Data Analysis and Statistical Methods Statistics 651

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Nicholas Hines
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1 Data Aalyi ad Statitical Method Statitic 65 Lecture 9 Suhaii Subba Rao Tetig o far We have looked at oe ample hypothei tet of the form H 0 : µ = µ 0 agait H A : µ µ 0 H 0 : µ µ 0 agait H A : µ > µ 0 H 0 : µ µ 0 agait H A : µ < µ 0. The bai of the tet i if the ample ize i large eough, the ample mea i approximately ormal thak to the cetral limit theorem o uder the ull (the ample mea X) X N(µ 0, σ /) (where i the ample ize ad σ i the variace of oe obervatio). If the ample ize i mall, ay 0 obervatio, the the above tet i oly valid if the ample itelf (ot jut the ample mea), that i X are ormally ditributed. Sometime it i realitic to aume that the obervatio come from a ormal ditributio, we ca alo check for thi by makig a QQplot. We alo looked at the otio of power. Typically a tet would be H 0 : µ 0 agait H A : µ > 0, more geerally it i H 0 : µ µ 0 agait H A : µ > µ 0. To calculate the power we calculate the probability of rejectig the ull whe the alterative µ = µ (where for thi tet µ > µ 0 ) i true (thi calculatio doe ot ivolve the ample mea X). Suppoe the power i large. I the cae that we oberve X ad are uable to reject the ull, (if the the power i large) we ca ay that the mea i ulikely to be over µ. Example A radom ample of flower petal are draw ad for each petal the legth i meaured i cm (a petal i part of a flower). Cotruct a 95% CI. We wat to ee whether there i igificat evidece to ugget that the mea legth of the petal i greater tha 0.8cm? Do the tet at the 5% level. The tet wa doe i SPSS ad the data i ummaried below. Oe Sample Statitic N Mea Std. Deviatio Std. Error Mea

2 Solutio The CI Iterpretatig the output: Treatig 0.5 a the true populatio tadard derivatio the 95% CI i The ample average i X =. The ample ize i = 4 (thi mea there were 4 flower petal). SPSS doe ot kow what the true populatio tadard deviatio σ (or variace σ ) i, hece it etimate it from the ample (uig the formula 4 i= (X i ) - i the ame way you evaluated the ample variace 40 i HW). I thi example, the ample tadard deviatio i = 0.5. Hece the ample variace of the populatio i = 0.5. The tadard error of the mea i = The tet: , ] = , ] = 0.847,.5]. We wat to tet omethig about the true populatio mea: H 0 : µ 0.8 agait H A : µ > 0.8. Note to do the tet we plot the ditributio uder the ull ad either calculate the p-value or the rejectio regio. 4 5 I thi example we will aume that 0.5 i the true populatio tadard deviatio. Of coure it i ot, ad it i a etimate of σ. I the ret of today lecture we will ee what happe whe it i actually etimated. p-value method Do the z-traform P( X ) = P(Z 0.8 ) = P(Z.56) = Sice level, there i eough evidece to reject the ull. rejectio regio Cotruct a oe-ided CI about 0.8. Hece if X i greater tha = 0.9, the we reject the ull. Sice > 0.9, there i eough evidece to reject the ull. REMEMBER we are aumig that 0.5 i the true tadard deviatio whe i fact thi i ot the cae. How etimatig the tadard deviatio effect our reult Uderlyig the work o far, we have aumed that the tadard deviatio σ i kow. Thi i ometime a plauible aumptio. For example if we wat to compare the ditace travelled lat year with thoe of thi year. Lat year the mea ditace travelled by a pero wa 000 km ad the tadard deviatio wa 500 km. Thi year baed o a ample of 50 people, the ample mea ditace travelled wa 00 km. It would be reaoable to aume that the variace ha ot chaged, oly the mea may or may ot have chaged. However i geeral we will ot have apriori kowledge of σ. σ will be ukow ad ha to be etimated from the data X,..., X. 6 7

3 Recall we ca etimate the variace uig the ample variace = i= (X i X). We ca ubtitute thi i all place where we eed a σ. Eg. whe we make a Z-traform we go from X µ σ X µ? error mea more a larger variace. We fid that the Z-traform i o loger a tadard ormal, but i i fact wider (ha a larger variace). What happe to the legth of the cofidece iterval? For example, the the petal example above we had to etimate the variace from the data but we completely igored the fact it wa a etimate whe we did the tet. A importat quetio to ak i whether etimatig the variace effect the reult of the tet. Recall i radom (recall before we had the populatio variace σ, which i ot radom), thi itroduce more error ito the ytem. More 8 9 Goett experimet We fid that whe we etimate the variace (rather tha ue the true variace) we eed to icreae the ize of the cofidece iterval become larger to accout for the greater variatio i the Z-traform. For each ample of ize 0 i cotructed the CI he cotructed the 95% CI: ] X.96 0, X Thi fact wa dicovered by William Goett, who wa a chemit, workig for Guie the brewery (i Irelad) ad had to judge the quality of everal brew. He wa workig with a mall ample ize X,..., X 0 (ample ize i 0), ad etimated the tadard deviatio from thi = 0 9 i= (X i 0). He kew that the ample mea wa µ 0 (ay µ 0 = 4). He wated to cotruct 95% CI for the mea. But, rather tha ue the true variace σ, he replaced it with the with the ample variace. He couted the umber of time the true mea µ 0 wa i the ize the iterval. You would expect that about 5% of the time the true mea hould be outide the iterval (ice it i a 95% CI). What William Goet oticed wa that the true mea wa outide the iterval more tha 5% of time. Hece the true i iide the iterval le tha 95% of the time. Thi 0

4 ugget that the iterval i ot log eough, ad we eed to ue a loger iterval for the 95% CI to be accurate. Why...what the problem? Itroducig the t-ditributio He ivetigated thi further, he howed that the Z-traform i ot from a tadard ormally ditributed, that i Z N(0,). Baically becaue the tadard deviatio ha to be etimated it add more radome (ucertaity) ito the ytem. More ucertaity mea that the z-traform X µ / i more likely to take large value (ice the ample variace i radom ad varie accordig to the ample). Whe we have to etimate the variace uig the ample variace = (X i X) i= 3 The t-ditributio The tadardiatio ha the followig ditributio X µ t( ), t( ) i a ditributio like the tadard ormal ditributio. differet we have a differet ditributio. We call t( ) the Studet t-ditributio with ( )-degree of freedom. Studet becaue William Goet wrote all hi paper uder the peudoym Studet. For i the umber of obervatio ued to etimate σ, eg = i= (X i X). Ulike the cae where σ i kow ad q X µ ha a tadard ormal σ ditributio the ditributio of q X µ deped o the ample ize. 4 5

5 Why the ditributio deped o the ample ize? Coider the followig ituatio. If you were to etimate the variace uig 00 obervatio, you would expect it to be far better tha a etimator calculated uig 0 obervatio. The idea here i exactly the ame a expectig the etimator of the mea ivolvig 00 obervatio to be far better tha a etimator ivolvig 0 obervatio. large X r 00 µ By more radom we mea that the ditributio of X r 0 µ 0 0. ha thicker taile tha the ditributio of X r 00 µ Thicker tail mea that we are more likely to draw value far from the zero. Therefore whe the ample ize i mall, the ample variace will itroduce far more radome ito the ytem tha whe the ample i large. Hece whe the ample ize i mall, it i reaoable to uppoe that the ditributio of X r 0 µ i more radom tha whe the ample ize i What doe thi mea for u? Cofidece iterval uig the t-ditributio We ca pretty much do everythig wa we did before, but whe we etimate the variace we eed to ue the t-ditributio itead of the tadard ormal. Replace every tadard ormal with the t-ditributio! Example If α = 0.05 rather tha look up z 0.05 i the Normal table we ue the t-table (i Table ) ad fid t 0.05 ( ) i the ample. We kow whe the variace σ i kow. The ( α)00% CI i ] σ X z α/, X σ + z α/. Whe the variace σ i ukow, etimate it from the data = i= (X i X) ad ue the CI ] X t α/ ( ), X + t α/ ( ). 8 9

6 Example: 95% Cofidece iterval We kow whe the variace σ i kow. σ.96,varx +.96 σ ]. Whe the variace i etimated, the 95% CI i = 3, =, t 0.05 () = The CI i X , X ]. = 0, = 9, t 0.05 (9) =.6. The CI i X.6 0, X ]. = 30, = 9, t 0.05 (9) =.045. The CI i X , X ]. =, = 0, t 0.05 (0) =.98. The CI i X.98, X +.98 ]. The 95% CI i X We kow that z 0.05 =.96 For the t-ditributio we have = 3, =, t 0.05 () = = 0, = 9, t 0.05 (9) =.6. = 30, = 9, t 0.05 (9) =.045. =, = 0, t 0.05 (0) =.98. We ee a get larger the value of t 0.05 () get cloer to z 0.05 =.96. I fact for > 50 we geerally do t ue the t-ditributio, but itead approximate thi by the ormal ditributio. Look at t ditributio lecture9.pdf 0 Hypothei tetig: Rejectio regio method we reject H 0 if X lie outide the iterval above. Whe variace i kow ad we were tetig H 0 : µ = µ 0 agait H 0 : µ µ 0 (or the oe-ided equivalet). At the level α µ 0 z α/ σ, µ 0 + z α/ ] σ we reject H 0 if X lie outide the iterval above. Whe the variace σ i ukow, etimate it from the data = i= (X i X). At the level α µ 0 t α/ ( ), µ 0 + t α/ ( ) ] Thi i ofte called the t-tet. Example Suppoe we do the tet H 0 : µ = 0 agait H 0 : µ 0 at the 5% level ad = 30. The the o-rejectio regio i ].045,.045. If X i ot i thi iterval we reject the ull. 3

7 Hypothei tetig: p-value method It i very hard to calculate p-value uig the t-ditributio by had. Thi i where the tatitical package become very ueful. The alway give the p-value (rather tha a rejectio regio). Take a look at the JMP output ad make ure you ca udertad it. Example A radom ample of flower petal are draw ad for each petal the legth i meaured i cm (a petal i part of a flower). Cotruct a 95% CI. We wat to ee whether there i igificat evidece to ugget that the mea legth of the petal i greater tha 0.8cm? Do the tet at the 5% level. The tet wa doe i SPSS ad the data i ummaried below. Oe Sample Statitic N Mea Std. Deviatio Std. Error Mea Solutio The quetio i idetical to the quetio at the tart of cla today. But ow we are able to take ito accout that 0.5 i the etimated tadard deviatio. There are 4 obervatio, hece we will be dealig with a t-ditributio with 4 = 40 degree of freedom. Look up.5% i the t-table with 40-degree of freedom. You hould fid.0. The 95% CI i We wat to tet H 0 : µ 0.8 agait H A : µ > 0.8. To do thi we cotruct rejectio regio, jut a we did before. Sice the tet i oe-ided we will look up 5% i the t-table with 40-degree of freedom. It give the value.684. We cotruct a oe-ided CI about 0.8 (uig.684). Hece if X i greater tha = 0.933, the we reject the ull. Sice > 0.933, there i eough evidece to reject the ull. Oberve that the CI i loger tha the oe obtaied i Solutio (due to our takig ito accout the radome of ) , ] = 0.84,.6] Thi mea we have 95% cofidece that the mea lie here. Compare thi with the awer i olutio. 6 7

8 Example 3 Propective alepeople are beig offered a ale traiig program. Previou data idicate that without traiig the average ale pero old 33 item. The compay wat to acce the impact the ale traiig program ha o ale. Oe moth after traiig tarted, 3 people had traiig ad it wa foud that o average 35 book were beig old, the ample tadard deviatio i = (calculated uig = 30 k (X i 35) ). cotruct a 95% CI for the mea umber of book old with traiig ad tet whether the traiig ha ehaced ale? Ue α = Solutio: CI We eed to cotruct a 5% CI, thi mea.5% o either ide of the ditributio. The ample ize i = 3. We ue Table (t-table) to look up t 0.05 ( ) = t 0.05 (3 ) =.04. The CI i: ] X t 0.05 ( ), X + t 0.05 ( ) ] = 35 t 0.05 (30) 3,35 + t 0.05(30) 3 = 35.04, ,] 8 9 Solutio: Hypothei tet The ull ad alterative are H 0 : µ 33 agait H A : µ > 33. Alway have i mid where the rejectio regio i. The above mea the rejectio regio i o the right of 33 (draw the ditributio). Uder the ull (ice the ample ize i quite large we have) X 33 3 t(30). The 5% rejectio regio i ] 3,. So for ay value of X greater tha = we reject the ull (draw the plot of the ditributio ad where the rejectio regio lie). Sice 35 i ot i the rejectio regio there i ot eough evidece to reject the ull at the 5% level. It i a oe ided tet o we look up i Table to fid: t 0.05 (30) =

9 Hypothei tetig: p-value method p-value ca alo be computed uig the ditributio. Typically JMP will alway evaluate the ample tadard deviatio from the data ad ue the t-tet (thi i what the oe ample mea tet doe i JMP - ee lecture 8 for a review of how to do it i JMP). But thi i quite difficult uig the table. I the t-table you eed to fid the probability P(t 0 x µ 0 ). which geerally i ot poible. If you are doig the tet by had, cotruct rejectio regio (like i a exam). However if you do the t-tet uig JMP, it will give you the actual p-value. It i a lot eaier to ue thi method! 3 33 Example 4 A airlie wat to evaluate the depth perceptio of pilot over the age of fifty. A radom ample of pilot over the age of fifty are ued. The ample data of the pilot error i lited below (a) Cotruct a 95% CI for the mea error. (b) The mea error for youg plot i, tet the hypothei that the mea error i larger for older pilot. 34

I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint).

I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). I. Meaure of Cetral Tedecy: -Allow u to ummarize a etire data et with a igle value (the midpoit.. Mode : The value (core that occur mot ofte i a data et. -Mo x Sample mode -Mo Populatio mode. Media : the