Hypothesis Testing for Two Population Means
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- Moris Dawson
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1 (0) Hypothesis Testig for Two Populatio Meas 0 Hypothesis Testig for Two Populatio Meas Hypothesis Testig for Two Populatio Meas QMI 0
2 (0) Hypothesis Testig for Two Populatio Meas Hypothesis Testig for Two Populatio Meas ample X estimate Idepedet samples estimate ample X mea μ =? σ =kow μ =? σ =kow Populatio Populatio The two populatios are Normal 3 Hypothesis testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) To test the hypotheses cocerig the differece betwee the two populatio meas, we will follow the same 7 steps i hypothesis testig as before. We start by formig the two hypotheses, amely, i oe of the followig forms: H o : μ μ = d H : μ μ d < d > d amplig Distributio of the differece of meas: (X - X ) ~N, 4 QMI 0
3 (0) Hypothesis Testig for Two Populatio Meas Hypothesis testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) Example: We could test the hypothesis that μ is less tha μ by 3 usig the followig hypotheses: H o : μ μ = 3 H : μ μ 3 < 3 > 3 If the differece betwee the two meas is zero. Tha, this meas that we are testig the equality of the two populatio meas. Ad the hypotheses will take oe of the followig forms: H o : μ μ = 0 H : μ μ 0 < 0 > 0 H o : μ = μ H : μ μ < > 5 Hypothesis testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) The test statistics is ample estiamtes of parameters X X ( ) ~ N (0,) Populatio parameters It follows the Normal Distributio whe the two populatios are ormally distributed, ad both stadard deviatios ( ad are kow), ad the two samples are idepedet. If the two populatio distributios are ukow, but both stadard deviatios ( ad are kow). The we should icrease the two sample sizes to have large samples ( should be more tha 30 from each populatio) ad apply the Cetral Limit Theorem. The test statistics i this case will be the same as the oe above, ad it will approximately follow the Normal Distributio. 6 QMI 0 3
4 (0) Hypothesis Testig for Two Populatio Meas Cofidece Iterval for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) The Cofidece Iterval for (μ - μ ): (whe the two populatios are ormal ad σ ad σ are kow) is: P X X Z X X Z 7 Cofidece Iterval for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) Example: To compare two types of baby formula, two samples of babies of the same age were take at radom with size 35 ad 56. The st sample of babies was fed with the st type of baby formula for 4 moths. The d were fed usig the d baby formula. After that, the mea weight gai for the two groups were foud to be: X =.560 kg, X =.795 kg. If it is kow that the baby weight gai i the two groups follows the Normal Distributio ad = kg ad = kg. Test the hypothesis that there are o differeces betwee the two baby formulas o the gai of weight usig α = QMI 0 4
5 (0) Hypothesis Testig for Two Populatio Meas Cofidece Iterval for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) - Example H o : μ - μ = 0 H : μ - μ 0 α = 0.0 The test statistics is : X X ( ) ~ N(0,) The value of the test statistics = -.39 is: Cofidece Iterval for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) - Example 0 QMI 0 5
6 (0) Hypothesis Testig for Two Populatio Meas Cofidece Iterval for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are kow) - Example Compute The Cofidece Iterval for (μ - μ ): P P X X Z X X Z Large samples. The two populatios are Normal. σ ad σ are kow μ μ X =.56 kg = kg = 35 X =.795 kg = kg = 56 P Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) Uder the above assumptios, we ca summarize the give coditios ad assumptios as: () The two populatios are ormally distributed. () σ ad σ are ukow. (3) σ = σ = σ (oe commo value). (4) mall sample sizes ( 30). ice the two tadard deviatios are assumed to be equal. Both equal to σ. The the two estimates for ad will be Pooled (combied) to obtai a better estimate for σ QMI 0 6
7 (0) Hypothesis Testig for Two Populatio Meas Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) Pooled Variace p ( ) ( ) Pooled estimated Variace. p x ( x ) y ( y ) 3 Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) The two hypotheses are: H o : μ μ = d H : μ μ d < d > d The test statistics is: X X ( ) ~ t ( ) p 4 QMI 0 7
8 (0) Hypothesis Testig for Two Populatio Meas Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) The Cofidece Iterval for (μ - μ ): [whe the two populatios are ormal, σ ad σ are ukow but equal, ad the sample sizes are small] is: P (X X ) t p (X X ) t p,, 5 Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) Example: A Factory wated to test a ew method for traiig its workers. Two samples of workers were take at radom. The st sample was subject to the old method of traiig. The d was subject to the ew method. was 9, was 5. After the ed of programs, the workers were put ito a experimet to fix idetical electroic devices, ad the completio time for each was recorded. It was foud that the mea completio time for the first sample ( X ) was 35.3 miutes, X was 3.6 miutes, with = mi ad =.444 mi. If it is kow that the two populatios are ormally distributed ad σ = σ. Does this iformatio support the claim that the ew method is better? Test this claim Use α = QMI 0 8
9 (0) Hypothesis Testig for Two Populatio Meas Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) H o : μ μ 0 H : μ μ > 0 The test statistic is: X X ( ) ~ t ( ) p = 9 = 5 X = 35.3 mi X = 3.6 mi = 4.445mi =.444 mi σ = σ α = 0.05 ( ) ( ) p ( 9 ) ( 5 ) This estimates the two values: ad sice they are equal 7 Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) The Value of the Test tatistics: = 9 = 5 X = 35.3 mi X = 3.6 mi = 4.445mi =.444 mi σ = σ α = QMI 0 9
10 (0) Hypothesis Testig for Two Populatio Meas 9 9 Hypothesis Testig for the differece of two populatio meas: (Case : The two populatios are Normal, ad ad are Ukow but equal [i.e. = ], ad the two sample sizes are small) Calculate The 95% Cofidece Iterval for (μ - μ ): = 9 = 5 X = 35.3 mi X = 3.6 mi = 4.445mi =.444 mi σ = σ α = 0.05 P(X X ) t, p (X X ) t, p P( ) ( ) P This meas: That we are 95% cofidet that the true differet betwee μ ad μ is betwee: ad QMI 0 0
11 (0) Hypothesis Testig for Two Populatio Meas HT ad CI for the differece of two populatio meas: (Case 3: The two populatios are Normal, ad are ukow ad ot equal (i.e. ), ad the two sample sizes are small) Uder the above coditios, we have the followig coditios ad assumptios: () The two populatios are ormally distributed. () σ ad σ are ukow. (3) σ σ. (4) mall sample sizes ( 30). The two hypotheses are: H O : μ μ = d H a : μ μ d < d > d HT ad CI for the differece of two populatio meas: (Case 3: The two populatios are Normal, ad are ukow ad ot equal (i.e. ), ad the two sample sizes are small) The test statistics is: X X ( ) approx. ~ t (*) where : *= + Roud dow to the ext smallest iteger QMI 0
12 (0) Hypothesis Testig for Two Populatio Meas HT ad CI for the differece of two populatio meas: (Case 3: The two populatios are Normal, ad are ukow ad ot equal (i.e. ), ad the two sample sizes are small) The Cofidece Iterval for (μ - μ ): (whe the two populatios are ormal, σ ad σ are ukow ad ot equal, ad the sample sizes are small) is: P ( ) : (X X ) t (,*) 3 HT ad CI for the differece of two populatio meas: (Case 3: The two populatios are Normal, ad are ukow ad ot equal (i.e. ), ad the two sample sizes are small) - Example olve the previous example assumig σ σ, ad α = 0.0 H o : μ μ 0 = 9 X = 35.3 = H : μ μ > 0 = 5 X = 3.6 =.444 The Test tatistics: X ( ).. X * ~ approx t ( df ) ~ t(*) 4 QMI 0
13 (0) Hypothesis Testig for Two Populatio Meas HT ad CI for the differece of two populatio meas: (Case 3: The two populatios are Normal, ad are ukow ad ot equal (i.e. ), ad the two sample sizes are small) - Example Ho : μ μ 0 = 9 X = 35.3 = Ha : μ μ > 0 = 5 X = 3.6 =.444 Decisio: ice the value of test statistics is i the o rejectio regio, we will ot reject H O : μ μ 0 with 99% cofidece QMI 0 3
14 (0) Hypothesis Testig for Two Populatio Meas HT ad CI for the differece of two populatio meas: (Case 3: The two populatios are Normal, ad are ukow ad ot equal (i.e. ), ad the two sample sizes are small) - Example The 99% Cofidece Iterval for (μ - μ ) is: P ( P ( ) : ) : ( ) (3.6) P HT ad CI for the differece of two populatio meas: I MINITAB Assumig the data are i colums: Populatio Populatio i C i C Whe σ = σ : The Miitab Commads are: MTB > Twosample C C; UBC> Alter ; for < alterative hypothesis 0 for alterative hypothesis + > for > alterative hypothesis UBC> pooled. If (σ = σ) is assumed Whe σ σ : The Miitab Commads are: MTB > Twosample C C; UBC> Alter. for < alterative hypothesis 0 for alterative hypothesis + > for > alterative hypothesis 8 QMI 0 4
15 (0) Hypothesis Testig for Two Populatio Meas HT ad CI for the differece of two populatio meas: I MINITAB A marketig research firm wishes to compare the prices charged by two supermarkets: City Ceter ad ulta Ceter. The research firm made idetical purchase for 0 radomly chose items from the weekly grocery list from each store. The shoppig expeses obtaied at the two stores are: (i files) City Ceter ulta Ceter City Ceter ulta Ceter HT ad CI for the differece of two populatio meas: I MINITAB - Example Commad ad Output for testig the equality of meas assumig the equality of the two variaces MTB > Twoample 'City' 'ulta'; UBC> Pooled. Two-ample T-Test ad CI: City, ulta Two-sample T for City vs ulta N Mea tdev E Mea City ulta Differece = mu City - mu ulta Estimate for differece: - 95% CI for differece: (-350, 37) T-Test of differece = 0 (vs ot =): T-Value = P-Value = DF = 8 Both use Pooled tdev = QMI 0 5
16 (0) Hypothesis Testig for Two Populatio Meas City Ceter ulta Ceter HT ad CI for the differece of two populatio meas: I MINITAB - Example Commad ad Output for testig the equality of meas assumig the uequality of the two variaces MTB > Twoample 'City' 'ulta'. Two-ample T-Test ad CI: City, ulta Two-sample T for City vs ulta N Mea tdev E Mea City ulta Differece = mu City - mu ulta Estimate for differece: - 95% CI for differece: (-35, 38) T-Test of differece = 0 (vs ot =): T-Value = P-Value = DF = 7 3 Hypothesis Testig for the Variace (or The tadard Deviatio of two populatios 3 QMI 0 6
17 (0) Hypothesis Testig for Two Populatio Meas Hypothesis Testig for the Variace (or the tadard Deviatio) of two populatios - Example To test the equality of variaces for two populatios. Two samples were radomly take with the followig results: = 9 X = 35.3 = = 5 X = 3.6 =.444 Test usig α = 0.0 H o : σ = σ H : σ σ The test statistics: max T. mi ~ F(8,4 ) Decisio: Do Not Reject H o : σ = σ with 98% cofidece QMI 0 7
18 (0) Hypothesis Testig for Two Populatio Meas Idepedet versus Depedet (Paired) ample Idepedet samples: The layout of the idepedet samples is preseted i the diagram below. Where we have two differet populatios from which we radomly select a sample. Each sample could have differet umber of elemets. From each sample we estimate the cocer populatio parameters. ample X estimate Idepedet samples estimate ample X mea μ =? σ =kow μ =? σ =kow Populatio Populatio The two populatios are Normal 35 Idepedet versus Depedet (Paired) ample The layout of the two Idepedet samples is as follows: ample ample X, X, X, X, X, X, The differece betwee ay two elemets i the two samples is caused by two factors: () They are differet items. (idividual differeces) () Each oe of them received a differet treatmet (treatmets differece) 36 QMI 0 8
19 (0) Hypothesis Testig for Two Populatio Meas Idepedet versus Depedet (Paired) ample Paired sample (Experimet): The layout of the paired sample is preseted i the diagram below. Where we have oe populatio, from which we radomly select a sample. Each elemet i the sample is subject to two treatmets. From each treatmet we estimate the cocer populatio parameters. ample Each sample elemet is subject to two treatmets Populatio Treatmet Treatmet 37 Idepedet versus Depedet (Paired) ample The layout of the oe sample with two treatmets is as follows: Item# Treatmet Treatmet X, X, X, X, The differece betwee ay two items i a row i the two treatmets is caused by oly oe factor (sice it is the same item): Each has received a differet treatmet (pure treatmets differece) X, X, 38 QMI 0 9
20 (0) Hypothesis Testig for Two Populatio Meas Hypothesis Testig for Depedet (Paired) ample The two hypotheses are: H o : μ μ = d H : μ μ d < > Paired sample (experimet) geeral layout: Uit # Trt Trt D X, X, X, - X, X, X, X, -X, X, X, X, - X, um Σ D If the populatio of the differeces (D) is ormally distributed, the: 39 Hypothesis Testig for Depedet (Paired) ample The Test statistics: is D - d D - d = ~t(-) D D w here D D= ad: D D D 40 QMI 0 0
21 (0) Hypothesis Testig for Two Populatio Meas Cofidece Iterval for Depedet (Paired) ample Cofidece iterval for the differece of two meas (Paired sample): D P D - t D α <μ-μ < D + t α =-α (,-) (,-) 4 Cofidece Iterval for Depedet (Paired) ample Example I a paired experimet, two treatmets were put ito a test usig a sample of size 8 from a ormal distributio populatio. Uit # T T Does the results above support the hypothesis that there is o differece i the two treatmets? Test this usig 95% cofidece. The costruct the 95% cofidece iterval for the differece of the two meas. 4 QMI 0
22 (0) Hypothesis Testig for Two Populatio Meas Cofidece Iterval for Depedet (Paired) ample Example Uit # T T Uit # T T Diff. (D) D From the above iformatio we have the stadard deviatio for the treatmets differece (D) equals: D -6 = D ΣD - 6- = 8-7 = Cofidece Iterval for Depedet (Paired) ample Example Uit # T T D = - =.070 α = 0.05 D The hypotheses are: H o : μ μ = 0 H : μ μ 0 The Test statistics value: ~ t CV =.365 Decisio: We will reject H O : μ μ = 0 with cofidece 95% 44 QMI 0
23 (0) Hypothesis Testig for Two Populatio Meas Cofidece Iterval for Depedet (Paired) ample Example Uit # T T The cofidece Iterval for the differece of the two treatmets meas: D = =0.05 D P μ -μ : D ± t = -α D α (,-).070 P μ -μ : - ±.365 ( ) = P QMI 0 3
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