Chapter 4 Tests of Hypothesis
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1 Dr. Moa Elwakeel [ 5 TAT] Chapter 4 Tests of Hypothesis 4. statistical hypothesis more. A statistical hypothesis is a statemet cocerig oe populatio or 4.. The Null ad The Alterative Hypothesis: The structure of hypothesis testig will be formulated with the use of the term ull hypothesis. This refers to ay hypothesis we wish to test that called H. The rejectio of H leads to the acceptace of a alterative hypothesis deoted by H. A ull hypothesis cocerig a populatio parameter,θ will always be stated so as to specify a exact value of the parameter, whereas the alterative hypothesis allows for the possibility of several values. We usually test the ull hypothesis: H : θ θ agaist oe of the followig alterative hypothesis: H θ θ : θ > θ θ < θ 5
2 Dr. Moa Elwakeel [ 5 TAT] 4.. Types of Errors: (i)type Oe Error I: Rejectio of the ull hypothesis whe it is true is called a type I error. The probability of committig a type I error also called the level of sigificace which is deoted by. ometimes is called the sie of the critical regio or the sie of the test. (ii)type Two Error II: Acceptace of the ull hypothesis whe it is false is called a type II error, which is deoted by β. Possible situatios i testig a statistical hypothesis H is true H is false Accept H Correct decisio Type II error,β Reject H Type I error, Correct decisio Type I error: rejectig H whe H is true. Type II error: acceptig H whe H is false. P (Type I error) P (rejectig H H is true). P (Type II error) P (acceptig H H is false) β. Ideally we like to use a test procedure for which both the type I ad type II errors are small. It is oticed that a reductio i β is always possible by icreasig the sie of the critical regio,. 53
3 Dr. Moa Elwakeel [ 5 TAT] For a fixed sample sie, decrease i the probability of oe error will usually result i a icrease i the probability of the other error. Fortuately the probability of committig both types of errors ca be reduced by icreasig the sample sie. (iii) Oe Tailed ad Two Tailed test: A test of ay statistical hypothesis where the alterative is oe sided such as: H : θ θ vs or H H : θ > θ : θ< θ is called a oe tailed test. The critical regio for the alterative hypothesis θ> θ lies etirely i the right tail of the distributio while the critical regio for the alterative hypothesis θ< θ lies etirely i the left tail. H H θ< : θ> θ : θ A test of ay statistical hypothesis where the alterative is two sided, such as: H θ θ vs H: : θ θ is called two tailed test sice the critical regio is split ito two parts havig equal probabilities placed i each tail of the distributio of the test statistic. 54
4 Dr. Moa Elwakeel [ 5 TAT] 4. P Values i Decisio Makig: A p value is the lowest level (of sigificace) at which the observed value of the test statistic is sigificat. P value P ( > obs ) whe H is as follows: H : θ θ p value P ( > obs ) whe H is as follows: H : θ> θ p value P ( < obs ) whe H is as follows: H : θ< θ H is rejected if P value otherwise H is accepted. Ex (): H : µ vs H : µ,.5.87 P value P( >.87 ) P( >.87) [ P(.87)] [.9693] (.37).64 ice P value > the H is accepted. 4.3 Testig Hypothesis Cocerig oe Populatio We have the followig steps to test ay hypothesis:. tatig the ull hypothesis H : θ θ.. Choosig a appropriate alterative hypothesis from oe of the alteratives H : θ < θ orθ > θ orθ θ. 55
5 Dr. Moa Elwakeel [ 5 TAT] 3. determiig the sigificace level of sie.,.5,.5 or.. 4. Determiig the rejectio or critical regio (R.R.) ad the acceptace regio (A.R.) of H. / 5. electig the appropriate test statistic ad establish the critical regio. If the decisio is to be based o a p value it is ot ecessary to state the critical regio. 6. Computig the value of the test statistic from the sample data. 7. Decisio rule: rejectig H if the value of the test statistic i the critical (rejectio) regio or if the p value also. acceptig H if the value of the test statistic i the acceptace regio or if the p value > also Testig hypothesis for a igle populatio Mea There are differet cases to study the populatio mea that is: (i)large amples( > 3) -data eeded:,,,,. -type of populatio(ormal or ot ormal) 56
6 Dr. Moa Elwakeel [ 5 TAT] - the variace of populatio is kow or ukow the usig - state the hypothesis : ( < > ) 3- the statistic: - if is kow x µ σ o - if is ukow x µ s o, where ~, 4- Determiig the rejectio or critical regio (R.R.) ad the acceptace regio (A.R.) of H, that is: i)if : >,reject if > ii)if : <,reject if < ( always egative) iii) if :, reject if > or < (where ) 5-Decisio rule: if the rejectio rule is hold the we reject H ad accept H if the rejectio rule is't hold the we accept H ad reject H E() We take a radom sample of 36 apples from a apples farms ad foud that the average weight of the apple is 9 gm. Assumig that the apples weight i this farm has ormal distribuo with stadard deviao 8 57
7 Dr. Moa Elwakeel [ 5 TAT] gm. Test that the average weight of apple i this farm less tha gm at.. olu. 36,.,, 8, 9 : : < x µ o σ ice 3.33 <. 33, thus we reject ad accept. i.e., the average weight of the apples i this farm is less tha gm at.. Ex (3): A radom sample of recorded deaths i the Uited tates durig the past year showed a average life spa of 7.8 years with a stadard deviatio of 8.9 years. Dose this seem to idicate that the average life spa today is greater tha 7 years? Use a.5 level of sigificace. olu.,.5, 7, 8.9, 7.8 : 7 : > 7 x µ o
8 Dr. Moa Elwakeel [ 5 TAT] ice.> , thus we reject ad accept i.e., the average life spa is greater tha 7 years at.5. or P value P ( >.) P (.) reject H sice > p value Ex (4): A maufacturer of sports equipmet has developed a ew sythetic fishig lie that he claims has a stadard deviatio of.5 kilogram. Test the hypothesis that µ 8 kilograms agaist the alterative that µ 8 kilograms if a radom sample of 5 lies is tested ad foud to have a mea breakig stregth of 7.8 kilograms. Use a. level of sigificace. olutio: H : µ 8 vs H : µ 8,. 5, 7.8, σ.5 µ σ ad.575 /.995 /.995 R. R.: >.575 or <.575 ice.83 R. R. we reject H at. 59
9 Dr. Moa Elwakeel [ 5 TAT] or p value P ( >.83 ) P ( >.83) ( P (.83) (.9977) (.3).46 H is rejected sice p value (ii)mall amples( < 3) I small samples we have three cases for the statistic as follows: )Normal populatio ad is kow, the the statistic is: x µ σ o Which has stadard ormal distributio, whe is true. ) Normal populao ad is ukow, the the statistic is: T x µ o Which has t-distributio with(-) degrees of freedom, whe is true. Rejectio rules for will be as follows: i) If : µ > µ o the, Reject if T > t, ii) if : µ < µ o the, Reject if T < t ( ), ( where ( ), t, t ) 6
10 Dr. Moa Elwakeel [ 5 TAT] iii) if : µ µ o the, Reject if T > t or, T < t, where t t,, 3)if the populao is ot Normal, it will be study at the ed of this course. Ex (5): If a radom sample of homes with a mea 4 icluded i a plaed study idicates that vacuum cleaers exped a average of 4 kilowatt hours per year with stadard deviatio of.9 kilowatt hours dose this suggest at the.5 level of sigificace that vacuum cleaers exped o the average less tha 46 kilowatt hours aually, assume the populatio of kilowatt - hours to be ormal? olu. H : µ 46 vs H : µ < 46,.5, 4,.9 T 4 46 µ v, t t.796 R. R.: T <.796 ice T.6 A. R. we accept H at.5 accept H sice the value of t is i the acceptace regio (A.R.) 6
11 Dr. Moa Elwakeel [ 5 TAT] 4.3. Testig hypothesis for a igle populatio Variace We will always assume that the populatio is ormally distributed.(there is o results for this by usig computer). teps for this test - Data : σ o,,,, - hypothesis : : < : > 3- the statistic: () 4- the table value:,,, 5- the decisio: we reject if,,( ) <, >, <, >,( ) Ex (6): h : < h : > h : : < : > : Chemically evaluated irrigatio water samples from 4 Qatif wells. The percet of Na الموجب) catios (األيون i the water was measured: 43, 47,4,45, 45,48, 47,47,46,5,5,5,5,49 6
12 Dr. Moa Elwakeel [ 5 TAT] olu. Assumig a ormal distributio, test whether the variace of the total Na catios is less tha at.5. - Data : σ, 4,.5, o - hypothesis : : : < 3- the statistic: () 4- the table value:, (.) 3.77, the decisio: we reject if <, Thus, we accept ad reject, i.e., we will reject the assumptio that the variace is less tha Testig hypothesis for a igle populatio Proportio We will assume that the sample sie is large ( > 3)ad the proportio sample is. teps for this test - Data : P o,,, - hypothesis : : < : > 63
13 Dr. Moa Elwakeel [ 5 TAT] 3- the statistic: () 4- the table value:, 5- the decisio: we reject if < > > < h : < h : > h : h : < h : > h : Ex (7): A radom sample of sie uit from a factory produco ad foud that it cotaied 8% defecve uits. Ca we say that the defecve proportio uits i the factor productio is more tha 7% at.5. olu. - Data : P o.7,,.5, r. 8 - hypothesis : :.7 : >.7 3- the statistic: () 4- the table value (.) 5- the decisio: we accept sice ( <
14 Dr. Moa Elwakeel [ 5 TAT] 4.4Testig Hypothesis Cocerig two idepedet Normal Populatios 4.4. tesg for the two populaos meas (i) Large amples(, > 3) -data eeded:,,, ad,, ( ), - the hypothesis: : 3- the statistic: < < : > > - if, is kow, the use σ σ + - if, is ukow, the use, where ~, + 4- Determiig the rejectio of H, that is: i)if : > >,reject if > ii) if : <,reject if < ( always egative) iii) if :, reject if > or < (ii) mall amples(, < 3) We will follow the same steps i the case of large samples except that we use t- distributio if populatios variaces are ukow ad equal, thus: 65
15 Dr. Moa Elwakeel [ 5 TAT] 3- the statistic: - if, is kow, the use σ + σ - if, is ukow(but equal), the use T p +, where P ( ) + ( + ) 4- Determiig the rejectio of H, that is: i)if : > >,reject if T> t,( + ) ii) if : <,reject if T< t ( + ), iii)if :, reject if T> t or,( + ) T< t,( + ) 5-Decisio rule: if the rejectio rule is hold the we reject H ad accept H if the rejectio rule is't hold the we accept H ad reject H Ex (8): To compare the icome level i two cities, a radomly two samples are selected. the first sample of sie 5 family from the first city has average icome 64 thousad dollar per year ad variace 6 thousad dollar ad a sample of sie 6 family from the secod city has a average icome 66 thousad dollar with variace 5 thousad dollar. is there exist a sigificat differece betwee the average icome for families i the two cities at.5. 66
16 Dr. Moa Elwakeel [ 5 TAT] olu. - Data: Hypothesis: 3-the statistic: : : reject if > or < ad Thus, reject sice <, i.e., there is a sigificat differece betwee the average of family icome i the two cities Testig for the two populatios Proportios If we have two idepedet samples of sie ad with proportios ad respectively. Thus, we will use the followig steps: -data eeded:, ad, - the hypothesis: : 67
17 Dr. Moa Elwakeel [ 5 TAT] < < : > > 3- the statistic: r r where rˆ( rˆ) + 4- Determiig the rejectio of H, that is: i)if : > >,reject if > ii) if : <,reject if < iii) if :, reject if > or < Ex (9): Two machie A ad B, a radom sample of sie 3 uits from machie A with defective proporo 8% ad aother sample of sie uits from machie B with defecve proporo 4%. The maager thik that the defective proportio from machie A is differ from the defective proportio from machie B, is he right?. use.5 olu. -data eeded: 3,.8 ad,.4,.5 - the hypothesis: : 3- the statistic: : (.64 )
18 Dr. Moa Elwakeel [ 5 TAT] where (.)(.) reject if < or. 96 Thus, we accept ad reject that says there is a differece betwee the defective proportios from machies A ad B Testig for the two populatios Variaces We have two idepedet radom samples of sie ad, with variaces ad respectively. We will assume that each populatio has Normal distributio. we will use the followig steps: -data eeded:, ad,, - the hypothesis: : 3- the statistic: < : > 4- Determiig the rejectio of H, that is: i)if : >,reject if F> F,, ii) if : <,reject if F < F,, F,, iii) if :, reject if F> F or,, F < F,, F,, 69
19 Dr. Moa Elwakeel [ 5 TAT] Ex (): For the data give i the followig table, test whether the variace of bacteria couts of pasteuried milk is less tha the variace of for couts for the raw milk. Use. ample sie Mea tadard deviatio Raw Milk Pasteuried Milk olu. -data eeded: 3, ad 3, 358.7,. - the hypothesis: : 3- the statistic: : > ,,.,, We reject ad accept, sice, >.,,.67.Thus, we coclude that the variace of bacteria couts of pasteuried milk is less tha the variace of for couts for the raw milk. Use. 7
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