After the invention of the steam engine in the late 1700s by the Scottish engineer

Transcription

1 Introduction to Statitic 22 After the invention of the team engine in the late 1700 by the Scottih engineer Jame Watt, the production of machine-made good became widepread during the However, it wa not until the 1920 that much attention wa paid to the quality control of the good being produced. In 1924, Walter Shewhart of Bell Telephone Laboratorie ued a tatitical chart for controlling product variable; in the 1940, quality control wa ued in much of wartime production. Quality control i one of the modern ue of tatitic, the branch of mathematic in which data are collected, diplayed, analyed, and interpreted. Today it i nearly impoible to read a newpaper or watch televiion new without eeing ome type of tudy, in area uch a medicine or politic, that involve tatitic. Other area in which tatitical method are ued include biology, phyic, pychology, ociology, reliability engineering, actuarial cience, economic, buine, and education, to name but a few. The firt ignificant ue of tatitic wa made in the 1660 by John Graunt, and in the 1690 by Edmund Halley (of Halley Comet), when each publihed ome concluion about the population in England baed on mortality table. There wa little development of tatitic until the 1800, when tatitical meaure became more widely ued. For example, important contribution were made by the cientit Franci Galton, who ued tatitic in the tudy of human heredity, and by the nure Florence Nightingale, who ued tatitical graph to how that more oldier died in the Crimean War (in the 1850) from unanitary condition than from combat wound. In uing tatitic, we generally collect and ummarize data (uing method from decriptive tatitic) to make inference baed on thoe data (uing method from inferential tatitic). The firt two ection of thi chapter are dedicated to decriptive tatitic. After a dicuion regarding the normal ditribution, we dedicate the ret of the chapter to introducing ome baic concept of inferential tatitic, including confidence interval, tatitical proce control, and regreion. David P. Lewi/Shuttertock Learning Outcome After completion of thi chapter, the tudent hould be able to: Undertand the baic concept of population, ample, parameter, tatitic, and variable Contruct frequency, relative frequency, and cumulative frequency table for data Draw a hitogram, a frequency polygon, and an ogive Calculate meaure of central tendency (mean, median, and mode) and meaure of pread (range and tandard deviation) Ue Chebychev theorem to draw concluion about a data et Calculate relative frequencie uing the normal ditribution Contruct large ample confidence interval for a population mean or for a population proportion Plot an x control chart, an R control chart, and a p control chart Find the equation of the leatquare line that bet fit a given et of data Find the equation of a curve that bet fit a given et of data by tranforming the independent variable and uing linear leat quare In Section 22.4 we ee how tatitical analyi wa ued in the deign of the 12.9-km-long Confederation Bridge that join New Brunwick and Prince Edward Iland. 615

2 616 Chapter 22 Introduction to Statitic 22.1 Tabular and Graphical Repreentation of Data Population Sample Variable Array Cla Frequency Ditribution Table Relative Frequency Frequency Polygon Hitogram Cumulative Frequency Ogive In tatitic, a population i the complete collection of element (people, DVD, houehold, temperature) that are of interet and about which information i deired. Typically, a reearcher i intereted in a numerical property of the population called a parameter. For example, if the population i all DVD produced by a certain manufacturer over the coure of a week, a parameter would be the proportion of defective DVD in that lot. If the population i all Internet uer, a parameter would be the average number of hour pent each week on the Internet. Becaue of contraint on time, money, and other carce reource, concluion about the population are uually drawn after oberving only a ubet of the population, called a ample. Quantitie computed from ample are called tatitic. Statitic are ued to etimate parameter in the population. For example, the average in a ample of Internet uer can be ued to etimate the average among all uer. Similarly, the proportion of defective in a ample can be ued to etimate the proportion of defective in the complete lot. We are uually only intereted in ome of the characteritic that element of the population have in common. A variable i any characteritic whoe value change from individual to individual in the population. A quantitative variable ha a value that repreent a numerical meaurement. Example of quantitative variable are weight, length, voltage, preure, and number of children in a family. When the value of a variable i non-numerical, it i called a qualitative variable, or an attribute. Example of attribute are colour, gender, and quality (meaured a defective or nondefective). Value of variable that have been recorded contitute data. Data that have been collected but not yet organized are called raw data. In order to obtain ueful information from the data, it i neceary to organize it in ome way. Normally, a firt tep in organizing the data i to arrange the numerical value in acending (or decending) order, forming what i called an array. Example 1 Illutrating an array Each uer in a ample of 50 home computer uer wa aked to etimate carefully the number of hour they pent each week on-line on the Internet. Following are the etimate. 12, 20, 15, 14, 7, 10, 12, 25, 18, 5, 10, 24, 16, 3, 12, 14, 28, 8, 13, 18, 15, 8, 11, 15, 14, 22, 14, 19, 6, 10, 18, 4, 16, 24, 18, 5, 13, 20, 12, 12, 25, 11, 8, 12, 20, 5, 10, 15, 13, 8 A we can ee, no clear pattern can be een from the raw data. Arranging thee in numerical order to form an array, we can ummarize the array by howing the number of peron reporting each etimate a follow: (hour@peron) 3@1, 4@1, 5@3, 6@1, 7@1, 8@4, 10@4, 11@2, 12@6, 13@3, 14@4, 15@4, 16@2, 18@4, 19@1, 20@3, 22@1, 24@2, 25@2, 28@1 In Example 1, although a pattern i omewhat clearer from the ummarized array than from the raw data, a till clearer pattern i found by grouping the data. In the proce of grouping, the detail of the raw data i lot, but the advantage i that a much clearer overall pattern of the data can be obtained. The grouping of data i done by firt defining what value are to be included in each group and then tabulating the number of value that are in each group. Each group i called a cla, and the number of value in each cla i called the frequency. The table i called a frequency ditribution table. Thi i illutrated in the following example.

3 22.1 Tabular and Graphical Repreentation of Data 617 Example 2 Frequency ditribution table In Example 1, we note that the etimate vary from 3 h to 28 h. We ee that if we form clae of 0 4 h, 5 9 h, etc., we will have five poible etimate in each cla and that there will be ix clae. Thi give u the following frequency ditribution table of value howing the number of peron (frequency) reporting the indicated etimate of the hour pent on the Internet. Etimate (hour) Frequency (peron) Thi table how u the frequency ditribution. The 0, 5, 10, and o on are the lower cla limit, and the 4, 9, 14, and o on are the upper cla limit. Each cla include five value, which i the cla width. A with the cla limit we have choen, it i generally preferable to have the ame width for each cla. We can ee from thi frequency ditribution table that the pattern of hour on the Internet by the peron reponding to the urvey i clearer. Guideline for Contructing Frequency Table Make ure that clae are mutually excluive, o that each obervation belong to one, and only one, cla. Ue between 5 and 20 clae. The principal conideration i that the relevant characteritic of the data hould be clear. Enure that all clae (except for open-ended clae) have the ame width. Ue cla limit with convenient number. The firt lower cla limit i elected a the lowet value, or a a convenient number le than the lowet value. Be ure to include all clae, even if their frequency i zero. At time, it i alo helpful to know the relative frequency of the cla, which i the frequency of the cla divided by the total frequency of all clae. The relative frequency can be expreed a a fraction, decimal, or percent. Example 3 Relative frequency The relative frequency of each cla for the data in Example 2 can be hown a in the following table: Practice Exercie 1. Auming the data in Example 1 are divided into clae of 0 3 h, 4 7 h, etc., for the 8 11 h cla find: (a) the frequency (b) the relative frequency A graphing calculator may be ued to diplay hitogram and frequency polygon. Etimated Hour on Internet Frequency Relative Frequency (,) >50 = 0.04 = 4, Total Uing graph i a very convenient method of repreenting frequency ditribution. There are everal ueful type of graph for uch ditribution. Among the mot important of thee are the hitogram and the frequency polygon. In order to repreent grouped data, where the raw data value are generally not all the ame within a given cla, we find it neceary to ue a repreentative value for each cla. For thi, we ue the cla mark, which i found by dividing the um of the lower and upper cla limit by 2. The following example illutrate the ue of a cla mark with a hitogram and a frequency polygon.

4 618 Chapter 22 Introduction to Statitic Frequency Hour Computer preadheet are very ueful for thi type of analyi. Frequency Hour Fig Complete polygon to next (empty) cla mark. 32 Example 4 Hitogram A hitogram repreent a particular et of data by diplaying each cla of the data a a rectangle. Each rectangle i labelled at the centre of it bae by the cla mark. The width of each rectangle repreent the cla width, and the height of the rectangle repreent the frequency of the cla. For the data in Example 2 on etimated hour on the Internet, the cla mark are >2 = 2, >2 = 7, and o on. Uing thee value, a hitogram repreenting thee data i hown in Fig Example 5 Frequency polygon A frequency polygon i ued to repreent a et of data by plotting the cla mark a abcia (x-value) and the frequencie a ordinate (y-value). The reulting point are joined by traight-line egment. A frequency polygon repreenting the data in Example 2 on etimated hour on the Internet i hown in Fig If the polygon i not completed a hown in Fig. 22.2, and the figure tart at the firt cla mark, and end at the lat cla mark, it i then referred to a a broken-line graph. Fig Another way of analying data i to ue cumulative total. The way thi i generally done i to change the frequency into a le than cumulative frequency. To do thi, we add the cla frequencie, tarting at the lowet cla boundary. The graphical diplay that i generally ued for cumulative frequency i called an ogive (pronounced oh-jive). Example 6 Cumulative frequency ogive 50 For the data on etimated hour on the Internet in Example 2 and 3, the cumulative frequency i hown in the following table: Cumulative frequency Hour Etimated Hour on Internet Cumulative Frequency Le than 5 2 Le than Le than Le than Le than Le than Fig Learning Tip Note that in a cumulative frequency table, the name of the cla can vary. For example, the firt cla in Example 6 could be defined a le than 5, or up to but not including 5, or le than or equal to 4. What i important i that the clae be cumulative and that there be no overlap nor ambiguity between the clae. The ogive howing the cumulative frequency for the value in thi table i hown in Fig The vertical cale how the frequency, and the horizontal cale how the cla boundarie. One important ue of an ogive i to determine the number of value above or below a certain value. For example, to approximate the number of repondent that ue the Internet le than 18 hour per week, we draw a line from the horizontal axi to the ogive and then to the vertical axi a hown in Fig From thi, we ee that about 37 repondent ue the Internet le than 18 hour per week. If the data with which we are dealing have only a limited number of value and we do not divide the data into clae, we can till ue the method we have developed, uing the pecific value rather than cla value. Conider the following example.

5 22.1 Tabular and Graphical Repreentation of Data 619 Example 7 Graph uing pecific value A tet tation meaured the loudne of the ound of jet aircraft taking off from a certain airport. The decibel (db) reading meaured to the nearet 5 db for the firt 20 jet were a follow: 110, 95, 100, 115, 105, 110, 120, 110, 115, 105, 90, 95, 105, 110, 100, 115, 105, 120, 95, 110 Frequency Decibel reading Fig Frequency Decibel reading Fig Since there are only even different value for the 20 reading, the bet idea of the pattern i found by uing thee value, a hown in the following frequency table: db Reading Frequency The hitogram for thi table i hown in Fig. 22.4, and the frequency polygon i hown in Fig Exercie 22.1 In Exercie 1 4, divide the data in Example 1 into five clae of hour (0 5, 6 11, etc.) on the Internet and then do the following: 1. Form a frequency ditribution table. 2. Find the relative frequencie. 3. Draw a hitogram. 4. Draw an ogive. In Exercie 5 12, ue the following data. An automobile company teted a new engine, and found the following reult in twenty tet of the number of litre of gaoline ued by a certain model for each 100 km travelled. 5.3, 5.8, 5.6, 5.4, 5.9, 5.4, 6.0, 5.8, 5.8, 5.4, 6.3, 5.6, 5.7, 5.6, 5.7, 5.9, 5.5, 6.1, 5.9, Form an ordered array and ummarize it by finding the frequency of each number of litre ued. 6. Find the relative frequency of each number of litre ued. 7. Form a frequency ditribution table with five clae. 8. Find the relative frequencie for the data in the frequency ditribution table in Exercie Draw a hitogram for the data of Exercie Draw a frequency polygon for the data of Exercie Form a cumulative frequency table for the data of Exercie Draw an ogive for the data of Exercie 7. In Exercie 13 32, find the indicated quantitie. 13. In teting a computer ytem, the number of intruction it could perform in 1 n wa meaured at different point in a program. The number of intruction were recorded a follow: 19, 21, 22, 25, 22, 20, 18, 21, 20, 19, 22, 21, 19, 23, 21 Form a frequency ditribution table for thee value. 14. For the data of Exercie 13, draw a hitogram. 15. For the data of Exercie 13, draw a frequency polygon. 16. For the data of Exercie 13, form a relative frequency ditribution table. 17. A trobe light i deigned to flah every 2.25 at a certain etting. Sample bulb were teted with the following reult: Time () between Flahe (cla mark) Number of Bulb Time () No. Bulb Draw a hitogram for thee data. 18. For the data of Exercie 17, draw a frequency polygon. 19. For the data of Exercie 17, form a cumulative frequency ditribution table. 20. For the data of Exercie 17, draw an ogive. 21. In teting a braking ytem, the ditance required to top a car from 110 km>h wa meaured in 120 trial. The reult are hown in the following ditribution table: Stopping Ditance (m) Time Car Stopped Stopping Ditance (m) Time Car Stopped Form a relative frequency ditribution table for thee data. 22. For the data in Exercie 21, form a cumulative frequency ditribution table. 23. For the data of Exercie 21, draw an ogive. 24. From the ogive in Exercie 23, etimate the number of car that topped in le than 57 m.

6 620 Chapter 22 Introduction to Statitic 25. The doage, in milliievert (msv), given by a particular X-ray machine, wa meaured 20 time, with the following reading: 0.425, 0.436, 0.396, 0.421, 0.444, 0.383, 0.437, 0.427, 0.433, 0.434, 0.415, 0.390, 0.441, 0.451, 0.418, 0.426, 0.429, 0.409, 0.436, Form a hitogram with ix clae and the lowet cla mark of msv. 26. For the data ued for the hitogram in Exercie 25, draw a frequency polygon. 27. The life of a certain type of battery wa meaured for a ample of batterie with the following reult (in number of hour): 34, 30, 32, 35, 31, 28, 29, 30, 32, 25, 31, 30, 28, 36, 33, 34, 30, 31, 34, 29, 30, 32 Draw a frequency polygon uing ix clae. 28. For the data in Exercie 27, draw a cumulative frequency ditribution table uing ix clae. Anwer to Practice Exercie 1. (a) 10 (b) 20, 29. The diameter of a ample of fibre-optic cable were meaured with the following reult (diameter are cla mark): Diam. (mm) No. Cable Diam. (mm) No. Cable Draw a hitogram for thee data. 30. For the data of Exercie 29, draw a hitogram with ix clae. Compare the pattern of ditribution with that of the hitogram in Exercie To four coin 50 time and tabulate the number of head that appear for each to. Draw a frequency polygon howing the number of toe for which 0, 1, 2, 3, or 4 head appeared. Decribe the ditribution. (I it about what hould be expected?) 32. Mot calculator can generate random number (between 0 and 1). On a calculator, diplay 50 random number and record the firt digit. Draw a hitogram howing the number of time for which each firt digit 10, 1, 2, c, 92 appeared. Decribe the ditribution. (I it about what hould be expected?) 22.2 Summarizing Data Median Arithmetic Mean Mode Range Standard Deviation Chebychev Theorem Learning Tip Note that all meaure of central tendency and pread are uually rounded off to one more decimal place than wa preent in the original data. Median Table and graphical repreentation give a general decription of data. However, it i alo ueful and convenient to find repreentative value for the location of the centre of the ditribution, and other number to give a meaure of the deviation from thi central value. In thi way, we can obtain a numerical ummary of the data. We tudy ome meaure of centre and deviation (or pread) in thi ection. MEASURES OF CENTRAL TENDENCY The tak of a meaure of central tendency i to decribe with a ingle value the location of the centre of the ditribution. Since there are different way of defining what centre i, there are everal meaure of central tendency. The firt of thee meaure of central tendency i the median. The median i the value that fall in the middle of an ordered array of data, leaving a many obervation above it a it doe below it. If there i no middle obervation, the median i the number halfway between the two number nearet to the middle of the array. Example 1 Median odd or even number of value Given the number 5, 2, 6, 4, 7, 4, 7, 2, 8, 9, 4, 11, 9, 1, 3, we firt arrange them in numerical order. Thi arrangement i middle number 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 9, 9, 11 Since there are 15 number, the middle number i the eighth. Since the eighth number i 5, the median i 5. If the number 11 i not included in thi et of number and there are only 14 number in all, the median i that number halfway between the eventh and eighth number. Since the eventh i 4 and the eighth i 5, the median i 4.5. Arithmetic Mean Another very widely applied meaure of central tendency i the arithmetic mean (often referred to imply a the mean). The mean i calculated by finding the um of all

7 22.2 Summarizing Data 621 the value and then dividing by the number of value. (The mean i the number mot people call the average. However, in tatitic the word average ha the more general meaning of a meaure of central tendency.) Example 2 Arithmetic mean The arithmetic mean of the number given in Example 1 i determined by finding the um of all the number and dividing by 15. Therefore, by letting x (read a x bar ) repreent the mean, we have x = um of value = = 5.5 number of value Thu, the mean i 5.5. If we wih to find the arithmetic mean of a large number of value, and if ome of them appear more than once, the calculation can be implified. The mean can be calculated by multiplying each value by it frequency, adding thee reult, and then dividing by the total number of value (the um of the frequencie). Letting x repreent the mean of the value x 1, x 2, c, x n, which occur with frequencie f 1, f 2, c, f n, repectively, we have Thi i called a weighted mean ince each value i given a weight baed on the number of time it occur. x = x 1 f 1 + x 2 f 2 + g + x n f n f 1 + f 2 + g + f n (22.1) Example 3 Arithmetic mean uing frequencie Uing Eq. (22.1) to find the arithmetic mean of the number of Example 1, we firt et up a table of value and their repective frequencie, a follow: Value Frequency We now calculate the arithmetic mean x by uing Eq. (22.1): multiply each value by it frequency and add reult x = = = 5.5 um of frequencie We ee that thi agree with the reult of Example 2. Summation uch a thoe in Eq. (22.1) occur frequently in tatitic and other branche of mathematic. In order to implify writing thee um, the ymbol g i ued to indicate the proce of ummation. (g i the Greek capital letter igma.) gx mean the um of the x.

8 622 Chapter 22 Introduction to Statitic Example 4 Summation ymbol g We can how the um of the number x 1, x 2, x 3, c, x n a Practice Exercie For the following number, find the indicated value: 12, 17, 16, 12, 14, 18, 14, 12, 15, The median 2. The arithmetic mean a x = x 1 + x 2 + x 3 + g + x n If thee number are 3, 7, 2, 6, 8, 4, and 9, we have a x = = 39 Uing the ummation ymbol g, we can write Eq. (22.1) for the arithmetic mean a x = x 1f 1 + x 2 f 2 + x 3 f 3 + g + x n f n f 1 + f 2 + f 3 + g + f n = a xf a f (22.1) n The ummation notation gx i an abbreviated form of the more general notation a x i. Thi more general form can be ued to indicate the um of the firt n number of i=l a equence or to indicate the um of a certain et within the equence. For example, for 5 a et of at leat 5 number, a x i indicate the um of the third through the fifth of thee number (in Example 4, a 5 i=3 i=3 the um of all the number being conidered. x i = 16). We will ue the abbreviated form gx to indicate The arithmetic mean i one of a number of tatitical meaure that can be found on a calculator. Mode Example 5 Arithmetic mean uing frequencie We find the arithmetic mean of the Internet hour in Example 1 of Section 22.1 (page 616) by x = a xf a f = = g g = 13.7 h (rounded off to tenth) Another meaure of central tendency i the mode, which i the value that appear mot frequently. If two or more value appear with the ame greatet frequency, each i a mode. If no value i repeated, there i no mode. Example 6 Mode (a) The mode of the number in Example 1 i 4, ince it appear three time, and no other value appear more than twice. (b) The mode of the number 1, 2, 2, 4, 5, 5, 6, 7 are 2 and 5, ince each appear twice and no other number i repeated. (c) There i no mode for the value ince none of the value i repeated. 1, 2, 5, 6, 7, 9 Example 7 Meaure of central tendency To find the frictional force between two pecially deigned urface, the force to move a block with one urface along an inclined plane with the other urface i meaured ten time. The reult, with force in newton, are 2.2, 2.4, 2.1, 2.2, 2.5, 2.2, 2.4, 2.7, 2.1, 2.5 Find the mean, median, and mode of thee force.

9 22.2 Summarizing Data 623 Learning Tip The mean i ueful for many tatitical method and i ued extenively. Neverthele, be aware that the mean i very enitive to extreme obervation o that a ingle extreme obervation can change the value of the mean dramatically and give the wrong impreion about the data. The median i not affected by extreme obervation. Therefore, it i a good choice a a meaure of centre in the preence of extreme value. The mean, the median, and the mode coincide when the ditribution of data i ymmetric. In thoe cae, the median or the mode (which are eay to calculate) can be ued a etimate of the mean. Range Standard Deviation To find the mean, we um the value of the force and divide thi total by 10. Thi give F = a F 10 = = = 2.33 N The median i found by arranging the value in order and finding the middle value. The value in order are 2.1, 2.1, 2.2, 2.2, 2.2, 2.4, 2.4, 2.5, 2.5, 2.7 Since there are 10 value, we ee that the fifth value i 2.2 and the ixth i 2.4. The value midway between thee i 2.3, which i the median. Therefore, the median force i 2.3 N. The mode i 2.2 N, ince thi value appear three time, which i more than any other value. MEASURES OF SPREAD Meaure of central tendency on their own are not very informative. They do not tell u whether value are grouped cloely together or how pread out they are. Therefore, we alo need ome meaure of the deviation, or pread, of the value from the centre. If the pread i mall and the number are grouped cloely together, the meaure of central tendency i more reliable and decriptive of the data than in the cae in which the pread i greater. In tatitic, there are everal meaure of pread that may be defined. The implet one i the range, which i the difference between the highet value and the lowet value in the data et. For example, the range of the data in Example 7 i = 0.6. We will ee how the range i applied to tatitical proce control in Section The mot widely ued meaure of pread i the tandard deviation. The tandard deviation of a et of ample value i defined by the equation a 1x - x22 = C n - 1 (22.2) The definition of how that the following tep are ued in computing it value. Learning Tip The tandard deviation i a poitive number. It i a deviation from the mean, regardle of whether the individual number are greater than or le than the mean. Number cloe together will have a mall tandard deviation, wherea number further apart have a larger tandard deviation. Therefore, the tandard deviation become larger a the pread of data increae. Step for Calculating Standard Deviation 1. Find the arithmetic mean x of the number of the et. 2. Subtract the mean from each number of the et. 3. Square thee difference. 4. Find the um of thee quare. 5. Divide thi um by n Find the quare root of thi reult. Following the tep hown above, we ue Eq. (22.2) for the calculation of tandard deviation in the following example.

10 624 Chapter 22 Introduction to Statitic Example 8 Standard deviation uing Eq. (22.2) Find the tandard deviation of the following number: 1, 5, 4, 2, 6, 2, 1, 1, 5, 3. A table of the neceary value i hown below, and tep 1 6 are indicated: tep 2 tep 3 Learning Tip The population tandard deviation, repreented by the Greek letter (read igma ), i computed uing n in the denominator of Eq. (22.2) intead of n - 1. The n - 1 in the denominator of Eq. (22.2) adjut o that it give good etimate of the parameter when the tandard deviation can only be meaured from a ample. Since we generally ue data coming from ample, in thi text we will alway ue Eq. (22.2), and we will refer to the ample tandard deviation imply a the tandard deviation. x x - x 1x - x tep 4 a 1x - x22 = n - 1 x = = 3 tep = 32 9 = A 32 9 tep 5 = 1.9 tep 6 (rounded off to tenth) If ome of the value in the data are repeated, we can ue the frequency of thoe value that occur more than once in calculating the tandard deviation. Thi i illutrated in the following example. Example 9 Standard deviation uing frequencie Find the tandard deviation of the number in Example 1. Since everal of the number appear more than once, it i helpful to ue the frequency of each number in the table, a follow: tep 2 tep 3 x f xf x - x 1x - x2 2 1x - x2 2 f tep 1 a 1x - x22 f n - 1 tep 4 x = = 5.5 = = = A tep 5 = 3.0 tep 6 It i poible to reduce the computational work required to find the tandard deviation. Algebraically, it can be hown (although we will not do o here) that the following equation i another form of Eq. (22.2) and therefore give the ame reult. = H n1 a x a x2 2 n1n - 12 (22.3)

11 22.2 Summarizing Data 625 Although the form of thi equation appear more involved, it doe reduce the amount of calculation that i neceary. Conider the following example. Example 10 Standard deviation uing Eq. (22.3) Uing Eq. (22.3), find for the number in Example 8. Practice Exercie 3. Find the tandard deviation of the firt eight number in Example 8. x x n = 10 gx 2 = gx2 2 = 30 2 = 900 = C = 1.9 Common Error It i a common error to confue gx 2 and1 gx2 2 in Eq. (22.3). Note that for gx 2, we quare the x value and then add the quare, wherea for 1 gx2 2, we firt add the x value and then quare the um. Example 11 Standard deviation application An ammeter meaure the electric current in a circuit. In an ammeter, two reitance are connected in parallel, with mot of the current paing through a very low reitance called the hunt. The reitance of each hunt in a ample of 100 hunt wa meaured. The reult were grouped, and the cla mark and frequency for each cla are hown in the following table. Calculate the arithmetic mean and the tandard deviation of the reitance of the hunt. Statitical meaure uch a x, gx,gx 2, x, x, and n can be obtained directly on a cientific or a graphing calculator. R (ohm) f Rf R 2 f R = = Ω 100 n = 100 gr 2 = gr2 2 = = C = The arithmetic mean of the reitance i Ω, with a tandard deviation of Ω. Example 12 Standard deviation application Find the tandard deviation of the etimated hour on the Internet a grouped in Example 2 of Section 22.1 (page 617). In doing thi, we aume that each value in the cla i the ame a the cla mark. The method i not exact, but with a large et of number, it provide a good approximation with le arithmetic work.

12 626 Chapter 22 Introduction to Statitic Interval x f xf x 2 f n = 50 gx 2 = gx2 2 = = C = 6.1 Thu, = 6.1 h. The mean and the tandard deviation together can help u draw concluion about the value in a data et. Thank to a reult known a Chebychev theorem, we can tate the percentage of data value that mut be within a pecific number of tandard deviation from the mean. Chebychev Theorem For any data et (population or ample), the proportion of obervation that mut be within k tandard deviation of the mean i alway at leat 1-1 k 2 1k For the particular value k = 2, 3, and 4, here i what the tatement of the theorem implie: At leat 75% of obervation are within two tandard deviation of the mean. At leat 89% of obervation are within three tandard deviation of the mean. At leat 94% of obervation are within four tandard deviation of the mean. Note that ince Chebychev theorem i o general, it will underetimate the percentage for ome ditribution. In Section 22.3, we will obtain more precie percentage for the important cae of the normal ditribution. Example 13 An application of Chebychev theorem A ample of computer of a certain brand had a mean time of 38 month without a hardware malfunction, with a tandard deviation of 2.5 month. What percentage of the computer in the ample lated between 33 and 43 month without a hardware malfunction? We can write 33 = , and 43 = , o 33 and 43 are 2 tandard deviation away from the mean, and we ue Chebychev theorem with k = 2. Therefore, at leat 75% of the computer in the ample lated between 33 and 43 month without a hardware malfunction. In uing the tatitical meaure we have dicued, we mut be careful in uing and interpreting uch meaure. Conider the following example. Example 14 Interpreting tatitical meaure (a) The number 1, 2, 3, 4, 5 have a mean of 3, a median of 3, and a tandard deviation of 1.6. Thee value fairly well decribe the centre and ditribution of the number in the et. (b) The number 1, 2, 3, 4, 100 have a mean of 22, a median of 3, and a tandard deviation of 44. The large difference between the median and the mean and the very large range of value within one tandard deviation of the mean (-22 to 66) indicate that thi et of meaure doe not decribe thi et of number well. In a cae like thi, the 100 hould be checked to ee if it i in error. Example 14 illutrate how tatitical meaure can be mileading in the preence of extreme value. Mileading tatitic can alo come from the proce of data collection.

13 22.2 Summarizing Data 627 Conider the probable reult of a urvey to find the percent of peron in favour of raiing income taxe for the wealthy if the urvey i taken at the entrance to a welfare office or if it i taken at the entrance to a tock brokerage firm. There are many other conideration in the proper ue and interpretation of tatitical meaure. Exercie 22.2 In Exercie 1 4, delete the 5 from the data number given for Example 1 and then do the following with the reulting data. 1. Find the median. 2. Find the arithmetic mean uing the definition, a in Example Find the arithmetic mean uing Eq. (22.1), a in Example Find the mode, a in Example 6. In Exercie 5 and 6, ue the data in Example 8. Change the firt 1 to 6 and the firt 2 to 7 and then find the tandard deviation of the reulting data a directed. 5. Find from the definition, a in Example Find uing Eq. 22.3, a in Example In Example 13, change 33 to 28 and 43 to 48 and then find the percentage. In Exercie 8 15, ue the following et of number. A: 3, 6, 4, 2, 5, 4, 7, 6, 3, 4, 6, 4, 5, 7, 3 B: 25, 26, 23, 24, 25, 28, 26, 27, 23, 28, 25 C: 0.48, 0.53, 0.49, 0.45, 0.55, 0.49, 0.47, 0.55, 0.48, 0.57, 0.51, 0.46, 0.53, 0.50, 0.49, 0.53 D: 105, 108, 103, 108, 106, 104, 109, 104, 110, 108, 108, 104, 113, 106, 107, 106, 107, 109, 105, 111, 109, 108 In Exercie 8 11, determine (a) the mean, (b) the median, and (c) the mode of the number of the given et. 8. Set A 9. Set B 10. Set C 11. Set D In Exercie 12 15, find the tandard deviation for the indicated et of number (a) uing Eq. (22.2), and (b) uing Eq. (22.3). 12. Set A 13. Set B 14. Set C 15. Set D In Exercie 16 30, the required data are thoe in Exercie Find the mean, the median, and the mode of L/100 km of fuel uage in Exercie Find the tandard deviation of L/100 km of fuel uage in Exercie 5 uing Eq. (22.2). 18. Find the tandard deviation of L/100 km of fuel uage in Exercie 5 uing Eq. (22.3). 19. Find the mean, the median, and the mode of computer intruction in Exercie Find the range and tandard deviation of computer intruction in Exercie Find the mean and the median of trobe light time in Exercie Find the tandard deviation of trobe light time in Exercie Find the mean and median of topping ditance in Exercie 21. (Ue the cla mark for each cla.) 24. Find the tandard deviation of topping ditance in Exercie 21. (Ue the cla mark for each cla.) 25. Find the mean, the median, and the mode of X-ray doage in Exercie Find the range and the tandard deviation of X-ray doage in Exercie Find the mean, the median, and the mode of battery live in Exercie Find the tandard deviation of battery live in Exercie Find the mean and the median of cable diameter in Exercie Find the tandard deviation of cable diameter in Exercie 29. In Exercie 31 47, find the indicated meaure of central tendency or of pread. 31. The weekly alarie (in dollar) for the worker in a mall factory are a follow: 600, 750, 625, 575, 525, 700, 550, 750, 625, 800, 700, 575, 600, 700 Find the median and the mode of the alarie. 32. Find the mean alary for the alarie in Exercie Find the range and the tandard deviation of the alarie in Exercie In a particular month, the electrical uage, rounded to the nearet 400 MJ, of 1000 home in a certain city wa ummarized a follow: Uage No. Home Find the mean of the electrical uage. 35. Find the median and mode of electrical uage in Exercie Find the tandard deviation of electrical uage in Exercie A tet of air pollution in a city gave the following reading of the concentration of ulfur dioxide (in part per million) for 18 conecutive day: 0.14, 0.18, 0.27, 0.19, 0.15, 0.22, 0.20, 0.18, 0.15, 0.17, 0.24, 0.23, 0.22, 0.18, 0.32, 0.26, 0.17, 0.23 Find the median and the mode of thee reading. 38. Find the mean of the reading in Exercie Find the range and the tandard deviation of air pollution data in Exercie The following data give the mean number of day of rain for Vancouver, Britih Columbia, for the 12 month of the year. 20, 17, 17, 14, 12, 11, 7, 8, 9, 16, 19, 22 Find the tandard deviation.

14 628 Chapter 22 Introduction to Statitic 41. The midrange, another meaure of central tendency, i found by finding the um of the lowet and the highet value and dividing thi um by 2. Find the midrange of the alarie in Exercie Find the midrange of the ulfur dioxide reading in Exercie 37. (ee Exercie 41.) 43. Add $100 to each of the alarie in Exercie 31. Then find the median, mean, and mode of the reulting alarie. State any concluion that might be drawn from the reult. 44. Multiply each of the alarie in Exercie 31 by 2. Then find the median, mean, and mode of the reulting alarie. State any concluion that might be drawn from the reult. 45. Change the final alary in Exercie 31 to $4000, with all other alarie being the ame. Then find the mean of thee alarie. State any concluion that might be drawn from the reult. (The $4000 here i called an outlier, which i an extreme value.) 46. Find the median and mode of the alarie indicated in Exercie 45. State any concluion that might be drawn from the reult. 47. The k, trimmed mean i a meaure of central tendency that avoid the influence of extreme obervation while till uing mot of the obervation in the data et. It i computed by finding the mean of the data after the mallet k, and the larget k, of the data have been dicarded. Find the 10, trimmed mean of X-ray doage in Exercie 25 of Section In Exercie 48 50, olve the given problem. 48. Ue Chebychev theorem to find the percentage of value that are between 175 and 195 in a data et with mean 185 and tandard deviation Ue Chebychev theorem to find the percentage of value that are between 55.7 and 68.3 in a data et with mean 62 and tandard deviation The mean compreive trength of a ample of teel beam wa N/cm 2, with a tandard deviation of 450 N/cm 2. What percent of the beam had compreive trength between and N/cm 2? Anwer to Practice Exercie normal Ditribution Normal Ditribution of a Population Standard Normal Ditribution Standard Score (z-score) Sampling Ditribution The firt derivation of the normal ditribution i due to Abraham de Moivre ( ), who wa intereted in approximating quantitie ariing in gambling problem. It wa alo derived independently by Pierre-Simon Laplace ( ), and by Carl Friedrich Gau ( ), both in the context of meaurement error. y Fig x In thi ection we dicu the normal ditribution, the mot important and mot widely ued ditribution in tatitic. The normal ditribution i a continuou ditribution, o we begin by dicuing ome generalitie of continuou ditribution. In Section 22.1 we learned that for variable that take a limited number of value, the relative frequency of a value i obtained by dividing the frequency of that value by the total frequency of all value. Let u now conider variable that can be regarded a having an infinite number of poible value, uch a weight, length, or duration for a very large population. (Such variable are aid to be continuou.) When uing the ame procedure a before, the denominator become infinite, giving a relative frequency of zero for all value. How are we then to compute the relative frequency of interval, if the relative frequency of all value i zero? The anwer lie in etablihing a correpondence between relative frequency and area. To each continuou variable we aociate a function, which we can graph a a curve on the plane. The relative frequency of a particular interval correpond to the area under the curve in that interval. Becaue the relative frequency for the complete population mut be 1, the total area under the curve mut be 1 (100% of the data). The normal ditribution i aociated with the ymmetric, bell-haped curve hown in Fig Uing advanced method, it equation i found to be y = e-1x - m22 >2 2 12p (22.4) Here, m i the population mean and i the population tandard deviation, and p and e are the familiar number firt ued in Chapter 2 and 12, repectively. From Eq. (22.4), we can ee that any particular normal ditribution depend on the value of m and. The horizontal location of the curve depend on m, and the hape (how pread out the curve i) depend on, but the bell hape remain. Thi i illutrated in general in the following example.

15 22.3 Normal Ditribution 629 Example 1 Location and pread of a normal ditribution In Fig. 22.7, for the left curve, m = 10 and = 5, wherea for the right curve, m = 20 and = 10. y Fig x See the chapter introduction y Fig x( C) Example 2 The normal ditribution applied to reliability The normal ditribution ha important application in probabilitic reliability technique for bridge deign. For intance, load factor developed for the deign of the Confederation Bridge are dicued in the article Deign criteria and load and reitance factor for the Confederation Bridge, by J. G. MacGregor et al. (Can. J. Civ. Eng., 24, (1997)). Quantitie that were found to be normally ditributed (or whoe logarithm were found to be normally ditributed) aroe in the analyi of dead load, live load due to vehicle, wind load, temperature load, and ice load. To give a pecific example, after analying record of daily average temperature in the region for 46 year, it wa concluded that the 3-day temperature drop that wa equalled or exceeded 100 time in 100 year i ditributed normally, with mean 26.9 C and tandard deviation 3.2 C. Thi ditribution i hown in Fig Propertie of the Normal Curve The curve i ymmetric about the mean. The curve i alway above the x-axi. ( y i alway poitive.) The x-axi i a horizontal aymptote. (A x increae numerically, y become very mall.) The total area under the curve i 1. The curve i bell-haped, a een in Fig A more complete and rigorou treatment of the material covered in thi and the remaining ection of thi chapter would require the tudy of probability theory, which i beyond the cope of thi introductory chapter. Fig how area under a normal curve with mean m and tandard deviation for particular region. Uing the correpondence between relative frequency and area, we can ue the given information to find the percentage of data value that fall within one, two, and three tandard deviation from the mean, a ummarized below m 3 m 2 m m m + m + 2 m + 3 x Fig. 22.9

16 630 Chapter 22 Introduction to Statitic Important Percentage for the Normal Curve About 68% of value are within one tandard deviation from the mean that i, between m - and m +. About 95% of value are within two tandard deviation from the mean that i, between m - 2 and m + 2. Almot all value (99.74%) are within three tandard deviation from the mean that i, between m - 3 and m + 3. We can compare thee percentage with thoe given by Chebychev theorem in Section 22.2 namely, that at leat 75% of obervation are within two tandard deviation from the mean and that at leat 89% of obervation are within three tandard deviation from the mean. We ee that Chebychev theorem heavily underetimate the true percentage in the cae of the normal ditribution. We can ue the area from Fig to calculate relative frequencie for other interval. Thi i illutrated in the following example. Example 3 Relative frequencie and area under the curve application Let u conider the normal ditribution of 3 day temperature drop from Example 2, o that the mean and tandard deviation are m = 26.9 C and = 3.2 C, repectively. The percentage of value that lie between one tandard deviation below the mean and two tandard deviation above the mean i 81.85%, ince the area between m - and m + 2 i = We have m - = = 23.7 and m + 2 = = 33.3 Therefore, about 81.85% of value for thi ditribution are within 23.7 C and 33.3 C y STANDARD NORMAL DISTRIBUTION A we have jut een, there are innumerable poible normal ditribution. However, there i one of particular interet. The tandard normal ditribution i the normal ditribution for which the mean i 0 and the tandard deviation i 1. Making thee ubtitution in Eq. (22.4), we have y = 1 12p e-x2 /2 (22.5) Fig x a the equation of the tandard normal ditribution curve. All the propertie of a normal curve are atified by the tandard normal ditribution. In particular, ince the mean i 0, the curve i ymmetric with repect to the y-axi. The curve i alo bell haped, a een in Fig A we dicu below, area under the tandard normal curve are ued to find relative frequencie for all other normal curve. We can find the relative frequency of value for any normal ditribution by ue of the tandard core z (or z-core), which i defined a z = x - m (22.6) For the tandard normal ditribution, where m = 0, if we let x =, then z = 1. If we let x = 2, z = 2. Therefore, we can ee that a value of z tell u the number of tandard deviation the given value of x i above or below the mean. From the dicuion above, we can ee that the value of z can tell u the area under the curve between the mean and the value of x correponding to that value of z. In turn, thi tell u the relative frequency of all value between the mean and the value of x.

17 22.3 Normal Ditribution 631 Table 22.1 give the area under the tandard normal ditribution curve between zero and the given value of z. The table include value only to z = 3 ince nearly all of the area i between z = -3 and z = 3. Since the curve i ymmetric to the y-axi, the value hown are alo valid for negative value of z. Table 22.1 Standard Normal (z) Ditribution z Area z Area z Area The following example illutrate the ue of Eq. (22.6) and z-core z = 0.8 z = 2.4 Fig Practice Exercie 1. For value of m = 40 and = 8, find the area between x = 36 and x = 48. x Example 4 Normal core (z-core) For a normal ditribution curve baed on value of m = 20 and = 5, find the area between x = 24 and x = 32. To find thi area, we ue Eq. (22.6) to find the correponding value of z and then find the difference between the area aociated with thee z-core. Thee z-core are z = = 0.8 and z = = 2.4 For z = 0.8, the area i , and for z = 2.4, the area i Therefore, the area between x = 24 and x = 32 (ee Fig ) i = Thi mean that the relative frequency of the value between x = 24 and x = 32 i 20.37,. If we have a large et of meaured value with m = 20 and = 5, we hould expect that about 20, of them are between x = 24 and x = z = z = 1.2 Fig x (day) Example 5 z-core application The lifetime of a certain type of watch battery are normally ditributed. The mean lifetime i 400 day, and the tandard deviation i 50 day. For a ample of 5000 new batterie, determine how many batterie are expected to lat (a) between 360 day and 460 day, (b) more than 320 day, and (c) le than 280 day. (a) For thi ditribution, m = 400 day and = 50 day. Uing Eq. (22.6), we find the z-core for x = 360 day and x = 460 day. They are z = = -0.8 and z = = 1.2 For z = -0.8, the area i to the left of the mean, and ince the curve i ymmetric about the mean, we ue the z = 0.8 value of the area and add it to the area for z = 1.2. Therefore, the area i = See Fig Thi mean that 67.30, of the 5000 batterie, or 3365 of the batterie, are expected to lat between 360 and 460 day. Becaue of variability within ample, not every ample will have exactly 3365 batterie that will lat between

18 632 Chapter 22 Introduction to Statitic z = z = 2.4 Fig Fig x (day) x (day) 360 and 460 day. On average, however, a ample of 5000 batterie will have 3365 that will lat that amount of time. (b) To determine the number of batterie that will lat more than 320 day, we firt find the z-core for x = 320. It i z = >50 = Thi mean we want the total area to the right of z = In thi cae, we add the area for z = 1.6 to the total area to the right of the mean. Since the total area under the curve i , the total area on either ide of the mean i Therefore, the area to the right of z = -1.6 i = See Fig Thi mean that * 5000 = 4726 batterie are expected to lat more than 320 day. (c) To find the number of batterie that will lat le than 280 day, we firt find that z = >50 = -2.4 for x = 280. Since we want the total area to the left of z = -2.4, we ubtract the area for z = 2.4 from , the total area to the left of the mean. Since the area for z = 2.4 i , the total area to the left of z = -2.4 i = See Fig Therefore, * 5000 = 41 batterie are expected to lat le than 280 day. We now ummarize the procedure for finding relative frequencie uing z-core. Finding Relative Frequencie Uing z-score 1. Sketch the normal curve, labeling the mean and the given x value. Identify the deired relative frequency a an area under the curve. 2. Ue Eq. (22.6) to find the z-core for each x. Identify the deired relative frequency a an area under the tandard normal curve. 3. Look up the abolute value of each z-core in Table 22.1 to find it aociated area. 4. Depending on the ituation, proceed a follow: Area Between two z-core of the ame ign Between two z-core of different ign To the right of a poitive z-core or to the left of a negative z-core To the right of a negative z-core or to the left of a poitive z-core Procedure Subtract the maller area from the larger one Add both area together Subtract the area from 0.5 Add the area to 0.5 ampling ditribution In Example 4, we aumed that the lifetime of the batterie were normally ditributed. Of coure, for any et of 5000 batterie, or any number of batterie for that matter, the lifetime that actually occur will not follow a normal ditribution exactly. There will be ome variation from the normal ditribution, but for a large ample, thi variation hould be mall. The mean and the tandard deviation for any ample will vary omewhat from that of the population. When we conider the relative frequency ditribution of the ample mean obtained from all poible ample of the ame ize, we obtain what i called the ampling ditribution of the ample mean. In the tudy of probability, it i hown that if we elect all poible ample of ize n from a population with a mean m and tandard deviation, the mean of the ample mean i alo m. Alo, the tandard deviation of the ample mean, denoted by x, and called the tandard error of the mean, i x = 1n (22.7)