Section 6 4 Application of the Normal Ditribution 307 The area between the two value i the anwer, 0.885109. To find a z core correponding to a cumulative area: P(Z z) 0.0250 1. Click the f x icon and elect the Statitical function category. 2. Select the NORMSINV function and enter 0.0250. 3. Click [OK]. The z core whoe cumulative area i 0.0250 i the anwer, 1.96. 6 4 Application of the Normal Ditribution Objective 4 Find probabilitie for a normally ditributed variable by tranforming it into a tandard normal variable. The tandard normal ditribution curve can be ued to olve a wide variety of practical problem. The only requirement i that the variable be normally or approximately normally ditributed. There are everal mathematical tet to determine whether a variable i normally ditributed. See the Critical Thinking Challenge on page 342. For all the problem preented in thi chapter, one can aume that the variable i normally or approximately normally ditributed. To olve problem by uing the tandard normal ditribution, tranform the original variable to a tandard normal ditribution variable by uing the formula value mean z tandard deviation or z X m Thi i the ame formula preented in Section 3 4. Thi formula tranform the value of the variable into tandard unit or z value. Once the variable i tranformed, then the Procedure Table and Table E in Appendix C can be ued to olve problem. For example, uppoe that the core for a tandardized tet are normally ditributed, have a mean of 100, and have a tandard deviation of 15. When the core are tranformed to z value, the two ditribution coincide, a hown in Figure 6 28. (Recall that the z ditribution ha a mean of 0 and a tandard deviation of 1.) Figure 6 28 Tet Score and Their Correponding z Value 3 2 1 0 1 2 3 55 70 85 100 115 130 145 z To olve the application problem in thi ection, tranform the value of the variable to z value and then find the area under the tandard normal ditribution, a hown in Section 6 3. 6 23
308 Chapter 6 The Normal Ditribution Example 6 14 The mean number of hour an American worker pend on the computer i 3.1 hour per workday. Aume the tandard deviation i 0.5 hour. Find the percentage of worker who pend le than 3.5 hour on the computer. Aume the variable i normally ditributed. Source: USA TODAY. Solution Step 1 Draw the figure and repreent the area a hown in Figure 6 29. Figure 6 29 Area Under a Normal Curve for Example 6 14 3.1 3.5 Step 2 Find the z value correponding to 3.5. z X m 3.5 3.1 0.5 0.80 Hence, 3.5 i 0.8 tandard deviation above the mean of 3.1, a hown for the z ditribution in Figure 6 30. Figure 6 30 Area and z Value for Example 6 14 0 0.8 Step 3 Find the area by uing Table E. The area between z 0 and z 0.8 i 0.2881. Since the area under the curve to the left of z 0.8 i deired, add 0.5000 to 0.2881 (0.5000 0.2881 0.7881). Therefore, 78.81% of the worker pend le than 3.5 hour per workday on the computer. Example 6 15 Each month, an American houehold generate an average of 28 pound of newpaper for garbage or recycling. Aume the tandard deviation i 2 pound. If a houehold i elected at random, find the probability of it generating a. Between 27 and 31 pound per month. b. More than 30.2 pound per month. Aume the variable i approximately normally ditributed. Source: Michael D. Shook and Robert L. Shook, The Book of Odd. 6 24
Section 6 4 Application of the Normal Ditribution 309 Solution a Step 1 Draw the figure and repreent the area. See Figure 6 31. Figure 6 31 Area Under a Normal Curve for Part a of Example 6 15 H itorical Note Atronomer in the late 1700 and the 1800 ued the principle underlying the normal ditribution to correct meaurement error that occurred in charting the poition of the planet. Step 2 Step 3 Find the two z value. z 1 X m z 2 X m 27 28 2 31 28 2 27 1 2 0.5 3 2 1.5 28 31 Find the appropriate area, uing Table E. The area between z 0 and z 0.5 i 0.1915. The area between z 0 and z 1.5 i 0.4332. Add 0.1915 and 0.4332 (0.1915 0.4332 0.6247). Thu, the total area i 62.47%. See Figure 6 32. Figure 6 32 Area and z Value for Part a of Example 6 15 27 0.5 28 31 0 1.5 Hence, the probability that a randomly elected houehold generate between 27 and 31 pound of newpaper per month i 62.47%. Solution b Step 1 Draw the figure and repreent the area, a hown in Figure 6 33. Figure 6 33 Area Under a Normal Curve for Part b of Example 6 15 28 30.2 6 25
310 Chapter 6 The Normal Ditribution Step 2 Find the z value for 30.2. z X m 30.2 28 2 2.2 2 1.1 Step 3 Find the appropriate area. The area between z 0 and z 1.1 obtained from Table E i 0.3643. Since the deired area i in the right tail, ubtract 0.3643 from 0.5000. 0.5000 0.3643 0.1357 Hence, the probability that a randomly elected houehold will accumulate more than 30.2 pound of newpaper i 0.1357, or 13.57%. A normal ditribution can alo be ued to anwer quetion of How many? Thi application i hown in Example 6 16. Example 6 16 The American Automobile Aociation report that the average time it take to repond to an emergency call i 25 minute. Aume the variable i approximately normally ditributed and the tandard deviation i 4.5 minute. If 80 call are randomly elected, approximately how many will be reponded to in le than 15 minute? Source: Michael D. Shook and Robert L. Shook, The Book of Odd. Solution To olve the problem, find the area under a normal ditribution curve to the left of 15. Step 1 Draw a figure and repreent the area a hown in Figure 6 34. Figure 6 34 Area Under a Normal Curve for Example 6 16 15 25 Step 2 Find the z value for 15. Step 3 Find the appropriate area. The area obtained from Table E i 0.4868, which correpond to the area between z 0 and z 2.22. Ue 2.22. Step 4 Subtract 0.4868 from 0.5000 to get 0.0132. Step 5 z X m 15 25 4.5 2.22 To find how many call will be made in le than 15 minute, multiply the ample ize 80 by 0.0132 to get 1.056. Hence, 1.056, or approximately 1, call will be reponded to in under 15 minute. 6 26
Section 6 4 Application of the Normal Ditribution 311 Note: For problem uing percentage, be ure to change the percentage to a decimal before multiplying. Alo, round the anwer to the nearet whole number, ince it i not poible to have 1.056 call. Finding Data Value Given Specific Probabilitie A normal ditribution can alo be ued to find pecific data value for given percentage. Thi application i hown in Example 6 17. Example 6 17 Objective 5 Find pecific data value for given percentage, uing the tandard normal ditribution. To qualify for a police academy, candidate mut core in the top 10% on a general abilitie tet. The tet ha a mean of 200 and a tandard deviation of 20. Find the lowet poible core to qualify. Aume the tet core are normally ditributed. Solution Since the tet core are normally ditributed, the tet value X that cut off the upper 10% of the area under a normal ditribution curve i deired. Thi area i hown in Figure 6 35. Figure 6 35 Area Under a Normal Curve for Example 6 17 10%, or 0.1000 Step 1 Step 2 Work backward to olve thi problem 200 X Subtract 0.1000 from 0.5000 to get the area under the normal ditribution between 200 and X: 0.5000 0.1000 0.4000. Find the z value that correpond to an area of 0.4000 by looking up 0.4000 in the area portion of Table E. If the pecific value cannot be found, ue the cloet value in thi cae 0.3997, a hown in Figure 6 36. The correponding z value i 1.28. (If the area fall exactly halfway between two z value, ue the larger of the two z value. For example, the area 0.4500 fall halfway between 0.4495 and 0.4505. In thi cae ue 1.65 rather than 1.64 for the z value.) Figure 6 36 Finding the z Value from Table E (Example 6 17) z.00.01.02.03.04.05.06.07.08.09 0.0 0.1 0.2 Specific value... 1.1 1.2 1.3 1.4 0.3997 Cloet value 0.4000 0.4015... 6 27
312 Chapter 6 The Normal Ditribution Intereting Fact American are the larget conumer of chocolate. We pend $16.6 billion annually. Step 3 Subtitute in the formula z (X m)/ and olve for X. 1.28 X 200 20 1.28 20 200 X 25.60 200 X 225.60 X 226 X A core of 226 hould be ued a a cutoff. Anybody coring 226 or higher qualifie. Intead of uing the formula hown in tep 3, one can ue the formula X z m. Thi i obtained by olving z X m for X a hown. z X z X X z Multiply both ide by. Add m to both ide. Exchange both ide of the equation. Formula for Finding X When one mut find the value of X, the following formula can be ued: X z m Example 6 18 For a medical tudy, a reearcher wihe to elect people in the middle 60% of the population baed on blood preure. If the mean ytolic blood preure i 120 and the tandard deviation i 8, find the upper and lower reading that would qualify people to participate in the tudy. Solution Aume that blood preure reading are normally ditributed; then cutoff point are a hown in Figure 6 37. Figure 6 37 Area Under a Normal Curve for Example 6 18 20% 60% 30% 20% X 2 120 X 1 6 28
Section 6 4 Application of the Normal Ditribution 313 Note that two value are needed, one above the mean and one below the mean. Find the value to the right of the mean firt. The cloet z value for an area of 0.3000 i 0.84. Subtituting in the formula X z m, one get X 1 z m (0.84)(8) 120 126.72 On the other ide, z 0.84; hence, X 2 ( 0.84)(8) 120 113.28 Therefore, the middle 60% will have blood preure reading of 113.28 X 126.72. A hown in thi ection, a normal ditribution i a ueful tool in anwering many quetion about variable that are normally or approximately normally ditributed. Determining Normality A normally haped or bell-haped ditribution i only one of many hape that a ditribution can aume; however, it i very important ince many tatitical method require that the ditribution of value (hown in ubequent chapter) be normally or approximately normally haped. There are everal way tatitician check for normality. The eaiet way i to draw a hitogram for the data and check it hape. If the hitogram i not approximately bellhaped, then the data are not normally ditributed. Skewne can be checked by uing Pearon index PI of kewne. The formula i PI 3 X median If the index i greater than or equal to 1 or le than or equal to 1, it can be concluded that the data are ignificantly kewed. In addition, the data hould be checked for outlier by uing the method hown in Chapter 3, page 143. Even one or two outlier can have a big effect on normality. Example 6 19 and 6 20 how how to check for normality. Example 6 19 A urvey of 18 high-technology firm howed the number of day inventory they had on hand. Determine if the data are approximately normally ditributed. 5 29 34 44 45 63 68 74 74 81 88 91 97 98 113 118 151 158 Source: USA TODAY. Solution Step 1 Contruct a frequency ditribution and draw a hitogram for the data, a hown in Figure 6 38. Cla Frequency 5 29 2 30 54 3 55 79 4 80 104 5 105 129 2 130 154 1 155 179 1 6 29
314 Chapter 6 The Normal Ditribution Figure 6 38 Hitogram for Example 6 19 Frequency 5 4 3 2 1 4.5 29.5 54.5 79.5 104.5 129.5 154.5 179.5 Day Since the hitogram i approximately bell-haped, one can ay that the ditribution i approximately normal. Step 2 Check for kewne. For thee data, X 79.5, median 77.5, and 40.5. Uing Pearon index of kewne give 3 79.5 77.5 PI 40.5 0.148 In thi cae, the PI i not greater than 1 or le than 1, o it can be concluded that the ditribution i not ignificantly kewed. Step 3 Check for outlier. Recall that an outlier i a data value that lie more than 1.5 (IQR) unit below Q 1 or 1.5 (IQR) unit above Q 3. In thi cae, Q 1 45 and Q 3 98; hence, IQR Q 3 Q 1 98 45 53. An outlier would be a data value le than 45 1.5(53) 34.5 or a data value larger than 98 1.5(53) 177.5. In thi cae, there are no outlier. Since the hitogram i approximately bell-haped, the data are not ignificantly kewed, and there are no outlier, it can be concluded that the ditribution i approximately normally ditributed. Example 6 20 The data hown conit of the number of game played each year in the career of Baeball Hall of Famer Bill Mazeroki. Determine if the data are approximately normally ditributed. 81 148 152 135 151 152 159 142 34 162 130 162 163 143 67 112 70 Source: Greenburg Tribune Review. Solution Step 1 Contruct a frequency ditribution and draw a hitogram for the data. See Figure 6 39. 6 30
Section 6 4 Application of the Normal Ditribution 315 Figure 6 39 Hitogram for Example 6 20 Frequency 8 7 6 5 4 3 Cla Frequency 34 58 1 59 83 3 84 108 0 109 133 2 134 158 7 159 183 4 2 1 33.5 58.5 83.5 108.5 133.5 158.5 183.5 Game U nuual Stat The average amount of money tolen by a pickpocket each time i $128. The hitogram how that the frequency ditribution i omewhat negatively kewed. Step 2 Check for kewne; X 127.24, median 143, and 39.87. PI 3 X median 3 127.24 143 39.87 1.19 Since the PI i le than 1, it can be concluded that the ditribution i ignificantly kewed to the left. Step 3 Check for outlier. In thi cae, Q 1 96.5 and Q 3 155.5. IQR Q 3 Q 1 155.5 96.5 59. Any value le than 96.5 1.5(59) 8 or above 155.5 1.5(59) 244 i conidered an outlier. There are no outlier. In ummary, the ditribution i omewhat negatively kewed. Another method that i ued to check normality i to draw a normal quantile plot. Quantile, ometime called fractile, are value that eparate the data et into approximately equal group. Recall that quartile eparate the data et into four approximately equal group, and decile eparate the data et into 10 approximately equal group. A normal quantile plot conit of a graph of point uing the data value for the x coordinate and the z value of the quantile correponding to the x value for the y coordinate. (Note: The calculation of the z value are omewhat complicated, and technology i uually ued to draw the graph. The Technology Step by Step ection how how to draw a normal quantile plot.) If the point of the quantile plot do not lie in an approximately traight line, then normality can be rejected. There are everal other method ued to check for normality. A method uing normal probability graph paper i hown in the Critical Thinking Challenge ection at the end of thi chapter, and the chi-quare goodne-of-fit tet i hown in Chapter 11. Two other tet ometime ued to check normality are the Kolmogorov- Smikirov tet and the Lilliefor tet. An explanation of thee tet can be found in advanced textbook. 6 31
316 Chapter 6 The Normal Ditribution Applying the Concept 6 4 Smart People Aume you are thinking about tarting a Mena chapter in your home town of Viiala, California, which ha a population of about 10,000 people. You need to know how many people would qualify for Mena, which require an IQ of at leat 130. You realize that IQ i normally ditributed with a mean of 100 and a tandard deviation of 15. Complete the following. 1. Find the approximate number of people in Viiala that are eligible for Mena. 2. I it reaonable to continue your quet for a Mena chapter in Viiala? 3. How would you proceed to find out how many of the eligible people would actually join the new chapter? Be pecific about your method of gathering data. 4. What would be the minimum IQ core needed if you wanted to tart an Ultra-Mena club that included only the top 1% of IQ core? See page 344 for the anwer. Exercie 6 4 1. The average admiion charge for a movie i $5.39. If the ditribution of admiion charge i normal with a tandard deviation of $0.79, what i the probability that a randomly elected admiion charge i le than $3.00? Source: N.Y. Time Almanac. 2. The average alary for firt-year teacher i $27,989. If the ditribution i approximately normal with $3250, what i the probability that a randomly elected firt-year teacher make thee alarie? a. Between $20,000 and $30,000 a year b. Le than $20,000 a year Source: N.Y. Time Almanac. 3. The average daily jail population in the United State i 618,319. If the ditribution i normal and the tandard deviation i 50,200, find the probability that on a randomly elected day the jail population i a. Greater than 700,000. b. Between 500,000 and 600,000. Source: N.Y. Time Almanac. 4. The national average SAT core i 1019. If we aume a normal ditribution with 90, what i the 90th percentile core? What i the probability that a randomly elected core exceed 1200? Source: N.Y. Time Almanac. 5. The average number of calorie in a 1.5-ounce chocolate bar i 225. Suppoe that the ditribution of calorie i approximately normal with 10. Find the probability that a randomly elected chocolate bar will have a. Between 200 and 220 calorie. b. Le than 200 calorie. Source: The Doctor Pocket Calorie, Fat, and Carbohydrate Counter. 6. The average age of CEO i 56 year. Aume the variable i normally ditributed. If the tandard deviation i 4 year, find the probability that the age of a randomly elected CEO will be in the following range. a. Between 53 and 59 year old b. Between 58 and 63 year old c. Between 50 and 55 year old Source: Michael D. Shook and Robert L. Shook, The Book of Odd. 7. The average alary for a Queen College full profeor i $85,900. If the average alarie are normally ditributed with a tandard deviation of $11,000, find thee probabilitie. a. The profeor make more than $90,000. b. The profeor make more than $75,000. Source: AAUP, Chronicle of Higher Education. 8. Full-time Ph.D. tudent receive an average of $12,837 per year. If the average alarie are normally ditributed with a tandard deviation of $1500, find thee probabilitie. a. The tudent make more than $15,000. b. The tudent make between $13,000 and $14,000. Source: U.S. Education Dept., Chronicle of Higher Education. 9. A urvey found that people keep their microwave oven an average of 3.2 year. The tandard deviation i 0.56 year. If a peron decide to buy a new microwave oven, find the probability that he or he ha owned the old oven for the following amount of time. Aume the variable i normally ditributed. a. Le than 1.5 year b. Between 2 and 3 year 6 32
Section 6 4 Application of the Normal Ditribution 317 c. More than 3.2 year d. What percent of microwave oven would be replaced if a warranty of 18 month were given? 10. The average commute to work (one way) i 25.5 minute according to the 2000 Cenu. If we aume that commuting time are normally ditributed with a tandard deviation of 6.1 minute, what i the probability that a randomly elected commuter pend more than 30 minute a day commuting one way? Source: N.Y. Time Almanac. 11. The average credit card debt for college enior i $3262. If the debt i normally ditributed with a tandard deviation of $1100, find thee probabilitie. a. That the enior owe at leat $1000 b. That the enior owe more than $4000 c. That the enior owe between $3000 and $4000 Source: USA TODAY. 12. The average time a peron pend at the Barefoot Landing Seaquarium i 96 minute. The tandard deviation i 17 minute. Aume the variable i normally ditributed. If a viitor i elected at random, find the probability that he or he will pend the following time at the eaquarium. a. At leat 120 minute b. At mot 80 minute c. Sugget a time for a bu to return to pick up a group of tourit. 13. The average time for a mail carrier to cover hi route i 380 minute, and the tandard deviation i 16 minute. If one of thee trip i elected at random, find the probability that the carrier will have the following route time. Aume the variable i normally ditributed. a. At leat 350 minute b. At mot 395 minute c. How might a mail carrier etimate a range for the time he or he will pend en route? 14. During October, the average temperature of Whitman Lake i 53.2 and the tandard deviation i 2.3. Aume the variable i normally ditributed. For a randomly elected day in October, find the probability that the temperature will be a follow. a. Above 54 b. Below 60 c. Between 49 and 55 d. If the lake temperature were above 60, would you call it very warm? 15. The average waiting time to be eated for dinner at a popular retaurant i 23.5 minute, with a tandard deviation of 3.6 minute. Aume the variable i normally ditributed. When a patron arrive at the retaurant for dinner, find the probability that the patron will have to wait the following time. a. Between 15 and 22 minute b. Le than 18 minute or more than 25 minute c. I it likely that a peron will be eated in le than 15 minute? 16. A local medical reearch aociation propoe to ponor a footrace. The average time it take to run the coure i 45.8 minute with a tandard deviation of 3.6 minute. If the aociation decide to include only the top 25% of the racer, what hould be the cutoff time in the tryout run? Aume the variable i normally ditributed. Would a peron who run the coure in 40 minute qualify? 17. A marine ale dealer find that the average price of a previouly owned boat i $6492. He decide to ell boat that will appeal to the middle 66% of the market in term of price. Find the maximum and minimum price of the boat the dealer will ell. The tandard deviation i $1025, and the variable i normally ditributed. Would a boat priced at $5550 be old in thi tore? 18. The average charitable contribution itemized per income tax return in Pennylvania i $792. Suppoe that the ditribution of contribution i normal with a tandard deviation of $103. Find the limit for the middle 50% of contribution. Source: IRS, Statitic of Income Bulletin. 19. A contractor decided to build home that will include the middle 80% of the market. If the average ize of home built i 1810 quare feet, find the maximum and minimum ize of the home the contractor hould build. Aume that the tandard deviation i 92 quare feet and the variable i normally ditributed. Source: Michael D. Shook and Robert L. Shook, The Book of Odd. 20. If the average price of a new home i $145,500, find the maximum and minimum price of the houe that a contractor will build to include the middle 80% of the market. Aume that the tandard deviation of price i $1500 and the variable i normally ditributed. Source: Michael D. Shook and Robert L. Shook, The Book of Odd. 21. The average price of a peronal computer (PC) i $949. If the computer price are approximately normally ditributed and $100, what i the probability that a randomly elected PC cot more than $1200? The leat expenive 10% of peronal computer cot le than what amount? Source: N.Y. Time Almanac. 22. To help tudent improve their reading, a chool ditrict decide to implement a reading program. It i to be adminitered to the bottom 5% of the tudent in the ditrict, baed on the core on a reading achievement exam. If the average core for the tudent in the ditrict i 122.6, find the cutoff core that will make a tudent eligible for the program. The tandard deviation i 18. Aume the variable i normally ditributed. 6 33
318 Chapter 6 The Normal Ditribution 23. An automobile dealer find that the average price of a previouly owned vehicle i $8256. He decide to ell car that will appeal to the middle 60% of the market in term of price. Find the maximum and minimum price of the car the dealer will ell. The tandard deviation i $1150, and the variable i normally ditributed. 24. The average age of Amtrak paenger train car i 19.4 year. If the ditribution of age i normal and 20% of the car are older than 22.8 year, find the tandard deviation. Source: N.Y. Time Almanac. 25. The average length of a hopital tay i 5.9 day. If we aume a normal ditribution and a tandard deviation of 1.7 day, 15% of hopital tay are le than how many day? Twenty-five percent of hopital tay are longer than how many day? Source: N.Y. Time Almanac. 26. A mandatory competency tet for high chool ophomore ha a normal ditribution with a mean of 400 and a tandard deviation of 100. a. The top 3% of tudent receive $500. What i the minimum core you would need to receive thi award? b. The bottom 1.5% of tudent mut go to ummer chool. What i the minimum core you would need to tay out of thi group? 27. An advertiing company plan to market a product to low-income familie. A tudy tate that for a particular area, the average income per family i $24,596 and the tandard deviation i $6256. If the company plan to target the bottom 18% of the familie baed on income, find the cutoff income. Aume the variable i normally ditributed. 28. If a one-peron houehold pend an average of $40 per week on grocerie, find the maximum and minimum dollar amount pent per week for the middle 50% of oneperon houehold. Aume that the tandard deviation i $5 and the variable i normally ditributed. Source: Michael D. Shook and Robert L. Shook, The Book of Odd. 29. The mean lifetime of a writwatch i 25 month, with a tandard deviation of 5 month. If the ditribution i normal, for how many month hould a guarantee be made if the manufacturer doe not want to exchange more than 10% of the watche? Aume the variable i normally ditributed. 30. To qualify for ecurity officer training, recruit are teted for tre tolerance. The core are normally ditributed, with a mean of 62 and a tandard deviation of 8. If only the top 15% of recruit are elected, find the cutoff core. 31. In the ditribution hown, tate the mean and tandard deviation for each. Hint: See Figure 6 5 a. c. and 6 6. Alo the vertical line are 1 tandard deviation apart. 60 80 100 120 140 160 180 b. 7.5 10 12.5 15 17.5 20 22.5 15 20 25 30 35 40 45 32. Suppoe that the mathematic SAT core for high chool enior for a pecific year have a mean of 456 and a tandard deviation of 100 and are approximately normally ditributed. If a ubgroup of thee high chool enior, thoe who are in the National Honor Society, i elected, would you expect the ditribution of core to have the ame mean and tandard deviation? Explain your anwer. 33. Given a data et, how could you decide if the ditribution of the data wa approximately normal? 34. If a ditribution of raw core were plotted and then the core were tranformed to z core, would the hape of the ditribution change? Explain your anwer. 35. In a normal ditribution, find when m 110 and 2.87% of the area lie to the right of 112. 36. In a normal ditribution, find m when i 6 and 3.75% of the area lie to the left of 85. 37. In a certain normal ditribution, 1.25% of the area lie to the left of 42, and 1.25% of the area lie to the right of 48. Find m and. 6 34
Section 6 4 Application of the Normal Ditribution 319 38. An intructor give a 100-point examination in which the grade are normally ditributed. The mean i 60 and the tandard deviation i 10. If there are 5% A and 5% F, 15% B and 15% D, and 60% C, find the core that divide the ditribution into thoe categorie. 39. The data hown repreent the number of outdoor drive-in movie in the United State for a 14-year period. Check for normality. 2084 1497 1014 910 899 870 837 859 848 826 815 750 637 737 Source: National Aociation of Theater Owner. 40. The data hown repreent the cigarette tax (in cent) for 30 randomly elected tate. Check for normality. 3 58 5 65 17 48 52 75 21 76 58 36 100 111 34 41 23 44 33 50 13 18 7 12 20 24 66 28 28 31 Source: Commerce Clearing Houe. 41. The data hown repreent the box office total revenue (in million of dollar) for a randomly elected ample of the top-groing film in 2001. Check for normality. 294 241 130 144 113 70 97 94 91 202 74 79 71 67 67 56 180 199 165 114 60 56 53 51 Source: USA TODAY. 42. The data hown repreent the number of run made each year during Bill Mazeroki career. Check for normality. 30 59 69 50 58 71 55 43 66 52 56 62 36 13 29 17 3 Source: Greenburg Tribune Review. Technology Step by Step MINITAB Determining Normality Step by Step There are everal way in which tatitician tet a data et for normality. Four are hown here. Contruct a Hitogram Inpect the hitogram for hape. 1. Enter the data for Example 6 19 in the firt column of a new workheet. Name the column Inventory. 2. Ue Stat>Baic Statitic>Graphical Summary preented in Section 3 4 to create the hitogram. I it ymmetric? I there a ingle peak? Check for Outlier Inpect the boxplot for outlier. There are no outlier in thi graph. Furthermore, the box i in the middle of the range, and the median i in the middle of the box. Mot likely thi i not a kewed ditribution either. 6 35
320 Chapter 6 The Normal Ditribution Calculate Pearon Index of Skewne The meaure of kewne in the graphical ummary i not the ame a Pearon index. Ue the calculator and the formula. 3 X median PI 3. Select Calc>Calculator, then type PI in the text box for Store reult in:. 4. Enter the expreion: 3*(MEAN(C1) MEDI(C1))/(STDEV(C1)). Make ure you get all the parenthee in the right place! 5. Click [OK]. The reult, 0.148318, will be tored in the firt row of C2 named PI. Since it i maller than 1, the ditribution i not kewed. Contruct a Normal Probability Plot 6. Select Graph>Probability Plot, then Single and click [OK]. 7. Double-click C1 Inventory to elect the data to be graphed. 8. Click [Ditribution] and make ure that Normal i elected. Click [OK]. 9. Click [Label] and enter the title for the graph: Quantile Plot for Inventory. You may alo put Your Name in the ubtitle. 10. Click [OK] twice. Inpect the graph to ee if the graph of the point i linear. Thee data are nearly normal. What do you look for in the plot? a) An S curve indicate a ditribution that i too thick in the tail, a uniform ditribution, for example. b) Concave plot indicate a kewed ditribution. c) If one end ha a point that i extremely high or low, there may be outlier. Thi data et appear to be nearly normal by every one of the four criteria! TI-83 Plu or TI-84 Plu Step by Step Normal Random Variable To find the probability for a normal random variable: Pre 2nd [DISTR], then 2 for normalcdf( The form i normalcdf(lower x value, upper x value, m, ) Ue E99 for (infinity) and E99 for (negative infinity). Pre 2nd [EE] to get E. Example: Find the probability that x i between 27 and 31 when m 28 and 2 (Example 6 15a from the text). normalcdf(27,31,28,2) To find the percentile for a normal random variable: Pre 2nd [DISTR], then 3 for invnorm( The form i invnorm(area to the left of x value, m, ) Example: Find the 90th percentile when m 200 and 20 (Example 6 17 from text). invnorm(.9,200,20) 6 36
Section 6 4 Application of the Normal Ditribution 321 To contruct a normal quantile plot: 1. Enter the data value into L 1. 2. Pre 2nd [STAT PLOT] to get the STAT PLOT menu. 3. Pre 1 for Plot 1. 4. Turn on the plot by preing ENTER while the curor i flahing over ON. 5. Move the curor to the normal quantile plot (6th graph). 6. Make ure L 1 i entered for the Data Lit and X i highlighted for the Data Axi. 7. Pre WINDOW for the Window menu. Adjut Xmin and Xmax according to the data value. Adjut Ymin and Ymax a well, Ymin 3 and Ymax 3 uually work fine. 8. Pre GRAPH. Uing the data from Example 6 19 give Since the point in the normal quantile plot lie cloe to a traight line, the ditribution i approximately normal. Excel Step by Step Normal Quantile Plot Excel can be ued to contruct a normal quantile plot to examine if a et of data i approximately normally ditributed. 1. Enter the data from Example 6 19 into column A of a new workheet. The data hould be orted in acending order. 1 2. Since the ample ize i 18, each core repreent 18, or approximately 5.6%, of the ample. Each data point i aumed to ubdivide the data into equal interval. Each data value correpond to the midpoint of the particular ubinterval. 3. After all the data are entered and orted in column A, elect cell B1. From the function 1 icon, elect the NORMSINV command to find the z core correponding to an area of 18 of the total area under the normal curve. Enter 1/(2*18) for the Probability. 4. Repeat the procedure from tep 3 for each data value in column A. However, for each conecutive z core correponding to a data value in column A, enter the next odd multiple 1 of 36 in the dialogue box. For example, in cell B2, enter the value 3/(2*18) in the NORMSINV dialogue box. In cell B3, enter 5/(2*18). Continue uing thi procedure to create z core for each value in column A until all value have correponding z core. 5. Highlight the data from column A and B, and elect the Chart Wizard from the toolbar. 6. Select the catter plot to graph the data from column A and B a ordered pair. Click Next. 7. Title and label axe a needed; click [OK]. The point appear to lie cloe to a traight line. Thu, we deduce that the data are approximately normally ditributed. 6 37