Measures of Central Tendency: Basic Statistics Refresher. Topic 1 Point Estimates

Transcription

1 Basc Statstcs Refresher Basc Statstcs: A Revew by Alla T. Mese, Ph.D., PE, CRE Ths s ot a tetbook o statstcs. Ths s a refresher that presumes the reader has had some statstcs backgroud. There are some easy parts ad there are some hard parts. The most useful refereces used to put ths materal together were: Egeerg Statstcs by Motgomery, Ruger ad Hubele, 4 rd Edto, Joh Wley & Sos, 7. A ecellet troducto to statstcs for egeers. Statstcal Models Egeerg by Hah & Shapro, Wley Classcs Lbrary, 994. Paperback 3 Itroducto to Error Aalyss by Joh R. Taylor Mster Wzard Uversty Scece Books, Sausalto, CA, Qualty Egeerg Statstcs, by Robert A. Dovch, ASQ Qualty Press, 99. paperback. The outle of topcs as well as some eamples used ths refresher was take from ths book by Dovch. 5 Practcal Egeerg Statstcs, by Schff & D Agosto, Wley Iterscece, 99. A ecellet small statstcs book. Topcs Covered: Topc Topc Topc 3 Topc 4 Topc 5 Topc 6 Apped A Apped B Apped C Apped D Apped E Apped F Apped G Pot Estmates Dstrbuto Fuctos Cofdece Itervals Hypothess Testg Testg for Dffereces Varaces Decso Errors, Type I ad Type II Probablty Dstrbutos Goodess of Ft. Samplg by Varables Lear Regresso Estmate of Epected Value ad Varace for Nolear Fuctos Basc cocepts Probablty some advaced materal Nocetral dstrbutos advaced Topc Pot Estmates Whe workg wth data, typcally a small sample from a large populato of data, we wsh to use ths sample to estmate parameters of the overall populato. The populato may be fte or fte. I descrbg a populato we typcally wsh to kow where the ceter resdes, how much varato there s the data about the cetral value, whether the dstrbuto s symmetrc or skewed to oe sde, ad how peaked or flat t s. Oe possble set of pot estmates for data would be the mea, varace, coeffcet of skewess, ad the coeffcet of kurtoss. Ths s eplored the followg sectos. Measures of Cetral Tedecy: There are three major measures of cetral tedecy of a populato; they are the mea, meda ad mode. We fd these parameters by calculatg statstcal estmators for these parameters usg sample data. Geerally, we wsh to have statstcal estmators that gve the best ubased estmates of these populato parameters. The populato mea s /78

2 Basc Statstcs Refresher estmated from the smple arthmetc average of the sample data. If the umber of data pots the sample s N the mea s calculated by N N where s the value of the th data pot the sample of sze N. Ths s the ubased estmator of the populato mea. The meda of the populato s estmated from the meda of the sample data, whch s the mddle data pot from a data sample that s sorted from smallest to largest values see eample below. For N odd, t s the mddle data pot. For N eve the meda s the average of the mddle two data pots. The mode s smply the most probable value ad s determed from the sample data by plottg the data usually as a hstogram ad determg the rage of the -values that most frequetly occurs; the ceter of that rage s called the mode. There ca be more tha oe mode, called mult-moded, for a populato. Eample. Test data value 33 Hstogram of data 3 bs Rage lower upper mdpot frequecy Hstogram mode mea meda 3 mode The meda s determed by sortg the data from smallest to largest values ad coutg to the mddle N+/ pot. Sortg the above data produces; The mddle value s the 9+/ 5 th value the sorted order whch s the umber 3. Thus, the meda s the pot for whch 5% of the umbers are bgger tha the meda ad 5% of the umbers are less tha the meda. If there are a eve umber of data pots the the meda s take to be the average of the mddle two data values. Creatg a hstogram of the data, as see above, ad fdg the value that represets the most frequetly occurrg rage of umbers determes the mode. For eample, oe easly sees there are more values that lay wth the terval 33 to 38 tha the other /78

3 Basc Statstcs Refresher tervals. We geerally choose the mdpot of the terval to represet the value of the ordate o the hstogram. I may samples of real data, there may ot be just oe peak value whe the data s plotted a hstogram. I these multmodal cases, the mode may ot be a useful or meagful measure of cetral tedecy. Whe we deal wth symmetrcal dstrbutos of data such as are represeted by a Gaussa or ormal or bell-shaped dstrbuto the mea meda mode. Attrbute data: Whe dealg wth attrbute data such as what fracto of M&Ms a package are blue, we are terested a rato # blue / total # the package. These proportos are called attrbute data. Aother eample would be fracto of ocoformg uts whch s used whe screeg uts from a suppler. I ths stace we would record p umber of ocoformg uts / umber of uts tested. For eample f we oly had uts the populato ad all uts were tested ad 4 faled the π 4/. s the populato fracto defectve. Measures of Varato or Dsperso. The four useful measures of dsperso are the varace σ, whch s estmated from the sample data by the statstc s, the stadard devato σ, whch s the square root of the varace, ad s estmated by s whch s based, 3 the rage, R largest sample value smallest sample value, ad 4 the average of the absolute value of the resduals. By far the most used measures are the varace ad the stadard devato. For a fte populato of sze N the varace s defed by N σ μ N ad μ s the mea of the populato μ N N. The stadard devato of the populato s foud by takg the square root of the populato varace. We seldom have the etre populato of a varable to use for calculatos so we must try ad fer formato about the populato from takg oe or more samples of data from the populato of terest. Whe we oly have a sample from the populato, the the sample varace s defed by 3 s, where s the sze of the sample ad s the mea of the sample. The factor of - s ecessary to make sure that s s a ubased estmator of the populato varace. More wll be sad of based ad ubased estmators Apped A. Oe usually wats ubased estmators of populato parameters. The square root of s s ot the ubased estmator of σ but t s usually close eough. Ecel has fuctos that calculate the mea ad the populato ad sample varaces ad stadard devatos. Mea AVERAGErage of cells cotag data, populato varace VARPrage of cells, sample varace VARrage of cells, STDEVPrage of cells stadard devato of populato, STDEVrage of cells stadard devato of sample based. Equato 3 ca be epaded to a more useful form for had calculato. 3/78

4 Basc Statstcs Refresher 4 N s The epected value of s, E[s ] s σ ad s a ubased estmator of the populato varace. The ubased estmate of σ s ot the sample statstc s square root of the sample varace. The epected value of s, E[s] C 4 σ where C 4 s a fucto of sample sze ad s gve Apped B. Thus the ubased estmate of σ s/c 4. Eample : 5, Sample data table From the above data mea.86, the meda.9,.77 N N N 46.47, , 59.6 s , s The ubased estmator of σ s s/c 4.53/ See Apped B for C 4 Measure of Dsperso for Attrbute Data: Whe workg wth attrbute data, such as fracto of ocoformg, the estmate of the varace s take from the bomal dstrbuto ad s gve by, 5 σ π π where π s the fracto ocoformg the populato. Eample 3: If π.34 the σ , σ.8. Measure of Asymmetry: Skewess Skewess s a measure of asymmetry of data. It s called the thrd momet about the mea. It s usually represeted by a quatty called the coeffcet of skewess ad s estmated from the data wth the formula skewess 3 6 Coef. of Skewess 3 3 stadard devato s If the data has a tal to the rght values > mea the the skewess wll be postve. If the data tals off to the left of the mea oe has egatve skewess. The data for eample gves Coef. Of Skewess We wll use the abbrevato Sk for the coeffcet of skewess. For the mathematcally cled the calculus defto s gve by 7 Sk μ 3 f d 3 σ 4/78

5 Basc Statstcs Refresher ad s called the thrd momet about the mea dvded by the stadard devato cubed order to produce a utless measure of asymmetry. Measure of Peakedess : Kurtoss A measure of the peaked ature of the data s called the kurtoss ad the usual measure s called the coeffcet of kurtoss ad s gve by the formula kurtoss Coef. of kurtoss stadard devato s 3 3 The skewess s called the fourth momet of the dstrbuto about the mea ad s gve by the calculus formula as 9 Coef. of kurtossku μ 4 f d 4 σ The value of Ku for a ormal or Gaussa dstrbuto s 3 ad s used as a base agast whch peaked behavor of a dstrbuto or data s measured..e. f oe substtutes a ormal dstrbuto for f to equato 9 ad tegrates oe fds Ku3. The data for eample gves Coef. of kurtoss.43 so t s a bt less peaked tha a ormal dstrbuto. 5/78

6 Basc Statstcs Refresher Topc : Dstrbuto Fuctos: There are dscrete dstrbutos such as the bomal dstrbuto ad there are cotuous dstrbutos such as the ormal dstrbuto. We wll work mostly wth cotuous dstrbutos. There are two types of dstrbuto fuctos. The frst s called the probablty desty fucto or pdf. The secod s the cumulatve dstrbuto fucto or CDF. More detaled dscussos are gve Apped A. Eamples of these two types for the famlar ormal dstrbuto are show below. Normal Dstrbuto: The sgle peaked bell-shaped curve blue s the pdf labeled fz, ad the S-shaped curve red s the CDF labeled Fz. The curves are related. At ay gve value of the.5 fz,pdf ad Fz, CDF for Normal Dstrbuto fz Fz pdf, f CDF, F Value of radom varable radom varable, say, the CDF s the area uder the pdf up to that value e.g. for the CDF reads.975 or there s 97.5% probablty of havg a X value. The formulas that descrbe these ormal curves are μ σ fn e N μσ, σ π zμ σ FN e dz Pr{ X } σ π z The pdf, f N represets the probablty per ut of value of the radom varable..e. fd s the probablty that the radom varable X s wth d of the value. The CDF beg the area uder f N from X - up to X, s the cumulatve probablty that the radom varable X s the value. For as useful as the ormal probablty fucto may 6/78

7 Basc Statstcs Refresher be there s o closed form aalytcal epresso for ts CDF. It s gve tables all statstcs books ad s avalable Ecel usg the NORMDIST commad. The pot of some ote s that the etre dstrbuto fucto ca be descrbed wth oly parameters μ, σ. The coeffcet of skewess ad the coeffcet of kurtoss 3 for every ormal curve. Estmatg the values of the parameters for the data s called pot estmato. Several other dstrbutos wll be descrbed below. A useful referece s CRC Stadard Probablty ad Statstcs Tables ad Formulas, Studet Edto, by Da Zwllger et al. Da s a seor fellow at Raytheo. Ut Normal Dstrbuto. A terestg property of the ormal dstrbuto s that oe ca trasform the varable usg the trasformato ZX-μ/σ ad ths trasformato produces a dstrbuto gve below ad ths shows a ormal dstrbuto wth a zero mea ad a varace. Thus ay ormal dstrbuto ca be trasformed to a ut ormal dstrbuto. I statstcs books the appedces you wll fd values oly for ut ormal dstrbutos. The symbols φ ad Φ have tradtoally bee used to sgfy a ut ormal pdf ad CDF respectvely. z f z e φ z, π F z F μ/ σ Φ z Cetral Lmt Theorem. The reaso for troducg the ormal dstrbuto so soo ths ote s because we wsh to troduce the most mportat theorem statstcs, called the cetral lmt theorem. If oe draws a sample of sze from a populato whose dstrbuto, g, ca be farly arbtrary shape, ad computes the average of ths sample, ad f oe repeats ths radom samplg aga ad aga, the dstrbuto of these sample averages ca be show to approach a ormal dstrbuto. The mea of ths dstrbuto of meas, s the mea of the populato from whch the samples were draw ad the stadard devato of the meas equals the stadard devato of the populato dvded by the square root of the sample sze. Ths s called the Cetral Lmt Theorem CLT. Note that the populato dstrbuto ca be arbtrary ad ot look a thg lke a ormal dstrbuto ever-the-less the dstrbuto of mea values from samples draw from ths arbtrary populato wll be dstrbuted as a ormal dstrbuto the lmt that s large. How large? A sample sze > s adequate f populato s umodal w/o too much skewess ad usually >3 to 5 s cosdered a farly good umber. The dstrbuto of sample meas,.e. s foud to be dstrbuted as a ormal dstrbuto as the sze of the sample becomes large. The dstrbuto s gve by the equato show below. 7/78

8 Basc Statstcs Refresher fn e π σ μ σ Ths may be the most useful theorem appled statstcs. Logormal Dstrbuto. The logormal dstrbuto s used etesvely for dstrbutos that caot for physcal reasos take o values less tha zero. A typcal case mght be a tme to falure or a tme to repar a product. The logormal dstrbuto looks somewhat lke a ormal dstrbuto. lyθ ω f y e, y >, θ E l y, Var l y πσ y [ ] ω [ ] ω ω θ +, mea y μ e Var y σ μ e A typcal graph of the pdf θ6,ω. s show below. ft Logormal Dstrbuto A referece to real data comes from Logormal Dstrbuto for Modelg Qualty Data Whe the Mea s Near Zero, Joural of Qualty Techology, 99, pp E+.E+ 4.E+ 6.E+ 8.E+.E+3.E+3 tme to falure, t Logormal Dstrbuto mea 89, stdev /78

9 Basc Statstcs Refresher The Webull Dstrbuto. Ths dstrbuto s very useful as t ca take o may forms. It ca have ether 3 parameters or more commoly two parameters. It s used etesvely relablty work as the dstrbuto that gves the tme to falure of a part, subsystem or a etre comple system. For a reasoable rage of mea values the Webull ca be made to look very much lke the ormal dstrbuto as ca be see the fgure below..6 Normal vs. Webull, equal momets Webull,3.634 pdf Webull,3.634, CDF ormal9.6,.758, pdf ormal9.6,.758, CDF f, pdf mea 9.6 stdev.758 alpha beta F, CDF The formulas for the Webull are gve below: X Two parameter: t α W The parameter α s called the characterstc lfe relablty work but more geerally s called a scale parameter. The parameter β s the so-called shape parameter. I the three parameter versos oe smple replaces t by t-γ, where γ locato parameter ad the pdf ad CDF are oly defed for t γ. The shapes of these β t α β t fw t e α α F t e Pr{ T t} β pdf, f β Webull Dstrbuto, pdf ad CDF f, pdf F, CDF Pr{X< Radom Varable, X> CDF, F 9/78

10 Basc Statstcs Refresher curves are show above. Ths chart shows a Webull wth α 7 ad β.5. For other parameter values look at the pdf curves below. The specal case β s called the epoetal dstrbuto ad s the dstrbuto that s assumed to gve the tme to falure for most electroc compoets a mssle system. It forms the bass for the old MIL-HDBK-7F ad several commercal codes. More wll be made of ths later. Webull pdf plots for varous beta values β.7 β β.5 β3.5 β5 pdf, f radom varable value Gamma Dstrbuto: Aother useful dstrbuto s the gamma dstrbuto ad ts pdf ad CDF are gve below. k λ λ λ fλ e y k k y Γ k λ λ λ y F e dy Λ Γ The shapes of the pdf curves are show below for varous values of k, the shape parameter, ad a fed value of λ.. Aga k gves the epoetal dstrbuto. /78

11 Basc Statstcs Refresher Gamma Dstrbuto pdf..8 pdf, f.6 k k k3 k radom varable value The dstrbutos defed so far are called ubouded dstrbutos as the radom varable ca take o values that ca eted to fty oe or more drectos. A useful bouded dstrbuto s called the beta dstrbuto ad s gve below Beta Dstrbuto y β α y B α, β α Γ α Γ β fβ, B α, β B α, β Γ α + β β y Fβ dy /78

12 Basc Statstcs Refresher If the terval of terest s a < z <b oe smply trasforms usg z-a/b-a ad the lmts become, stead of a,b. Beta Dstrbuto 7 pdf, f a7,b a,b7 a4,b4 a,b6 a6,b a,b.e+.e-.e- 3.E- 4.E- 5.E- 6.E- 7.E- ormalzed radom varable 8.E- 9.E-.E+ Whe α β oe has a symmetrc dstrbuto. αβ produces a uform dstrbuto The questo ow becomes are there 3 parameter dstrbutos other tha addg a locato parameter to the two parameter dstrbutos such as the gamma or Webull dstrbutos? The aswer s yes. The most wdely used s the tragle dstrbuto where oe specfes the mmum value, the mamum value ad the value at whch the tragle peaks. There are also 4 parameter ad 5-parameter dstrbutos. The Johso famly of dstrbutos has four parameters ad oe procedure for determg those four parameters s to match the aalytcal values for the 4 momets to the values of the four momets determed from the data tself. At RMS s doe usg a tool called JFt. Other 4-parameter dstrbutos clude the geeralzed lambda ad geeralzed kappa dstrbutos. /78

13 Basc Statstcs Refresher Tragle Dstrbuto. a, a< X < m b a m a fδ b, b> X > m ba bm a lower lmt b upper lmt m most probable or mode, < a a, a m ba ma FΔ b, m b ba bm, > b a m b I may smulato stuatos oe ca use a tragle dstrbuto f there s o data avalable to costruct a more sutable dstrbuto. 3/78

14 Basc Statstcs Refresher Emprcal Dstrbuto EDF. May tmes we wsh to costruct a o-parametrc dstrbuto from the data. Such a dstrbuto s called a emprcal dstrbuto. Cosder the case whch oe has data values {,,, }.. The EDF s produced by Plottg F j j/ vs. j. See the plot below. The set of step fuctos represets the EDF ad the red cotuous curve s the CDF for a ormal dstrbuto that has bee matched wth some measure of goodess-of-ft to the data. There are data pots. The program EasyFt was used to geerate ths plot. j jdata j-.5/ zjf - j-.5/ To geerate the plot lst the data values from smallest to largest ad set up the followg table, The plot colum 4 vs colum. The result s show o the et graph. 4/78

15 Basc Statstcs Refresher ordered Z values from Normal dstrbuto Normal Probablty Plot Probablty Plot assumg a ormal dstrbuto for the data. Blue dots are actual data pots ordered X values from data Dstrbuto of Mama. Largest Order Statstc advaced topc The mamum value of a gve radom varable, say the mamum wd speed ay gve year over a perod of years, has a dstrbuto just as oe mght plot the dstrbuto of the mea wd speed due to all storms a seaso for seasos. The CDF ad pdf for the mamum value s determed by, [ ] [ ] F X F X, f X F X f Dstrbuto of Mama.. f, Value of radom varable e.g. Mamum Wd Speed 5/78

16 Basc Statstcs Refresher Where f s the dstrbuto of wd speeds, X, for a seaso ad s presumed to be the same type of dstrbuto based o hstorcal data. If there were 3 years of hstorcal data the ths data would be used to determe f ad F. Suppose we are terested the dstrbuto of mamum wd speeds over the et 5 years. The 5 would be used the above formulas. The plot of the pdf s show below for the case of f beg a ormal dstrbuto wth mea4 ad stdev. A plot of the CDF s show below alog wth a le showg the 95 th percetle values for X for varous values of. CDF of Mama CDF, F th percetle Value of radom varable e.g. mamum wd speed Dstrbuto of Mma. Smallest Order Statstcadvaced topc Smlarly oe mght be terested the dstrbuto of the mma over years where hstorcal data has show that X s dstrbuted as f wth a CDF, F. The dstrbuto of the lowest mmum value of X s gve by,, [ ] [ ] F X F X f X F X f, A plot of the pdf s show below. Ths dstrbuto s used to predct the relablty of a seres of compoets Let X represet tme. R relablty -F therefore, f -dr/d ad,, [ ] [ ] R X R X f X R X f 6/78

17 Basc Statstcs Refresher Dstrbuto of Mma.. pdf, f Value of radom varable Smlarly the CDF for the mma s plotted below. CDF of Mma FX X 7/78

18 Basc Statstcs Refresher Topc 3: Cofdece, Tolerace & Predcto Itervals A terval estmate for a populato parameter e.g. the populato mea μ s called a cofdece terval. We caot be certa that the terval cotas the true, ukow populato parameter we oly use a sample from the full populato to compute the pot estmate, e.g., ad the terval. However, the cofdece terval s costructed so that we have hgh cofdece that t does cota the ukow populato parameter. Cofdece tervals are wdely used egeerg ad the sceces. A tolerace terval s aother mportat type of terval estmate. For eample, the chemcal product vscosty data mght be assumed to be ormally dstrbuted. We mght lke to calculate lmts that boud 95% of the possble vscosty values. For a ormal dstrbuto, we kow that 95% of the dstrbuto s the terval μ.96σ, μ+.96σ. However, ths s ot a useful tolerace terval because the parameters μ ad σ are ukow. Pot estmates such as ad s ca be used the terval equato for μ ad σ. However, we eed to accout for the potetal error each pot estmate. The result s a terval of the form -ks, +ks, where k s a approprate costat that s larger tha.96 to accout for the estmato error. As for a cofdece terval, t s ot certa that ths equato bouds 95% of the dstrbuto, but the terval s costructed so that we have hgh cofdece that t does. Tolerace tervals are wdely used ad, as we wll subsequetly see, they are easy to calculate for ormal dstrbutos. Cofdece ad tolerace tervals boud ukow elemets of a dstrbuto. It s mportat to lear ad apprecate the value of these tervals A predcto terval provdes bouds o oe or more future observatos from the populato. For eample, a predcto terval could be used to boud a sgle, ew measuremet of vscosty aother useful terval. Wth a large sample sze, the predcto terval for ormally dstrbuted data teds to the tolerace terval equato, but for more modest sample szes the predcto ad tolerace tervals are dfferet. Keep the purpose of the three types of terval estmates clear: A cofdece terval bouds populato or dstrbuto parameters such as the mea vscosty. A tolerace terval bouds a selected proporto of a dstrbuto. A predcto terval bouds future observatos from the populato or dstrbuto. Cofdece Itervals. I the prevous secto, pot-estmates were calculated from the data draw from a populato of terest. If aother sample were to be take from the same populato the pot estmates would most lkely tur out wth dfferet values. To compesate for samplg varatos we use the cocept of cofdece tervals. A typcal form for such a terval s gve by, P{lower lmt true value of populato parameter upper lmt} cofdece level Eample: P{ Z α/ σ/ / < μ < + Z α/ σ/ / }α 8/78

19 Basc Statstcs Refresher Ths s read the probablty of the populato parameter, μ, beg betwee the upper ad lower lmt s greater tha or equal to the cofdece level -α, a fracto <. Ths s a two sded or two-taled cofdece terval sce we are lookg for lmts o both sdes of the parameter. The procedure that wll be used to geerate the upper lmt ad lower lmt wll some sese guaratee that the terval so calculated usg a sample of data sze wll cota the true but ukow populato parameter for a percetage, - α%, of the samples chose, as log as the sample was draw usg a radom samplg techque. These percetages are called cofdece levels ad they are typcally.9,.95, or hgher whe ecessary. A 95% cofdece level.e. cofdece level.95 would mply the followg. If I sample data from the populato of terest the the prescrpto I use to calculate the upper ad lower lmts of the cofdece terval wll produce a terval that wll clude the true ukow populato parameter 95% of the samples that are possble to draw. Therefore 5% of the possble samples would ot clude the parameter. Ths may soud a bt obtuse. There s a temptato to smply say that a 95% cofdece terval, oce calculated, has oly a 5% chace of ot cludg the true ukow parameter of terest. These two statemets soud the same but are ot. Ths may be a subtlety ot worth emphaszg the frst tme oe eplores statstcs but t has bee cluded to provde the real terpretato for those who are stckler s for the truth. Note: The secod statemet, the oe we would lke to have be correct, IS correct f we use Bayesa methods to create what s called a credblty terval. Ths Bayesa stuff wll be addressed later. Let s try a eample. Remember oce a sample of data s take ad the terval calculated the ether the true populato parameter s sde that terval or t s ot! How are tervals upper& lower lmts calculated? Cofdece Iterval for the Mea. To calculate the cofdece terval oe eeds to kow the dstrbuto of the samplg statstc. I the case of the mea we kow the samplg dstrbuto for the mea s the ormal dstrbuto whe s large. Ths s due to the cetral lmt theorem. If we were to estmate some other populato parameter such as the varace we eed to fd the samplg dstrbuto for s. Whe a sample sze s large >3 ad the populato stadard devato s kow, we make use of the Cetral Lmt Theorem whch says that X s dstrbuted appromately as a ormal dstrbuto. Remember f we do ot kow how X s dstrbuted the we caot proceed further. The dscusso of ths s well outled Referece [] by Motgomery et al. σ σ The terval s gve by X Z, X Z α + α however we may tmes wrte ths terval a more meagful way as a probablty statemet. σ σ 6 P X Z α μx X Z α α + 9/78

20 Basc Statstcs Refresher Ths s read as the probablty that the populato mea ukow & ukowable s betwee the lower ad upper lmts show above s greater tha or equal to -α where the cofdece level s defed as -α%. The reaso for defg cofdece level as -α, where α s called the sgfcace level, s mostly hstorcal but t s the coveto used all statstcs tets. A eample for a 95% cofdece level α.5. The value α represets the probablty that the mea of the populato would le outsde the calculated upper ad lower lmts. Z -α/ s the umber of stadard devatos from the mea that we are addg to or subtractg from the mea to have a cofdece level of -α, half above the mea ad half below the mea, ad σ s the stadard devato of the mea value, X. Ths rato s called the stadard error of the mea, SEM or σ. The radom X varable X s the sample mea calculated from the sample of sze. If the CL s 95% the Z almost stadard devatos ad we would be 95% cofdet that the populato mea would le betwee the sample mea plus.96 tmes σ ad the sample X mea mus.96 tmes. Some typcal Z-values are show below for the usual array σ X of α values. Oe ca use the ecel fucto NORMSINVprobablty to fd these values. For a two-sded cofdece terval we look for Z-values evaluated at α/ probablty. α Z α/ Eample 4: Aga takg the data from the table Eample ad assumg the stadard devato of the populato.8, we obta X.5, σ.8, 5 ad calculate the 95% cofdece terval to.8.8 be.5.96, ,.464. The value.96 ca be foud usg 5 5 the Ecel fucto NORMSINV.975. What does oe obta usg NORMSINV.5? If we do ot kow the stadard devato of the populato but stead have to use a sample stadard devato calculated from a small sample, the we use a t-value from the studet t-dstrbuto stead of the stadard ormal Z-values. The t-dstrbuto [Ref ] looks much lke the ormal dstrbuto but has fatter tals tha a ormal dstrbuto ad also depeds o the sample sze. For a sample sze of 5 the values of t for varous alpha values are show below. α t α/, Z -α/ Note that we use t-values evaluated at -5-4 degrees of freedom. /78

21 Basc Statstcs Refresher Eample 5: Usg the same formato as Eample 4 but stead usg the sample stadard devato s.85 o obtas the 95% cofdece terval to be, , ,.488. The value.5 ca be foud 5 5 usg the Ecel fucto TINV.5, 4. A pecularty of Ecel s that for the t- dstrbuto oly, the verse values are gve assumg a two taled test from the start..e. teral to Ecel t calculates the.5 value ad gves that aswer for TINV. The t- dstrbuto s parameterzed by the so-called degrees of freedom dof whch for ths terval calculato s gve by the sample sze. The oe s subtracted to accout for the fact that oe degree of freedom was used to calculate the sample varace tself. Note that due to the ucertaty the stadard devato the cofdece terval s larger for the same cofdece level. Whe the sample sze >3 the t-dstrbuto s well appromated by the ormal dstrbuto. For >3 oe ca stll use the Z-values whch releves some complcatos. Oe assumpto behd the t-dstrbuto s that the samplg dstrbuto of mea values s ormally dstrbuted, whch by the cetral lmt theorem s true the lmt that s large. Cofdece Iterval for the Varace. Ulke the cofdece terval for the mea, the cofdece terval for the varace s ot symmetrcal about the pot estmate. The varace s dstrbuted as a ch-square radom varable. The formula for calculatg the cofdece terval for the varace s gve by the probablty statemet, s s 7 P σ α χα/, χ α/, The ch-square dstrbuto s gve tables or ca be computed ecel usg the CHIINVprobablty,dof fucto. Ths CI assumes that the varable tself.e. s dstrbuted ormally or at least comes from a bell-shaped dstrbuto. The dof - to accout for calculato of sample varace. Eample 6: Usg data from Eample where we foud that s.673, the CI calculato proceeds as follows; σ.443 σ.66 for a 9% cofdece terval for the populato varace. The Ecel fucto CHIINV.5,436.4 ad CHIINV.95, Note ths s a broad rage of values. It s ot mmedately apparet that a creased sample sze helps. I fact, f the sample stadard devato s the same the havg that value usg a larger sample makes the cofdece terval eve broader. If however, the real case, where creasg the sample sze decreases the sample varace the the reduced sample varace outweghs the - ch-square verse ad the cofdece terval becomes smaller. /78

22 Basc Statstcs Refresher Cofdece Iterval for Fracto Nocoformg ormal dstrbuto. Ay estmate of fracto ocoformg s dstrbuted as a radom varable wth assocated cofdece lmts. If the sample sze s large eough >3 the oe ca use the ormal dstrbuto. Let Z-m/s where s ay value of terest, such as a specfcato lmt. Eample 7: Suppose the UCL for a process s 9, the process mea s 88.5 ad the stadard devato s.3. Both of these values were estmated from a relatvely large sample 5. Usg the formula gves Z /.3.. A value of. mples a probablty PZ>.. whch says the ocoformg fracto s.%. I actualty the estmates of the mea ad stadard devato are pot estmates take from the sample data ad usg the z formula may seem somewhat dssatsfyg as there s ucertaty the parameters of the dstrbuto tself. Fortuately there are some appromatos that ca led more accuracy to the soluto to ths dlemma. Wegarte [J.Qual.Tech., 4, #4 Oct.,98,pp7-] has developed formulas that are more accurate e.g. 8 Z Z UCL LCL Z α + s s + Z α + s s Aga specfyg a terest a upper spec lmt of 9 ad usg the process mea value of 88.5 ad sample stadard devato of.3 we fd.. Now s defg Z UCL Z-value for upper cofdece lmt, Z LCL Z-value for lower cofdece lmt. Desrg a 95% cofdece level, oe fds Z UCL.9, Z LCL.43. For a momet, these values may seem correct or reversed from what they should be. The formulas are correct. If I use Z.9 ad go to the stadard ormal tables I fd PZ>.9.56 or the fracto of ocoformg s 5.6%. Ths s a upper cofdece lmt about the spec lmt 9. Usg the lower cofdece lmt of Z.43, I fd PZ> or 7.6% of the uts are ocoformg. Ths s a lower cofdece lmt about the spec lmt 9. Thus.76 P>9.56 wth a cofdece of 95%. Ths s a subtle pot but s very mportat. Obvously f the sample sze s very large the lttle error occurs usg the pot estmate e.g. P>9PZ>.. for fracto ocoformg. Cofdece Iterval for Proporto: The pot estmate for a proporto was gve before as p umber of occurreces / total sample sze /. Whe the sample sze,, s large ad *p >5 ad *-p>5 we /78

23 Basc Statstcs Refresher ca use the ormal dstrbuto to calculate cofdece tervals. The formula for the terval s show below. p 9 p p p pz α π p+ Z α where p s the proporto calculated from the data. Note: The epected value for the sample varace, p p /. Eample 8: p proporto, sample sze, Z ormal dstrbuto z-value depedet of cofdece level. Usg values from Eample 3 where p 6/7.3 ad 7, for a 9% two sded cofdece terval Z ad the cofdece terval for π becomes, π π whch s the 9% cofdece terval for the populato proporto π. Small Sample Sze Cofdece Itervals. If the sample sze s small to moderate there s more accuracy gaed usg the followg formulas for cofdece lmts o proportos or fracto ocoformg. Z α Zα p + Z α p p+ + pl Z α + Z α Zα p+ + + Z α p+ p + pu Z α + where r umber of occurreces, sample sze, p r/, Z calculated from ut ormal dstrbuto Eample 9: take 5, r7, Z -α/.96.e. 95% two sded cofdece terval, Φ.63, Φ.74 L U Ths leads to the terval.63 π.74 at a 95% cofdece level ad the pot estmate for π ~ p r/ 7/5.4. Cotuty Correcto Factor p I actualty the dstrbuto of the proporto estmator,, s the dstrbuto of X, the umber of evets of terest whch s the scaled by dvdg by, the sample sze. X s dstrbuted as a seres Beroull varable epermets X s success, X s falure ad s therefore govered by the bomal dstrbuto. Ths s a dscrete dstrbuto ad for larger sample szes >3 t s appromated by the ormal dstrbuto whch s 3/78

24 Basc Statstcs Refresher cotuous. It has bee foud over the years that oe ca obta better umercal results for appromate cofdece tervals by usg the followg smple rules. for cases P{X } use P{X +.5} P{Z +.5-p/p-p / } for cases P{X } use P{X -.5} P{Z -.5-p/p-p / } The cofdece Iterval the becomes a somewhat smplfed verso of equatos ad show above. p p p p p Z α π p+ + Z α Ths cofdece terval s sad to have the cotuty correctos appled. Cofdece Iterval for Posso Dstrbuted Data. Whe dealg wth coutg types of data such as the umber of defectve parts a sample oe typcally s dealg wth a Posso process or Posso dstrbuted data. To calculate a cofdece terval for the average umber of defects preset a populato that has bee sampled wth a sample sze, oe ca use the ch square dstrbuto as follows: To calculate the upper cofdece lmt frst calculate the umber of degrees of freedom dof *r+. For the lower lmt dof*r. Whe calculatg the 9% two-sded cofdece terval use the./ ad the -./ probablty values to fd the ch square values. The cofdece terval s foud to be χ α,r χ α, r+ r Eample : Suppose a comple assembly oe foud r3 ocoformaces. Fd the 9% cofdece terval for the umber of ocoformaces. χ α, r+ χ.5,8 UCL /.67 χ α, r χ.95,6 LCL / r.67 The cofdece terval becomes Noparametrc Cofdece Itervals: Oe ca calculate cofdece tervals for fracto ocoformg uts wthout havg to presuppose a ormal or ay other dstrbuto. The advatage of ths oparametrc approach s that t releves us of tryg to substatate a populato dstrbuto. The pealty s that the resultat terval s larger tha the parametrc estmates. If there are tests ad r uts are foud to fal the p r/ s a estmate of the falure probablty ad -r/ s a estmate of the relablty at the ed of the test. Sce falure of a operatg ut s cosdered to be a bary evet relablty takg to accout degradato s the topc of aother paper ad thus ca be represeted by a Beroull tral for each of the uts uder test, oe ca ask the followg questos. What s the upper 4/78

25 Basc Statstcs Refresher lmt o the populato proporto of falures, p U, that wll allow for r or fewer falures wth a α/ probablty. I equatoal form we have r Cp r U pu α /. Smlarly, f we ask what s the lowest value for the proporto, p L, whch would allow for there to be r or greater falures to occur wth probablty α/, oe would obta r Cp r L pl α /, or, Cp r L pl α /. r These equatos ca be solved by terato Ecel. The results of such terato are show below. PL Use sum -a/.4 Solver r 33 α.5 α/.975 PU sum a/ The results ca also be determed usg the F-dstrbuto or the complete Beta dstrbuto see below. Eact o parametrc Cofdece Iterval for proportos. Two-sded cofdece terval the eact terval that s depedet of ay dstrbuto s gve by the followg formulas. p L Betaα/, -r,r+ p U Betaα/, -r+,r umber of tests, r umber of falures, α cofdece level 5/78

26 Basc Statstcs Refresher Topc #4: Hypothess Testg Testg a sample mea versus a hypotheszed mea whe s kow. Whe the stadard devato s kow or ca be assumed, the dstrbuto of sample averages draw from a populato wll be dstrbuted ormally wth a stadard devato of σ. Ths s may tmes called the stadard error of the mea or SEM. Ths result comes from the Cetral Lmt Theorem. Usg ths formato we ca develop ad test hypotheses to determe the locato of the mea of a populato μ wth respect to a hypotheszed mea value μ. The hypotheses that ca be tested are; H : μ μ Two sded test. H: μ μ The ull hypothess, H, s that the populato mea s equal to the hypotheszed mea. The alteratve hypothess H s that the two meas are ot equal. REMEMBER THE FOLLOWING! Rule : The statemet we wsh to prove s always placed the alterate hypothess, that s, you wat to see f the data you have collected from the populato of terest wll allow you to reject the ull hypothess favor of the alterate hypothess. The hypotheses statemets are always stated terms of the parameters of the populato, the hypotheses are ot statemet about statstcs such as comparg sample mea values or sample stadard devatos. Oe-sded test. H : μ μ H: μ > μ or H : μ μ H: μ < μ Rule : The equalty sg s always placed the statemet of the ull hypothess by coveto ot ecessty. Ths may seem cofusg at frst but smply follow these two rules ad you caot go wrog. Note: Tetbooks dffer the maer whch they set up hypothess testg ad some do ot follow ths equal sg placemet coveto, so read carefully. How cofdet do you wat to be that H s cosstet wth the data? Say you wsh to be 95% cofdet, the we eed to set up some kd of statstcal test that wll gve us ths level of cofdece. X μ Let Z* where we are usg the mea of the sample data, X, as a estmator of σ the populato mea, μ. The stadard devato s assumed kow ad s the sample sze. The varable Z s called a statstc because t cotas oly data ad kow costats. The mea of the populato, μ, s ot a statstc; t s a kow parameter 6/78

27 Basc Statstcs Refresher hypotheszed that characterzes a populato. We wat to kow f data comes from a populato whose mea s μ. The test wll be formulated by establshg a crtcal value for the radom varable beg tested,.e. Z. I ths problem, the crtcal Z-value, labeled Z crt s presumed to come from a ut ormal dstrbuto, N,. If we dd ot kow how Z was dstrbuted we could ot perform ths test! How s the value for Z crt determed? Remember from prevous eamples that f Z.96 ths dcates that PZ< , so for the sake of argumet pck.96 as the crtcal Z-value. The calculate a test statstc Z* ad f that test statstc s greater tha.96 the the data from whch I calculated the test statstc s from a populato who s mea s far away at least.96 stadard devatos of the mea from the hypotheszed mea, μ. It s so far away that the probablty of the data comg from a populato whose mea s μ s oly 5%. Smply put, ths data t s ot lkely to have come from a populato whose mea was equal to the hypotheszed mea f Z* s greater tha Z crt. Eample : To esure a ew producto le has the capacty to meet demad, t s ecessary to mata the holdg tme the bulk taks to a average of 8 mutes. A smlar process s le s beg used at aother locato. Usg the orgal process as a referece, test whether or ot the holdg tak the ew le averages 8 mutes at the 5% level of rsk. The sample sze s. Assume the process the ew le has a stadard devato of 3 mutes. H : μ 8 The ull ad alterate hypotheses are. Ths s a two-sded test that we H : μ 8 wsh to see f the ew le s ether above or below the mea of the kow referece le. The crtcal Z value s Z sce we wsh to take oly a 5% rsk.e. have a 95% cofdece level. Assume the data from the ew le s gve below mea 84.6 test Z 4.85 It ca clearly be see that Z84.6-8/3 ad Z 4.85 whch s > Z crt.96. Therefore we must reject the ull hypothess ad say that the ew producto le does ot have the same mea as the referece le. Techcally we caot say that the ew le has a greater average because we performed a two-sded test. We eed to perform a oe sded test. If we dd a oe-sded test wth H : μ 8, H: μ > 8 the Z crt Z ad sce Z4.85 >.645 we would reject H at ths 95% cofdece level ad the we ca techcally say that the ew le has a mea that s greater tha the mea of the referece le. Testg Sample Mea Versus Hypotheszed Mea Whe Stadard Devato Is Estmated From Sample Data. Whe σ s ukow ad must be estmated from the sample data we would epect there to be more ucertaty tha whe s s kow. Ths ucertaty wll mafest tself our thkg ad tests. Thus, we wll use the t-dstrbuto sometmes called the studet-t dstrbuto to perform our hypothess tests stead of the ormal dstrbuto. The t 7/78

28 Basc Statstcs Refresher dstrbuto assumes that the radom varable beg eamed comes from a ormal dstrbuto but the test s stll reasoably accurate f the populato, from whch the sample s draw, s symmetrc ad somewhat bell-shaped. The t statstcs s defed for ths problem by X μ t s The t dstrbuto looks much lke the ormal dstrbuto but has fatter tals. I fact, the t-dstrbuto goes over to the ormal dstrbuto for large sample szes. Eample : Usg the data from the prevous eample let us test the ull hypothess that the ew le has a mea holdg tme of 83 mutes ad stll requre a 95% level of cofdece. The problem sets up as follows; H : μ 83, H: μ 83 α.95, α /.5 dof 9 I usg the t-dstrbuto we eed to use aother parameter called the degrees of freedom dof ad for ths problem the dof-. The crtcal value of the t-statstc, t crt, for a two-sded test at α.5 ad dof9 s foud from the tables or usg ecel fucto TINVa,dof to be t.5,9.6. From the sample data, we fd X 84.6, s.547 whch gves a test statstc X μ t.987. s.547 / Sce t.987 <.6 t crt we caot reject the ull hypothess so we caot reject the possblty that the ew le has a mea holdg tme of 83 mutes o the bass of the data we have take. We are ot acceptg that 83 mutes s the populato mea of the populato from whch we have sampled, we smply caot reject that possblty. Ths s s subtle pot but s mportat. We could ot have rejected, o ths bass of ths same data, a ull hypothess that μ 84 or 85. There are a fte umber of hypothess values that would ot have bee rejected o the bass of ths data. The power ths test s whe you ca reject the ull hypothess. Testg for Dfferece Betwee Two Populatos Meas - σ s Kow. The test statstc s gve by X X Z σ + σ where the subscrpts deote the sample averages from populatos ad respectvely ad smlarly for the varaces from the populatos uder test ecept that we are assumg the varaces are kow. 8/78

29 Basc Statstcs Refresher Eample 3: Usg the data below test the ull hypothess that the mea of process oe s less tha or equal to the mea of process two, versus the alteratve hypothess that the mea of process oe s greater tha the mea of process two. Test at the 99% cofdece level α. ad assume varace of process oe s 4.3 ad of process two s.5. H : μ μ H : μ > μ Data process process Calculatg the relevat formato we obta, X 86., X , 6 σ 4.3, σ ad the test statstc becomes Z The crtcal value of Z from the ormal dstrbuto for a oe-sded test at a. s foud to be from tables or Ecel NORMSINV.99 Z crt Z Sce Z.93 <.36Z crt we caot reject the ull hypothess that the mea of process s less tha or equal to the mea of processes. Testg for Dfferece Two Populato Meas whose Stadard Devatos are Ukow but Presumed Equal. Here we wll use the t-dstrbuto as the varaces are ukow ad calculated from data. The test statstc s gve by, XX s + s t, sp s p + + The varable s p s called the pooled stadard devato ad t s just a weghted average of the calculated varaces from each sample. Oe ca pool ths maer f oe beleves the data should come from processes that have the same varace. The approprate dof + - for ths test. 9/78

30 Basc Statstcs Refresher Eample 4: Usg data from Eample 4, we wll test the hypothess of equal meas. H : μ μ H : μ μ at a cofdece level of 9% α.. the two-sded crtcal t value s foud to be t crt t.5,9 +/-.833, usg X 86., X 88.3, 5, 6, ad calculatg the varaces of both samples oe obtas s 8.47, s 4.8 that produces a pooled varace s p Usg ths to evaluate the test statstc t oe otes that.833 < -.97 < so oe caot reject the ull hypothess ad thus the two mea values caot be clamed to be uequal based upo the gve test data. Testg for Dfferece Betwee Two Proportos: Whe oe uses large sample szes such that *p>5 ad -p*>5 the oe ca use ormal dstrbuto theory. p p Let Z, sp / p p p + / sp p Ad the value p + p p s a weghted proporto + Eample 5: Assume oe has two techologes to produce the same product. We wll test the proporto ocoformg of the product from both processes. Determe f the fracto ocoformg for each process s the same at a 95% cofdece level. H : p p H: p p α.5, α.5 Zcrt Z.975 ±.96 The data for each process s stated to be as follows: Process : umber uder test 3, umber of defects d, p d/ /3.4. Process : umber uder test 35, umber of defects d, p d/ / Usg ths data oe fds p.5 ad the stadard devato /78

31 Basc Statstcs Refresher sp.5.949/ 3 / 35.7 p + ad the test statstcs becomes.4.6 Z.76.7 Sce.96 < -.76 <.96 we caot reject the ull hypothess at the 95% cofdece level. Note: Had we chose the 7% cofdece level the Z crt +/-.36 ad we could reject the ull hypothess at ths lower level cofdece.e. we are wllg to accept a 3% chace of beg wrog. Testg for Dffereces Cout Data. Suppose we have Y ocoformaces sample ad Y ocoformaces sample We wsh to kow f the umber of ocoformaces ca be cosdered essetally equal or stated aother way s the dfferece umber of defects betwee the two samples statstcal sgfcat? Whe the sample szes are uequal ad large we ca use the ormal appromato as the Posso dstrbuto ca be appromated by the ormal for large sample szes ad small probabltes. Y Y The test statstc s Z ote the trasposto of the sample szes the Y+ Y umerator. Oe fds a crtcal value from the stadard ormal dstrbuto. Eample 6: A sample of 4 uts foud there to be 67 ocoformaces of a partcular type. After a desg chage to ad assembly a sample of 3 uts foud oly 3 ocoformaces. Determe f the egeerg chage resulted a decrease oe-sded test at a cofdece level of 9%. H : μ μ H: μ > μ α. Zcrt Z Calculatg the test statstc gves Z Sce Z.55 >.8 Z crt oe ca reject the ull hypothess ad coclude that the desg process dd deed reduce the umber of ocoformaces. The secod process comes from a populato that has a lower umber of ocoformaces per ut. 3/78

32 Basc Statstcs Refresher Topc #5: Testg for Dffereces Varaces There are three commo tests for varaces: testg to determe f the varace of a populato as estmated by a sample equals a hypotheszed or kow varace, testg to determe f two varaces estmated from two samples could come from the same populato havg equal varaces, ad 3 testg for the equalty of 3 or more varaces as ormally doe a Aalyss of Varace ANOVA procedure. Testg Varace calculated from a Sample Compared to a Hypotheszed Varace. The ull hypothess becomes H : σ σ, H: σ σ for a two sded test or H : σ σ, H: σ > σ or H : σ σ, H: σ < σ for oe-sded test. The test statstcs ths case s the Ch-square dstrbuto, so ths s called a χ test. The formula for the test statstc s; s χ σ where s the sample sze ad s s the calculated sample varace ad σ s the hypotheszed or kow varace. Eample 7: Process varace s kow to be.68. A ew method to reduce process tme wll be cosdered f t does ot crease the varace. Seems reasoable! The ew method resulted a varace of.87. Usg a 5% level of rsk α, test the ull hypothess of there beg o dfferece the varaces versus the ew varace s greater tha the curret process varace. Assume a sample sze of for the ew process. H : σ σ, H: σ > σ ad α.5 the ch-square statstc also has a dof parameter whch for ths problem s equal to -. Dof -- ad the crtcal value χ crt χ.5, Ecel CHIINV.5, The test statstcs s computed to be; s.87 χ σ.68 Sce 4.74 < we caot reject the ull hypothess. There s ot suffcet evdece that the varace of the ew hypothess s greater tha the curret process varace. Testg for Dfferece Two Observed Varaces Usg Sample Data. Whe comparg varaces of two populatos usg sample data, the F dstrbuto s the approprate dstrbuto f both populatos from whch the samples are draw are ormal. Some authors dcate that off-ormal populatos ca sgfcatly affect the accuracy of the results, others cludg Motgomery dcate that havg ormal dstrbutos s ot as crtcal as oce supposed. The other assumpto s that the two samples are depedet of oe aother, e.g. they are ot correlated. 3/78

33 Basc Statstcs Refresher The test statstc s gve by; s F, s > s s dof, dof To fd crtcal F-values oe ca use tables or Ecel. Tables of F dstrbuto values are cumbersome to read. Use the Ecel fucto FINVα,dof,dof stead of tables. Eample 8: Two competg mache tool compaes preset equpmet proposals to a customer. After proposals have bee revewed, a test s establshed to determe whether the varaces of the products produced by the two maches are equvalet. A sample of 5 uts from mache produced a varace of. ad 3 uts from mache produced a varace of.7. At a 9% cofdece level, test the hypothess that the varaces are equal. The hypotheses for ths two-sded test are, H : σ σ, H: σ σ ad 5, dof 5-4, 3, dof3-9. The crtcal F value F crt F.5,4,9.9. Usg s., s.7 s > s oe calculates the test statstc,. F.73.7 Sce F.73 <.9 F crt we caot reject the ull hypothess ad therefore we coclude there s ot eough evdece to say oe tool has better varace tha the other tool. Oe may woder s t there a lower F value that would be a lower cofdece lmt for ths two-sded test ad the aswer s yes. However, sce we ssted that the varace the umerator be the larger of the two varaces there s oly oe crtcal F value that we eed cocer ourselves wth ad that s the oe computed above. If sample szes are large > the oe ca use a ormal dstrbuto appromato to test the equalty of two varaces. The test statstc s s + s Z s s ad the crtcal values are take from the stadard ormal tables. E.g. Z.975 +/-.96. If Z falls outsde ths rage the you ca reject the hypothess that the varaces are equal. Note that the stadard devatos are used the umerator of ths test statstc. Testg for Dffereces Several Observed Varaces. Ths test s dscussed legth by Motgomery ad s called Bartlett s test. It uses the ch-square dstrbuto wth k- degrees of freedom where there are k samples whose varaces are beg tested. It s a oe-sded test. The ull hypothess s H : σ σ σ L σ, H : at least oe varace s uequal. 3 k 33/78

34 Basc Statstcs Refresher The test statstcs s q χ.36, dof k c k p q N klog s log s k c + N k 3 k ad N k, ad s k p s N k s the pooled varace. Eample 9: We draw four samples of s parts each oe sample from each treatmet ad calculate the varace of each sample. s.96, s 9.45, s3 7., s Test for equalty at α.5 ad dofk-4-3. χ crt χ.5, s p q 4 4 log 8.95 log s.5 4 c The test statstcs the evaluates to χ Sce.45 <7.85 we caot reject the ull hypothess so we coclude that the varaces are homogeeous based upo the data etat. 34/78