# Thus far. Inferences When Comparing Two Means. Testing differences between two means or proportions

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1 Inference When Comparing Two Mean Dr. Tom Ilvento FREC 48 Thu far We have made an inference from a ingle ample mean and proportion to a population, uing The ample mean (or proportion) The ample tandard deviation Knowledge of the ampling ditribution for the mean (proportion) And it matter if the ample ize i large or mall Teting difference between two mean or proportion The ame trategy will apply for teting difference between two mean or proportion With a few twit Mean Large ample Small ample pool the variance Proportion When teting H o : we need to check if p p Teting difference between two mean or proportion We will alo need to come up with: An etimator of the difference of two mean/proportion The tandard error of the ampling ditribution for our etimator With two ample problem we have two ource of variability and ampling error We alo mut aume the ample are independent random ample What are independent, random ample? Independent ample mean that each ample and the reulting variable do not influence the other ample If we ampled the ame ubject at two different time we would not have independent ample If we ampled huband and wife, they would not be independent However, we have a trategy to ae change over time of the ame ubject paired difference tet Deciion Tree for Two Mean Target Aumption Tet Statitic Independent random ample z, uing Large ample ize (n, n >3) ample variance H : µ -µ D Independent random ample Small ample ize t, uing Population appr. normal pooled variance Equal variance S p

2 Deciion Tree for two Proportion Teting Aumption Tet Statitic H :p -p Independent random ample Large ample ize (n, n >3) Known that p p under H H :p -p D Independent random ample z Large ample ize When D z, uing pooled ample proportion P p Example Problem Two group of tudent were urveyed about lecture note 86 tudent in a promotional trategy cla that required the purchae of the lecture note 35 tudent enrolled in a ale/retailing cla which didn t offer lecture note At the end of the emeter, tudent in both clae were aked if Having a copy of the lecture wa [would be] helpful in undertanding the material. The quetion wa meaured on a nine point cale where trongly diagree and 9 trongly agree Lecture Note problem Cla with Lecture Note n Cla without Lecture Note n Do the ample provide ufficient evidence to conclude that there i a difference in mean repone of the two group? Ue ". Lecture Note problem Null hypothei H : (: -: )? Alternative H a : (: -: )? two-tailed tet Aumption Two independent ample, n i Large Tet Statitic z* Rejection Region z "./.575 Calculation z* z*? z Concluion ".5??? H : (: -: ) We need to figure out the ampling ditribution ( - ) The mean of the ampling ditribution for ( - ) Will equal (: -: ) D We uually deignate the expected difference a D under the the null hypothei Mot often we think of D ; no difference Standard Error of the difference of two mean The Standard Error of the difference of two mean i given a: ( x x ) n n The ampling ditribution of ( - ) i approximately normal for large ample under the Central Limit Theorem Page 454

3 The Standard Error for the difference of two mean I baed on two independent random ample We typically ue the ample etimate of F and F Which are and ( x x ) n n The Tet Statitic for our problem ( ) z* Z* (.68 )/ Comparion of two mean Null hypothei Alternative Aumption Tet Statitic Rejection Region Calculation Concluion H : (: -: ) H a : (: -: ) two-tailed tet Two independent ample, n i Large z* ( )/[(.94/86)(.99/35)].5 z "./.575 z*.9 z* < z ".5.9<.575 Cannot reject H : (: -: ) 99% Confidence Interval for the Difference of Two Mean ( x x) ± zα / ( x x ) ( )" z./ [(.94/86) (.99/35)].5.68 ".575(.34).68 ".8 -. to.48 Notice the 99% C.I. contain the null hypothei value, zero Deciion Tree for Two Mean Target Aumption Tet Statitic Independent random ample z, uing Large ample ize (n, n >3) ample variance H : µ -µ D Independent random ample Small ample ize t, uing Population appr. normal pooled variance Equal variance S p What about when n i mall? We will ue a t-tet and the t ditribution Aumption Both ample are approximately normal The population variance are equal Random ample elected independently of each other 3

4 The tandard error for a mall ample difference of mean Since we aume F F thu ( ) We hould pool our etimate of the tandard error of the ampling ditribution Uing information from both ample etimate POOLED ESTIMATE OF THE VARIANCE Then our formula will be a weighted average of and p ( n ) ( n ( n ) ( n ) ) Note: the denominator reduce to (n n ) which i the d.f. for the t ditribution Page 458 ˆ Next, we ue the Pooled Etimate of the Variance to calculate the etimate of the tandard error n p p ( x ) x p n n ˆ ( x x ) p n n n What doe pooling do for u? Pooling generate a weighted average a the etimate of the variance The weight are the ample ize for each ample A pooled etimate i thought to be a better etimate if we can aume the variance are equal And our degree of freedom are larger - d.f. n n Which mean the t-value will be maller Problem, Tapeworm in heep An experiment wa done to compare the mean number of tapeworm in the tomach of heep that had been treated for worm veru thoe not treated. There were 7 heep in the Treatment group and 7 in the Control Group I the number of tapeworm lower in the treatment group at ".5? Tapeworm in heep The mean and tandard deviation are: Treatment Control n 7 n 7 4

5 Tapeworm in heep We aume the variance are equal o we make a pooled etimate p ( n ) ( n ) ( n n ) Tapeworm in heep We aume the variance are equal o we make a pooled etimate p p (7 )98.6 (7 ) , Tapeworm in heep Then we ue our pooled etimate to etimate the tandard error of the ampling ditribution for the difference of mean ˆ ( x x ) p n n Tapeworm in heep Then we ue our pooled etimate to etimate the tandard error of the ampling ditribution for the difference of mean ( x ) x ( x 4.387(.5345) 7.69 x ) Tapeworm in heep Null hypothei Alternative Aumption Tet Statitic Rejection Region Calculation Concluion H : (: -: ) H a : (: -: ) < one-tailed tet, lower Small independent ample, approx normal, variance are equal t* ( )/7.69 -t.5, d.f t* -.48 t* > -t.5, d.f > -.78 We cannot reject H : (: -: ) The 9% C.I. For the Tapeworm example ( x x) ± t α /, n.. ( x x ) n d f ( )" t./, nn- d.f. [4.387 (/7/7).5 ] -.43 ".78(7.69) -.43 " to.73 Thi C.I. contain zero analogou to null hypothei, one-tailed tet, at ".5 5

6 What Do I need for a Difference of Mean Tet? Two independent random ample Determine if I am dealing with a mean or proportion If a Mean, large ample or mall ample Large ample I need not aume the ditribution of the population, or anything about the variance I calculate the tandard error and make my tet uing a z-value Small ample I mut aume Independent random ample From population that are normally ditributed The variance are equal Small Sample Step Calculate pooled etimate of Variance Ue pooled etimate to calculate the tandard error Conduct the tet ue a t-value with n n df Deciion Tree for two Proportion Teting Aumption Tet Statitic H :p -p Independent random ample Large ample ize (n, n >3) Known that p p under H H :p -p D Independent random ample z Large ample ize When D z, uing pooled ample proportion P p What about the difference in proportion? Baed on large ample only Same trategy a for the mean We calculate the difference in the two ample proportion Etablih the ampling ditribution for our etimator Calculate a tandard error of thi ampling ditribution Conduct a tet Difference of Proportion For the null hypothei E( ) (p -p ) D The Standard Error for ( p ) p q ˆ p q ( p ˆ ) n n For Proportion Since thi i a large ample problem, we could ue the ample etimate of p and p to etimate the tandard error ( p ˆ ) n n Page 476 6

7 For Proportion Note: Under the Null Hypothei where p p The book ugget that we ue a weighted average for p p baed on adding the total number of uccee and divide by the um of the two ample ize (P477-78) p p (x x )/(n n ) where x # of uccee Note : the book ue intead of p p So for Proportion For a confidence interval, ue the ample etimate in thi manner to generate the tandard error ince there i no aumption that the variance are equal ( p ˆ ) n n So for Proportion For a Hypothei Tet where the proportion are equal, pool the information and ue thi approach p p (x x )/(n n ) where x # of uccee q p - p p n ( p p ) p p n Gender gap in politic Over half the vote will be cat by women The quetion i if the Democratic platform i viewed more favorably by women Suppoe 5 men and 5 women tated their party preference for Republican Data urvey of 3 5 men and 5 women 8 men favor Republican 7 women favor Republican Conduct a tet uing ".5 that a lower proportion of women favor Republican Calculate the tandard error The ample proportion are Men 8/5.54 Women 7/5.467 Pooled etimate p p (87)/(55).533 q p Calculate the tandard error Standard Error for the problem F (p-p) [(.533)(.4967)(/5 /5)].5 F (p-p) [(.49989)(.333)].5 F (p-p)

8 What will be the tet tatitic? Gender Gap in Politic z* [( )- ]/ Where the numerator how the difference in voting Republican for men and women Note that I ubtract out the hypotheized value D o I uually try to keep the z* for a difference of proportion (or mean) poitive a a matter of convenience Null hypothei Alternative Aumption Tet Statitic Rejection Region Calculation Concluion H : (p m -p w ) H a : (p m -p w ) > one-tailed tet, upper Large ample proportion, ue normal, aume variance equal z* [( )-]/ z.5,.645 z*.64 z* < z.5.64 <.645 Cannot reject H : (p m -p w ) Confidence Interval for the Gender in Politic Problem We will ue ". to relate to the previou onetailed hypothei tet of ".5 Standard error (not auming equal variance) i: F (p-p) [(.54)(.46)/5 (.467)(.533)/5] % C.I. ( )".645(.5758).73 " to.68 Section 9.3: Paired Difference Tet When you have a ituation where we record a pre and pot tet for the ame individual We cannot treat the ample a independent In thee cae we can do a Paired Difference Tet. It called Matched Pair Tet in the book. (page467-7) Paired Difference Tet The trategy i relatively imple We imply create a new variable which i the difference of the pre-tet from the pot tet Thi new variable can be thought a a ingle random ample For thi new variable we calculate ample etimate of the mean and tandard deviation Paired Difference Tet And then calculate C.I. or conduct a hypothei tet on thi new variable Often time the mean difference i referred to a D And the hypothei i often: H : : D Thi i no different than any ingle mean tet, large ample or mall ample 8

9 Example of paired difference data Alzheimer tudy Patient Time 5 Time 5 3 Difference Twenty Alzheimer patient were aked to pell 4 homophone pair given in random order Homophone are word that have the ame pronunciation a another word with a different meaning and different pelling Nun and none; doe and dough The number of confuion were recorded The tet wa repeated one year later Alzheimer tudy The reearcher poed the following quetion: Do Alzheimer patient how a ignificant increae in mean homophone confuion over time? Ue an alpha value of.5 Alzheimer tudy Statitic Time Time Difference Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtoi Skewne Range 6 Minimum -3 Maximum 6 9 Sum Count Alzheimer tudy Null hypothei Alternative Aumption Tet Statitic Rejection Region Calculation Concluion H : : D H a : : D > one-tailed tet, upper mall ample, normal t* (.65 )/(3.//) t ".5, 9 d.f..79 t*.3 t* > t ".5, 9 d.f..3 >.79 Reject H : : D 9

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