/4 11/1615/16or 93.8% of the velocities will fall withi 4 stadard deviatios of the mea. This iterval is: x 4s 7,117 4(1,8) 7,117 5,1 (1,997, 3,37) At least 93.75% of the velocities will fall betwee 1,997 ad 3,37 km per secod. c. Sice the data look approximately symmetric, the mea would be a good estimate for the velocity of galaxy cluster A14. Thus, this estimate would be 7,117 km per secod.. a. Usig MINITAB, the boxplot is: 35 Boxplot of Freckle 3 5 Freckle Copyright 17 Pearso Educatio, Ic.