What is a ypothesis? ypothesis is a statemet about a populatio parameter developed for the purpose of testig. What is ypothesis Testig? ypothesis testig is a proedure, based o sample evidee ad probability theory, used to determie whether the hypothesis is a reasoable statemet ad should ot be rejeted, or is ureasoable ad should be rejeted. Why we odut the ypothesis Testig? Ad ow? We odut the hypothesis test to fid out is the differee betwee sample statisti (whih is used to estimate the populatio parameter) ad the value of populatio parameter (hypothesized value) is differee due to : - Samplig error (estimatio error) whih is our whe we draw a radom sample from populatio. - Or is it sigifiat differee? (Meas atually there is differee whih is has meaig i the study). By usig the sample iformatio (mea, size, ad variae) I simple word ypothesis Test is Statisti Test (Z test, t test, F test ad hi square) Example: If we eed to kow the value of IQ sore of studets i the KSU we draw radom sample from this populatio (KSU) ad ompute the mea of IQ from sample data whih is equal =, but there is A researher hypotheses that the IQ of the whole populatio is equal to 5 (μ = 5),here we eed to kow if the differee betwee the sample mea ad the hypothesized mea 5 is it Sigifiat differee Or it is differee due to samplig error (by hae)? To aswer this questio we a odut test statisti whih is alled ypothesis Test, ad we a ostrut Cofidee iterval whih is already kow.
Step (): State the Null ( ) ad alterate ( ) hypothesis Null hypothesis ( ): Is the hypothesis beig tested, here we write the hypothesized value whih is alled referee value (the value whih we assume that equal to populatio parameter) ad i ull hypothesis oly those operators are allowed ( =,, )beause i this hypothesis always we assume that there is o differee at all (beause we otie that the equal operator ). Alterate hypothesis ( ): A statemet that is aepted if the sample data provide suffiiet evidee that the ull hypothesis is false. (e.g. < ad > ) The alterative hypothesis whih is alled the researher hypothesis it otais the opposite mathematial operatio that are foud i meas always, we say that there is sigifiat differee (meas the populatio parameter will be <,>, ) ad i this hypothesis we write what the researher hypothesis about populatio parameter. Whe we rejet the Null ypothesis that meas we aept the alterative hypothesis ad Vie Versa (this will be our deisio at the ed of test) Importat Thigs to Remember about ad : subzero or ot. ad are mutually exlusive ad olletively exhaustive is always presumed to be true has the burde of proof A radom sample () is used to rejet If we olude 'do ot rejet ', this does ot eessarily mea that the ull hypothesis is true, it oly suggests that there is ot suffiiet evidee to rejet ; rejetig the ull hypothesis the, suggests that the alterative hypothesis may be true. Equality is always part of (e.g. =,, ). < ad > always part of 3
Keywords Iequality Symbol Part of: Larger (or more) tha > Smaller (or less) < No more tha At least as ireased > Is there differee? as ot haged = as improved, is better tha. is more effetive See left text Types of Tests Aordig the alterative hypothesis the hypothesis tests a be oe or two tailed. Case: Oe- tailed test (Right-tailed) For Example: : : Or : : Case: Oe- tailed test (left-tailed) For Example : : Or : : Case3: Two-tailed test For Example: : : Or : : Step (): Selet a level of sigifiae: Level of sigifiae (α) : The probability of rejetig the ull hypothesis whe it is true. e.g. We ofte take about testig at a % sigifiae level (probability. of rejetig true ) Relatioship betwee the alterative hypothesis ad the level of sigifiae (α): Case& : If the alterative hypothesis is oe- tailed test (Right or lefttailed); (α) appear i the oe side of the urve. 4
Right tailed Left tailed Case3: If the alterative hypothesis is two- tailed test (α) appear i the left & right sides of the urve, we divide (α) by (α/) Step3: Selet the Test Statisti (omputed value) The test Statisti: A value, determied from sample iformatio, used to determie whether to rejet the ull hypothesis. There are may differet statistial tests. The hoie of whih test to use depeds o several fators: - The type of data. The distributio of the data. - The type of study desig. Example of test statisti used to test hypothesis: Z,t, F Step (4): Seleted the Critial value Critial value: The dividig poit betwee the regio where ull ypothesis is rejeted ad the regio where it is ot rejeted. The ritial value is read from the statistial tables ad determied by: - α. - Type of test. - Sometimes by the degrees of freedom. Step (5): Formulate the Deisio Rule ad Make a Deisio Deisio rule: We will ompare the omputed value with ritial value (table value) ad make our deisio: Rejet whe observed test-statisti equals or exeeds Critial value. (By other mea; whe the test statisti falls i the rejetio regio). Otherwise, Fail to Rejet (Retai) 5
Case: Rejet if T.S > (C.V) Case: Rejet if T.S <-(C.V) or > (C.V) Case 3: Rejet if > (C.V) whih meas T.S> (C.V) or T.S<-(C.V) Results of a ypothesis test : Either: Aept the ull It is A week olusio ; ot sigifiat result hypothesis ( ( as reasoable meas the differee oly ause of samplig possibility error Or: Rejet the ull hypothesis ad Aept the researher hypothesis ( ( It is A strog olusio ;a sigifiat result (meas there is differee ) 6
Step (): State the ull ( ) ad alterate ( ) hypothesis : : : or or : : : Step (): Selet a level of sigifiae. Step (3): Selet the Test Statisti (omputed value) Kow populatio Stadard Deviatio X Z ukow populatio Stadard Deviatio( 3) X S Z ukow populatio Stadard Deviatio( 3) X S t Step (4): Seleted the Critial value Types of test Z t The oe tailed test (Right) Z α ( ) The oe tailed test (left) - Z α ( ) The two tailed test ±Z α/ ( ) Step (5): Formulate the Deisio Rule ad Make a Deisio Case: Rejet if Z Z t t, v, Case: Rejet if Z Z, t tv, This meas 7
Case 3: Rejet if that is: Z or Z Z Z, t, t t v, t v, Kow populatio Stadard Deviatio Example () Jamestow Steel Compay maufatures ad assembles desks ad other offie equipmet at several plats i wester New York State. The weekly produtio of the Model A35 desk at the Fredoia Plat follows the ormal probability distributio with a mea of ad a stadard deviatio of 6. Reetly, beause of market expasio, ew produtio methods have bee itrodued ad ew employees hired. The vie presidet of maufaturig would like to ivestigate whether there has bee a hage i the weekly produtio of the Model A35 desk. Is the mea umber of desks produed at Fredoia Plats differet from at the. sigifiae level? IF the vie presidet takes sample of 5 weeks ad he fids the mea is 3.5, use the statistial hypothesis testig proedure to ivestigate whether the produtio rate has haged from per week. Solutio: Step ():State the ull hypothesis ad the alterate hypothesis. : = : This is Two-tailed test (Note: keyword i the problem has haged ) Step (): Selet the level of sigifiae. α =. as stated i the problem 8
Step (3): Selet the test statisti Use Z-distributio sieσis kow, X 3.5, 6, 5 X Z 3.5 6 5 3.5.67.55 Step (4): Formulate the deisio rule (Critial value) Z Z. Z.5.5 =.495.5 Z.5.58 Rejet if Z. 58or Z. 58 Step (5): Make a deisio ad iterpret the result. Rejet if Z Z or Z Z.55.58 Z Beause.55 does ot fall i the rejetio regio, is ot rejeted at sigifiae level.. We olude that the populatio mea is ot differet from. So we would report to the vie presidet of maufaturig that the sample evidee does ot show that the produtio rate at the Fredoia Plat has haged from per week. The differee of 3.5 uits betwee the historial weekly produtio rate ad that last year a reasoably be attributed to samplig error. Example () Suppose you are buyer of large supplies of light bulbs. You wat to test at the 5% sigifiat level, the maufaturer s laim that his bulbs last more tha 8 hours, you test 36 bulbs ad fid that the sample mea is 86 hours, with stadard deviatios 7 hours.should you aept the laim? Solutio: Step : State the ull hypothesis ad the alterate hypothesis. : 8 : 8 9
This is oe-tailed test (right), (Note: keyword i the problem more tha ) Step : Selet the level of sigifiae. α =.5 as stated i the problem Step 3: Selet the test statisti. Use Z-distributio sie σ is kow 8, X 86, S 7, 36 X Z S 86 8 7 36 6.6667.37 Step 4: Formulate the deisio rule. Rejet if Z Z Z Z.5 =.5.5 =.45 Z.5.65 Step 5: Make a deisio ad iterpret the result. Rejet if Z. 65 Z.37.65 Beause.37 does ot fall i the rejetio regio, is ot rejeted. We olude that the populatio mea is less tha or equal 8. The deisio that we do t rejet the ull hypothesis at sigifiat level.5, but we rejet the alterative hypothesis the maufaturer s laim that μ>8. The differee of 6 hours betwee sample ad produtio a reasoably be attributed to samplig error.
Example (3) The MFarlad Isurae Compay Claims Departmet reports the mea ost to proess a laim is $6. A idustry ompariso showed this amout to be larger tha most other isurae ompaies, so the ompay istituted ost-uttig measures. To evaluate the effet of the ost-uttig measures, the Supervisor of the Claims Departmet seleted a radom sample of 6 laims proessed last moth. The sample iformatio is reported below. 6, X 56.4, S.4 At the. sigifiae level is it reasoable a laim is ow less tha $6? Solutio: Step : State the ull hypothesis ad the alterate hypothesis. : $6 : $6 This is oe-tailed test (Left) (Note: keyword i the problem ow less tha ) Step : Selet the level of sigifiae. α =. as stated i the problem Step 3: Selet the test statisti. Use t-distributio sie σ is ukow 6, X 56.4, S.4, 6 X 56.4 6 3.58 t.8 S.4.969 6 Step 4: Formulate the deisio rule. Rejet if t t, t.485 t,.,5 Step 5: Make a deisio ad iterpret the result. Rejet if t. 485 t.8.485
Beause -.8 does ot fall i the rejetio regio, is ot rejeted at the. sigifiae level. We have ot demostrated that the ost-uttig measures redued the mea ost per laim to less tha $6. The differee of $3.58 ($56.4 - $6) betwee the sample mea ad the populatio mea ould be due to samplig error.
The p-value or observed of sigifiae level of a statistial test is the smallest value of for whih a be rejeted. It is the atual risk of ommittig a type I error, if is rejeted based o the observed value of the test statisti. The p-value measures the stregth of the evidee agaist. The probability of observig samples data by hae uder the Null hypothesis (i.e. ull hypothesis is true). I testig a hypothesis, we a also ompare the p-value to with the sigifiae level (). - If the p-value < sigifiae level(), is rejeted ( meas sigifiat result) - If the p-value sigifiae level (), is ot rejeted. (meas ot sigifiat result). Compute the P- value (Oly we used the Z distributio) Case : If the oe tailed test (Right) Z. p value P z 5 z Case : If the oe tailed test (left) Z. p value P z 5 z Case3: If the two tailed test Z z PZ z.5 z. z = - p value P 5 Or. p value P Z z 5 z Example (4) : Refer to example () : :, X 3.5, 6, 5,. z 3
p value P P Z z.5 z.5.55.5.4394.66. Z.55 P value.. Do ot rejet 4
Step () : State the Null( ) ad alterate( ) hypothesis : : : or or : : : Step (): Selet a level of sigifiae: Step (3): Selet the Test Statisti (omputed value) p Z Step (4): Seleted the Critial value The oe tailed test (Right) Z α The oe tailed test (left) - Z α The two tailed test ±Z α/ Step (5): Formulate the Deisio Rule ad Make a Deisio Case: Rejet if Z Z Case: Rejet if or Z Z Case3:Rejet if 5
Z Z orz Z Example (5) Suppose prior eletios i a ertai state idiated it is eessary for a adidate for goveror to reeive at least 8 peret of the vote i the orther setio of the state to be eleted. The iumbet goveror is iterested i assessig his haes of returig to offie. A sample survey of, registered voters i the orther setio of the state revealed that 54 plaed to vote for the iumbet goveror. Usig the hypothesis-testig proedure, assess the goveror s haes of reeletio (the level of sigifiae is.5). Solutio: Step : State the ull hypothesis ad the alterate hypothesis. :.8 :.8 This is oe-tailed test (Left) (Note: keyword i the problem at least ) Step : Selet the level of sigifiae. α =.5 as stated i the problem Step 3: Selet the test statisti. Use Z-distributio sie the assumptios are met ad ad (-) 5 6
54.8, p.77, p.78.8.3 Z 3.37.8..89 Step 4: Formulate the deisio rule. Rejet if Z Z Z.5.65 Step 5: Make a deisio ad iterpret the result. Rejet if Z. 65 Z 3.37.65 Rejet The omputed value of (-3.37) is i the rejetio regio, so the ull hypothesis is rejeted at the.5 level. The differee of.5 peretage poits betwee the sample peret (77 peret) ad the hypothesized populatio peret (8) is statistially sigifiat. The evidee at this poit does ot support the laim that the iumbet goveror will retur to the goveror s masio for aother four years. 7
8 Step () : State the Null ( ) ad alterate ( ) hypothesis Case : : : Case : : : Case 3: : :
Step (): Selet a level of sigifiae Step (3): Selet the Test Statisti (omputed value) Step (4): Seleted the Critial value The oe tailed test (Right) ( ) The oe tailed test (left) ( ) The two tailed test ( ) & ( ) Step (5): Formulate the Deisio Rule ad Make a Deisio Case: The oe tailed test (Right) Rejet if ( ) Case: The oe tailed test (left) Rejet if ( ) Case3: The two tailed test; rejet if Or ( ) ( ) Example (6) A sample of size produed a variae of 4.Is this suffiiet to rejet the ull hypothesis that is equal to 6 Whe tested usig a.5 level of sigifiae? Step : State the ull hypothesis ad the alterate hypothesis. : : 6 6 This is two-tailed test (Note: keyword i the problem "that is equal to 6") 9
Step : Selet the level of sigifiae. α =.5 as stated i the problem.5.5.5.975 Step 3: Selet the test statisti. S 4 6 6 6 Step 4: Formulate the deisio rule (Critial value) 9.,. 73.5 ;.9.975; 9
Step 5: Make a deisio ad iterpret the result. Rejet if.5 ;.9 9.,, v Or. 73, v.975; 9 The deisio is to rejet the ull hypothesis, beause the omputed Value () is larger tha the ritial value (9.). We olude that there is a differee
Null ypothesis is true is false Does Not Rejet Rejets Do ot rejetig The ull hypothesis,,whe It is true( ) {Corret deisio} Do ot rejetig The ull hypothesis,,whe It is false( ) {Type II error} rejetig The ull hypothesis,,whe It is true( ) {Type I error} rejetig The ull hypothesis,,whe It is false (Power) ( ) {Corret deisio} Note: The quatity ( ) is alled the power of the test beause it measures the probability of takig the atio that we wish to have our-that is,rejetig the whe it is false ad is true. ( ) = P(rejetig the whe it is false) Ideally, you would like ( ) to be small ad the power ( ) to be large. Example (7) A maufaturer purhases steel bars to make otter pis. Past experiee idiates that the mea tesile stregth of all iomig shipmets is greater tha, psi ad that the stadard deviatio, σ, is 4 psi. I order to make a deisio about iomig shipmets of steel bars, the maufaturer set up this rule for the quality-otrol ispetor to follow: Take a sample of steel bars, at the.5 sigifiae level. Suppose the ukow populatio mea of a iomig lot, desigated is really psi. Fid a.the type I error (Rejetig the ull hypothesis,, whe It is true ( ). b. The orret deisio (Do ot rejetig the ull hypothesis,, whe It is true ( )..The type II error (Do ot rejetig the ull hypothesis,, whe It is false ( ). d.the orret deisio (Rejetig The ull hypothesis,, whe It is false ( ).
Solutio: : Z.645 : 4 Z.645 65.8 66 a.the type I error (where rejetig the ull hypothesis,, whe it is true ( ))..5 b. The orret deisio (Do ot rejetig The ull hypothesis,, whe It is true ( )..5.95.The type II error (where aeptig the ull hypothesis,, whe it is false ( )). X P Z 66 54 P Z PZ.5 P.35 Z 4 4.5.35.5.45.. 885 885 d.the orret deisio (where is false ad rejet it ( ))..885.95
Example (8) A maufaturer purhases steel bars to make otter pis. Past experiee idiates that the mea tesile stregth of all iomig shipmets is less tha, psi ad that the stadard deviatio, σ, is 4 psi. I order to make a deisio about iomig shipmets of steel bars, the maufaturer set up this rule for the quality-otrol ispetor to follow: Take a sample of steel bars, at the.5 sigifiae level. Suppose the ukow populatio mea of a iomig lot, desigated is really 99psi.fid: a.the type I error (Rejetig the ull hypothesis,, whe It is true ( )). b. The orret deisio (Do ot rejetig the ull hypothesis,, whe It is true ( ))..The type II error (Do ot rejetig the ull hypothesis,, whe It is false ( )). d.the orret deisio (Rejetig the ull hypothesis,, whe It is false ( )). Solutio: : :. Z.645 4 Z.645 65.8 9934 a.the type I error (where rejetig the ull hypothesis,, whe it is true ( ))..5 3
b. The orret deisio (Do ot rejetig the ull hypothesis, whe It is true ( ))..5.95,.The type II error (where aeptig the ull hypothesis,, whe it is false ( )). X P Z 9934 99 34 P Z P Z.5 P Z.85 4 4.5 (.85).5.33.977.977 d.the orret deisio (where is false ad rejet it ( ))..977.83 4
Example (9) If 55, : 4 : 4 Complete the followig statemets: a. The probability of ot rejetig the ull hypothesis whe it is true is the shaded area i diagram... b. The probability of Type I error is the shaded area i diagram.. The probability of Type II error is the shaded area i diagram. d.the probability of rejetig the ull hypothesis whe it is false is the shaded area i. diagram... Solutio: a. Diagram D b. Diagram C. Diagram A d. Diagram B 5
Example () If 3, : 4 : 4 Complete the followig statemets: a. The probability of ot rejetig the ull hypothesis whe it is true is the shaded area i diagram... b. The probability of Type I error is the shaded area i diagram.. The probability of Type II error is the shaded area i diagram. d.the probability of rejetig the ull hypothesis whe it is false is the shaded area i diagram... Solutio: a. Diagram B b. Diagram A. Diagram C d. Diagram D 6
Example () Null ypothesis Does Not Rejet Do ot rejetig The ull hypothesis,,whe It is true( ) Example Rejets rejetig The ull hypothesis,,whe It is true( ) Example is true If the deisio is Do ot rejet Let 8 The deisio is orret Do ot rejetig The ull hypothesis,,whe It is true( ) Do ot rejetig The ull hypothesis,,whe It is false( ) Example If the deisio is rejet Let 8 The deisio is iorret rejetig The ull hypothesis,,whe It is true( ) rejetig The ull hypothesis,,whe It is false (Power)( ) Example is false If the deisio is Do ot rejet Let 5 The deisio is iorret Do ot rejetig The ull hypothesis,,whe It is false( ) If the deisio is rejet Let 5 The deisio is orret rejetig The ull hypothesis,,whe It is false (Power)( ) 7
8 Note about P : : :, Z, = P Z P : :, Z, = P Z P
Amia Ali Saleh Professor of Statistis Test of hypothesis about sigle populatio mea Example (h) I the followig example, the populatio distributio is ormal with kow stadard deviatio ad the hypothesis test is right-tailed. A hypothesis test is to be performed to determie whether the mea waitig time durig peak hours for ustomers i a supermarket has ireased from the previous mea waitig time of 8. miutes. Previous experiee idiates that the waitig time follows a ormal distributio with stadard deviatio equal 3.8 miutes. To test the hypothesis, a radom sample of 5 ustomers will be seleted yields mea x 9. 75.. Aswer the questios to. Questio The ull ad alterative hypotheses are... (A) : 8. & : 8. (B) : 8. & : 8. (C) : 8. & : 8. (D) : 8. & : 8. Questio : This hypothesis test is lassifies as... (A) Right-tailed (B) Two-tailed (C) Multi-tailed (D) left-tailed Questio 3 The appropriate test statisti ad its distributio uder the ull hypothesis is... (A) (C) X Z ~ N(,) (B) / X T ~ t 4 (D) S / X Z ~ N (, ) / X Z ~ N (, ) S / 9
Amia Ali Saleh Professor of Statistis Questio 4 The ritial regio is best desribed by figure... (A) (B) (C) (D) Questio 5 With sigifiae level equal.5, the deisio riterio for the hypothesis test i terms of the omputed value of the test statisti is... (A) Rejet if z < -.645 (B) Rejet if z >.96 (C) Rejet if Z >.645 or Z.645 (D) Rejet if z >.645 Questio 6 With sigifiae level equal.5, the deisio riterio for the hypothesis test i terms of the omputed value of the estimator of the pupatio parameter( x )is (A) ) 9.45 (B) 3.98 (C) 8.9 (D) 6.95 Solutio: X 3.8 Z 8..645 9.45 5
Questio 7: The omputed value of our test statisti is... (A) -.4 (B) 3.98 (C).4 (D).54 9.75 8. Solutio: z. 4 3.8/ 5 Questio 8 The deisio would be to... (A) Caot be determied (B) Do ot rejet the ull hypothesis. (C) Rejet the ull hypothesis. (D) Rejet the alterative hypothesis. Questio 9 Suppose that i fat the waitig time is ireased to 9 miutes ( 9. 9 ), the the deisio has bee made is... (A) Committig Type I error (B) Committig Type II error (C) Corret deisio( ) (D) Corret deisio( ) Questio The power of our test at = 9.9 miutes is... (A).4 (B).74 (C).7776 (D).776 Solutio: Power equals the probability of rejetig the ull hypothesis while the alterative hypothesis is false. We ompute it as follows: At 9. 9 P X 9.45 9.9 Z Z P Z P Z PZ.59 =.5-.4=.776 Power=- =-.776=74 3.8/ 5
Questio Amia Ali Saleh Professor of Statistis A 95% ofidee iterval is8.3.. The ull hypothesis is (A) Rejet the ull hypothesis. (B) Do ot Rejet the ull hypothesis. (C) Ca ot be determied (D) Rejet the alterative hypothesis. Ed of example : 8. : 8. What is the deisio?
Amia Ali Saleh Professor of Statistis Example (h) I the followig example, the populatio distributio is approximately ormal with kow stadard deviatio ad the hypothesis test is left-tailed The studets at the uiversity laim that the average studet must travel for at least 5 miutes i order to reah the uiversity. The uiversity admissios offie obtaied a radom sample of 36 oe-way travel times from studets. The sample had a mea of 9.4 miutes. Assume that the populatio stadard deviatio is 9.6 miutes. Does the admissios offie have suffiiet evidee to rejet the studet's laim? Aswer the followig questios Questio The ull ad alterative hypotheses are... (A) : 5 & : 5 (B) : 5 & : 5 (C) : 5 & : 5 (D) : 5 & : 5 Questio : This hypothesis test is lassifies as... (A) Right-tailed (B) Two-tailed (C) Multi-tailed (D) left-tailed Questio 3: The appropriate test statisti ad its distributio uder the ull hypothesis is... P (A) Z ~ N(,) (B) ) (C) ( X Z ~ N(,) (D) / X T ~ N(,) / X Z ~ N (, ) / 3
Amia Ali Saleh Professor of Statistis Questio 4 With sigifiae level equal., the deisio riterio for the hypothesis test i terms of the omputed value of the test statisti ( Z ) is... (A) Rejet if Z < -.5 (B) Rejet if Z >.96 (C) Rejet if Z >.5 or Z. 5 (D) Rejet if Z <-.645 Questio 5 With level of sigifiae %, the ritial regio is best desribed by figure... (A) (B) (C) (D) Questio 6: With sigifiae level equal., the deisio riterio for the hypothesis test i terms of the omputed value of X, x is... (A) 4.67 (B) 3.98 (C).37 (D).7 X 9.6 Z 5.5.7 36 4
Amia Ali Saleh Professor of Statistis Questio 7: The omputed value of our test statisti is... (A) -3.5 (B).4 (C) 3.5 (D) -.36 Questio 8: The deisio would be to... (A) Caot be determied (B) Do ot Rejet the ull hypothesis. (C) Rejet the ull hypothesis. (D) Rejet the alterative hypothesis. Questio 9: Suppose that i fat the travelig time is miutes ( ), the the deisio has bee made is... (A) Committig Type I error (B) Committig Type II error (C) Corret deisio( ) (D) Corret deisio( Questio : The p-value of this test statisti is... (A). (B).5 (C).5 (D).5 ) Solutio: P-value= PZ z = P ( Z 3.5 5) PZ 3.5.5 P( Z.5.4998. p-value=.< =. Rejet 3.5) 5
Questio : The power of our test at = miutes is... (A).4 (B).8599 (C).7776 (D).74 Solutio: X.7.7 P Z Z PZ P Z 9.6 P Z.6 6 PZ.8.5.8.5.3599. 4 power.4.8599 Ed of example 6
Amia Ali Saleh Professor of Statistis Test of ypothesis about Sigle Populatio Proportio (large Samples) Example 3(h) I the followig example, the sample is large suh that 5& ( ) 5 ad the hypothesis test is Right-tailed. It assumed from last experiee that 75% of sports viewers are male. A famous sport ewspaper reports that this proportio is greater tha.75. A radom sample of 4 seaso tiket holders reveals that 35 are male. We wish to test the above hypothesis. Aswer the followig questios Questio : The ull ad alterative hypotheses are... (A) :. 75 & :. 75 (B) :. 75 & :. 75 (C) :. 75 & :. 75 (D) :. 75 & :. 75 Questio: This hypothesis test is lassifies as... (A) Two-tailed (C) Opposite-tailed (B) Right-tailed (D) left-tailed Questio 3: The appropriate test statisti ad its distributio uder the ull hypothesis is... P (A) Z ~ N(,) (B) ) / (C) ( X T ~ t 8 S / X P Z ~ N(,) (D) ~ / ( ) / Questio 4: With sigifiae level equal.5, the deisio riterio for the hypothesis test i terms of the omputed value of the test statisti ( Z ) is... (A) Rejet if Z <-.96 (B) Rejet if Z >.96 (C) Rejet if Z >.96 or Z.96 (D) Rejet if Z >.645 7
Amia Ali Saleh Professor of Statistis Questio 5: With level of sigifiae 5%, the ritial regio is best desribed by figure... Questio 6: With sigifiae level equal.5, the deisio riterio for the hypothesis test i terms of the omputed value of P, p is... (A).78 (B).978 (C).786 (D).6734 Solutio: We rejet the ull hypothesis if either z >.645 z >.645 P Z.75 Questio 7: The omputed value of our test statisti is... (A). (B) 5.99 (C).3 (D) -. Solutio: z 35 / 4.75.875.75.88.75 5.99 (.75)(.5) / 4.7.7.75.5 P.645 P. 786 4 8
Questio 8: The deisio would be to... (A) Do ot Rejet the ull hypothesis (B) Caot be determied. (C) Rejet the ull hypothesis. (D Rejet the alterative hypothesis. Questio 9: Suppose that i fat the true proportio is.85, the the deisio has bee made is... (A) Rejetig the true hypothesis( ) type error (C) Do ot rejetig the true hypothesis( )Corret deisio (B)Do ot Rejetig the false hypothesis ( ) type error. (D) Rejetig the false hypothesis( )Corret deisio Questio: Suppose that i fat the true proportio is.74, the the deisio has bee made is... (A) Rejetig the true hypothesis( ) type error (C) Do ot rejetig the true hypothesis( )Corret deisio (B)Do ot Rejetig the false hypothesis ( ) type error. (D) Rejetig the false hypothesis( )Corret deisio Questio: The power of our test at =.85 is... (A).87 (B).37 (C).35 (D).9998 Solutio: P.786 P P.786.85 P PZ 3.58.85.5 4.5 3.58.5.4998. Power=.. 9998 9
Amia Ali Saleh Professor of Statistis Questio: A 95% ofidee iterval is.5. 87. The ull hypothesis is :.96 What is the deisio? (A) Rejet the ull hypothesis. (B) Do ot Rejet the ull hypothesis. (C) Caot be determied (D) Rejet the alterative hypothesis. Questio3: A 95% ofidee iterval is.5. 87. The ull hypothesis is :. 77 What is the deisio? (A) Rejet the ull hypothesis. (B) Do ot Rejet the ull hypothesis. (C) Caot be determied (D) Do ot Rejet Rejet the alterative hypothesis. Ed of example 3
Amia Ali Saleh Professor of Statistis Test of hypothesis about Tow populatio meas (Idepedet Samples) Kow variaes Example 4(h) Exerise 3 page 393 A offee maufaturer is iterested i whether the mea daily osumptio of regular-offee drikers is less tha that of deaffeiated-offee drikers. Assume the populatio stadard deviatio for those drikig regular offee is. ups per day ad.36 ups per day for those drikig deaffeiated offee. A radom sample of 5 regular-offee drikers showed a mea of 4.35 ups per day. Aother radom sample (idepedet of the first) of 4 deaffeiated-offee drikers showed a mea of 5.84 ups per day. Aswer questios to 5. Remark: It is preferable to summarize the iformatio as follows: Regular-offee drikers () deaffeiated-offee drikers () i.. 36 5 4 i x i x 4. 35 x 5. 84 Questio The parameter(s) uder testig is (are)... (A) Oe populatio mea (B) Two populatio meas (C) Oe populatio variae Questio The ull hypothesis is... (A) (C) Questio 3: (B) (D) This hypothesis test is lassifies as... (A) Righttailed (B) Twotailed (D) Two populatio proportios. (C) Multitailed (D) lefttailed
Questio 4: The appropriate test statisti ad its distributio uder the ull hypothesis is... (A) Z ~ N(,) P P P ( P ) P ( P ) (B) Z X X ~ N (, ) (C) T ~ N(,) (D) Z ~ N(,) S P X X X X Questio 5: With sigifiae level equal.5, the deisio riterio for the hypothesis test i terms of the omputed value of the test statisti is... (A) Rejet if Z < -.645 (B) Rejet if Z >.96 (C) Rejet if Z >.645 or Z.645 (D) Rejet if Z >.645 Questio 6: A poit estimatio of the differee is... (A) -.6 (B) - (C) -.49 (D).49 Questio 7: The stadard deviatio of the statisti X X is give by the formula... (A) (C) P ( P ) P ( P ) (B) S P (D) Questio 8: The value of the stadard deviatio of the statisti X X equals... (A).74 (B).68 (C).58 (D).4 Questio 9: The omputed value of our test statisti is... (A) - 3.76 (B) 3.76 (C) - 5.438 (D) 5.936
Solutio: x x.49 z 5.438.74 Questio : The deisio would be to... (A) Ca ot be determied (B) Do ot rejet the ull hypothesis. (C) Rejet the ull hypothesis. (D) Rejet the alterative hypothesis. Questio : Suppose that i fat the mea daily osumptio of regular-offee drikers is less tha that of deaffeiated-offee drikers by.5 ups, the the deisio has bee made is... (A) Committig Type I error (B) Committig Type II error (C) Corret deisio( ) (D) Corret deisio( ) Questio : Suppose that i fat the mea daily osumptio of regular-offee drikers is greater tha that of deaffeiated-offee drikers by.5 ups, the the deisio has bee made is... (A) Committig Type I error (B) Committig Type II error (C) ) Corret deisio( ) (D) Corret deisio( ) Ed of example 4 3