Unit 20 Hypotheses Testing

Transcription

1 Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect data ad calculate the value of a appropriate test statistic i a hypothesis test about a populatio proportio To uderstad how to defie a rejectio regio ad obtai the p-value i a hypothesis test about a populatio proportio To uderstad how to summarize the results of a hypothesis test about a populatio proportio We have itroduced several terms ad cocepts importat i hypothesis testig. We ow wat to defie a hypothesis test formally ad begi to apply hypothesis testig i a variety of differet situatios. We ca formally defie a hypothesis test as the followig four steps: Step 1: State the ull ad alterative hypotheses, ad choose a sigificace level. Step 2: Collect data, ad calculate the value of a appropriate test statistic. Step 3: Defie the rejectio regio, decide whether or ot to reject the ull hypothesis, ad obtai the p-value (probability value) of the test. Step 4: State the results, ad perform ay further aalysis which may be required. We shall begi to illustrate the applicatio of these four steps by cosiderig hypothesis tests about a populatio proportio. Whe we first defied the ull hypothesis ad the alterative hypothesis, we cosidered a hypothesis test to decide whether or ot to believe the claim of a maufacturer that a lighter will igite o the first try 75% of the time. Recall that a hypothesis test is very much aalogous to a court trial, where we assume that the ull hypothesis, which states The defedat is iocet, is true, util ad uless there is sufficiet evidece to believe the alterative hypothesis, which states The defedat is guilty. I a hypothesis test cocerig a populatio parameter, a ull hypothesis is a statemet about equality, ad a alterative hypothesis is a statemet about iequality. I a hypothesis test about a populatio proportio, the statemet of the ull hypothesis cotais a hypothesized value for the populatio proportio, ad the alterative hypothesis is a statemet that the hypothesized value is ot correct. Oe of the reasos the ull hypothesis is so amed is because the meaig of the word ull suggests o differece, o chage, o effect, etc. By imagiig that the maufacturer s claim is o trial, our ull hypothesis is "The lighter will igite o the first try 75% of the time," ad the alterative hypothesis is "The lighter will ot igite o the first try 75% of the time." We shall fid it coveiet to itroduce some widely-used abbreviatios i hypothesis testig. First, we use H to represet a ull hypothesis ad H 1 to represet a alterative hypothesis. Also, whe writig a hypothesis, we may choose to use symbols i place of some words. The populatio proportio λ is the focus of the hypothesis test cocerig the lighter, ad the hypothesized value for λ is.75. A much shorter way of statig the ull hypothesis "The lighter will igite o the first try 75% of the time" is to simply say "λ =.75," ad a much shorter way of statig the alterative hypothesis "The lighter will ot igite o the first try 75% of the time" is to simply say "λ.75." As part of the first step of our hypothesis test, we eed to choose a sigificace level; we shall choose α =.1, which is what we origially chose whe we first cosidered the illustratio cocerig the lighter. This eables us to complete the first step of our hypotheses test by writig H : λ =.75 vs. H 1 : λ.75 (α =.1). Havig ow completed the first step i our hypothesis test, we move o to the secod step. The secod step i a hypothesis test is to collect data ad to calculate the value of a appropriate test statistic o which our coclusio will be based. I a court trial, this secod step correspods to collectig ad presetig evidece. I 138

2 the hypothesis test cocerig the lighter, this secod step correspods to obtaiig a simple radom sample of observatios ad basig our coclusios o a statistic that we obtai from our sample; the sample proportio p of times the lighter will igite o the first try certaily seems a reasoable statistic o which to base our coclusios about the lighter. I hypothesis testig, a test statistic is a statistic which is used to decide whether to believe the H or to believe the H 1. Remember that i hypothesis testig, we assume the ull hypothesis to be true uless the evidece is sufficietly strog to suggest that we should believe the H 1. Decidig whether or ot the evidece i a court trial is sufficietly strog to retur a verdict of guilty is ultimately a subjective decisio, sice there is o mathematical formula which will weigh all the evidece preseted i a typical court trial. I hypothesis testig, however, decidig whether or ot the evidece is sufficietly strog to believe the H 1 is based o doig a appropriate probability calculatio. Figures 19-1 ad 19-2 are illustratios of the probability calculatio we did with each of two differet data sets i order to decide whether or ot the evidece i the data was strog eough to covice us to believe the H 1. The probability calculatios illustrated i Figures 19-1 ad 19-2 were doe uder the assumptio that H : λ =.75 is true. Uder this assumptio, with a sufficietly large sample size, we are able to treat the samplig distributio of p as a ormal distributio. The z-score of p, which is z = p µ σ p p = p λ λ = p.75 ( 1 λ).75( 1.75), tells us how may stadard deviatios of differece there is betwee p ad the hypothesized.75. This is exactly the z-score we eeded to calculate to do the probability calculatios illustrated i Figures 19-1 ad I geeral, wheever we have a ull hypothesis H : λ = λ, where we use λ to represet the hypothesized value for λ, the test statistic is z = λ p λ ( 1 λ ), which we shall refer to as the z statistic about a populatio proportio. To complete the secod step i our hypothesis test about the lighter, we must collect data, ad calculate the value of our test statistic. To collect data, we observe a simple radom sample of attempts at igitig the lighter. Let us imagie that we observe a simple radom sample of = 3 attempts, ad we fid that the lighter igites o the first try i 214 of these attempts (which is exactly the same data we used i Figure 19-1). The sample proportio of successes is p = 214/3 =.7133, from which we calculate the value of our z statistic as follows: z = λ p λ ( λ ).75( 1.75) 1 = 3 = The third step i a hypothesis test is where we decide whether to believe H or H 1. I our hypothesis test cocerig the lighter, we are basig this decisio o whether the umber of stadard deviatios of differece betwee p ad the hypothesized.75, as measured by the z statistic, is beyod what we would expect from samplig error. Our chose sigificace level α determies what values of the z statistic are beyod what we should expect from samplig error. I our hypothesis test about the lighter, we chose α =.1, which implies that values of the z statistic which have less tha a.1 probability of occurrig (i either directio) provide us with sufficiet evidece to believe H

3 Figure 2-1 displays a stadard ormal curve with two shaded areas: oe cotaiig the lower 5% (.5) of the area uder the curve ad oe cotaiig the upper 5% (.5) of the area uder the curve. Together, these areas defie a set of values for the z statistic which has a.1 (.5+.5) probability of occurrig if H : λ =.75 is true. Notice that i Figure 2-1 the z-score above which.5 of the area lies has bee desigated by z.5 ; also, sice the stadard ormal curve is symmetric, we kow that the z-score below which.5 of the area lies must be z.5 (the egative of z.5 ). If our z statistic is either larger tha z.5 or smaller tha z.5, the we have sufficiet evidece to believe that H 1 : λ.75 is correct I hypothesis testig, a rejectio regio (sometimes also called a critical regio) is a set of test statistic values which lead to rejectig the ull hypothesis i favor of the alterative hypothesis. I order for us to defie our rejectio regio i the hypothesis test about the lighter, we must fid what the value of z.5 is. From Table A.2, you will fid that the area uder the stadard ormal curve above the z-score is.55, ad that the area uder the stadard ormal curve above the z-score is.495. From this, we deduce that the z-score above which lies.5 of the area must be betwee ad At the bottom of Table B.2, you will fid that z.5 = It is coveiet to use the otatio z A to represet the z-score above which lies whatever area A is the subscript. For istace, you will fid at the bottom of Table A.2 that z.25 = 1.96 ad z.5 = 2.576, which tell us that.25 of the area lies above the z-score ad.5 of the area lies above the z-score I geeral, the rejectio for the hypothesis test H : λ = λ vs. H 1 : λ λ with sigificace level α will be defied by z z α/2 or z z α/2 (i.e., z z α/2 ). Figure 2-1 graphically displays the rejectio regio i our hypothesis test about the lighter, ad we ca defie this rejectio regio algebraically as z or z (i.e., z 1.645). If the value of the z statistic is i the rejectio regio, the we would cosider our data ulikely eough to make us doubt that H : λ =.75 is true; however, if the value of the z statistic is ot i the rejectio regio, the we have o reaso to doubt that H : λ =.75 is true. Sice the H is assumed to be true uless sufficiet evidece is foud agaist it, the custom i statistical termiology has bee to state the decisio i terms of the H. If we decide that there is sufficiet evidece agaist the H, we say that we reject the H, ad, of course, sayig that we reject H is equivalet to sayig that we accept H 1. O the other had, if we decide that there is ot sufficiet evidece agaist the H, the oe might thik that we could say that we accept the H ; however, istead of sayig that we accept the H, it has become customary i statistical termiology to say that we do ot reject H. The reaso for sayig "do ot reject the H " istead of sayig "accept the H," eve though these statemets ca be cosidered equivalet, is to emphasize that our decisio to believe the H is ot based o evidece suggestig that H is true. Whe we decide to believe H, it is because there is ot sufficiet evidece agaist H ad ot because there is evidece to support H. Remember that we assume H is true at the outset of a hypothesis test, ad the purpose of the hypothesis test is to look for evidece idicatig that H 1 is true. Oce agai, we see the parallels betwee a hypothesis test ad a court trial. The two possible verdicts i a court trial are stated as "guilty" ad "ot guilty." The reaso for sayig "ot guilty" istead of sayig 14

4 "iocet," eve though these statemets ca be cosidered grammatically equivalet, is to emphasize that the verdict of "ot guilty" is based o a lack of sufficiet evidece suggestig guilt ad ot based ecessarily o evidece idicatig iocece. As part of the third step i our hypothesis test cocerig the lighter, we must decide whether or ot to reject the ull hypothesis. Sice our test statistic, calculated i the secod step, was foud to be z = 1.468, which is ot i the rejectio regio, our decisio is ot to reject H : λ =.75; i other words, our data does ot provide sufficiet evidece agaist H : λ =.75. I order to complete the third step i a hypothesis test, we must obtai what is called a probability value for the test. The probability value, usually just called the p-value, of a hypothesis test is the probability of obtaiig a test statistic value more supportive of the alterative hypothesis tha the test statistic value actually observed, uder the assumptio H is true. I our hypothesis test about the lighter, the p-value is the probability of obtaiig a sample proportio farther away from (below or above) the hypothesized.75 tha that actually observed, uder the assumptio H : λ =.75. This is the probability of obtaiig a z statistic which represets a larger distace away from the hypothesized proportio tha does the observed z = 1.468, ad this probability is represeted by the shaded regio i Figure 2-2. I Figure 2-2, the observed test statistic value z = has bee located o the horizotal axis, ad, i additio, the value has bee located; the shaded area below z = together with the shaded area above correspod to test statistic values which represet a larger distace away from the hypothesized proportio tha does z = We have actually already calculated this p--value, sice this is exactly the probability represeted i Figure To obtai the shaded area above i Figure 2-2 from Table A.2, we must first roud to The, we look for the table etry correspodig to the row labeled 1.4 ad the colum labeled.7. This table etry is.78, ad from the symmetry of a ormal distributio we kow that this is the area above ad also the area below Cosequetly, the total shaded area i Figure 2-2 must be = The p-value of this hypothesis test the is.1416, which we ca desigate by writig p-value = This is exactly the same probability we obtaied with Figure Whe we foud this probability to be.1416 with Figure 19-1, we the kew that we did ot have sufficiet evidece to reject H, because.1416 was ot less tha the chose sigificace level α =.1. Whe the p-value is larger tha the chose sigificace level, the we have ot foud sufficiet evidece to reject H ; whe the p-value is smaller tha the chose sigificace level, the we have foud sufficiet evidece to reject H. You should ow realize, however, that the p-value ca give us more iformatio tha simply whether or ot we should reject H with the chose sigificace level. The p-value.1416 just obtaied tells us ot oly that H is ot rejected with a.1 sigificace level, but also tells us that H would ot be rejected with ay of the commoly chose sigificace levels,.1,.5, ad.1 (because.1416 is larger tha each of these). Suppose for the sake of argumet, though, that we had obtaied p-value =.454; the we would see that H would have bee rejected with a sigificace level of.5 or.1, but H would ot have bee rejected with a sigificace level of.1. Remember that the choice of sigificace level α is made by the perso(s) performig the hypothesis test. Those who read the results of a hypothesis test may wish to kow whether or ot the coclusio would have bee the same with a differet sigificace level. The p-value of a hypothesis test is ofte treated as a measure of how strogly the observed data supports the H 1. For istace, if we had obtaied p-value =.2, the we would see that ot oly do we have sufficiet evidece to reject H with α =.1, but also with α =.5 ad with α =.1. The smaller the p-value is, the stroger we might cosider the evidece agaist H to be. After havig defied our rejectio regio, havig decided ot to reject H, ad havig obtaied the p-value, we have ow completed the first three steps i our hypothesis test about the lighter. The fourth step 141

5 cosists of statig results ad performig ay further aalysis which may be required. A clear, cocise statemet of the results of a hypothesis test is certaily importat. The results of hypothesis testig will be of o value to ayoe uless others ca read ad uderstad the results. We suggest that every hypothesis test ca ad should be summarized i a few seteces which iclude the observed test statistic value, the tabled value which defies the rejectio regio, the coclusio, ad the p-value; also, eve though we may use symbols as coveiet abbreviatios i our statemets of H ad H 1, for the sake of clarity it is preferable to summarize the results of a hypothesis test with complete seteces ad words istead of symbols. For istace, we ca summarize the results of the hypothesis test cocerig the lighter as follows: Sice z = ad z.5 = 1.645, we do ot have sufficiet evidece to reject H. We coclude that the proportio of times the lighter will igite o the first try is ot differet from.75 (p-value =.1416). I the statemet of results, we cocluded that the populatio proportio "is ot differet from.75." This appears to be a rather egative way of sayig that the populatio proportio "is equal to.75." Eve though these two statemets are grammatically equivalet, we chose to state the egative versio i our results i order to emphasize that cocludig H is true is based o a lack of sufficiet evidece to support H 1 ad ot based ecessarily o evidece suggestig that H is true. It is correct to state the coclusio either way, however. The use of the test statistic z i a hypothesis test about λ is based o the assumptio that we may treat the samplig distributio of p as a ormal distributio. The Cetral Limit Theorem for sample proportios says that this assumptio is reasoable whe λ 5 ad (1 λ) 5. I the illustratio just completed, the sample size was = 3, ad we ca use the hypothesized value of λ (.75) to verify that this sample size was sufficietly large to treat the samplig distributio of p as a ormal distributio. Sice (3)(.75) = 225 ad (3)(1.25) = 75 are both cosiderably larger tha 5, the use of the test statistic z was appropriate. As aother illustratio of a hypothesis test, we shall have you perform the four steps of the hypothesis test described i Table 2-1. The first step is to state H ad H 1, ad choose a sigificace level α. Remember that H is what we assume to be true util ad uless there is sufficiet evidece agaist it, ad H 1 is the statemet we are lookig for evidece to support; also, H is geerally a statemet ivolvig equality, ad H 1 is geerally a statemet ivolvig iequality. Usig the appropriate symbols, write H ad H 1 uder Step 1 i Table 2-1 to complete the first step of the hypothesis test. (As the first step of the hypothesis test, you should have H : λ =.7, H 1 : λ.7, α =.5.) The secod step is to collect data ad calculate the value of the test statistic. We ca verify that the use of the z statistic is appropriate by observig that (25)(.7) = 17.5 ad (25)(.93) = are both larger tha 5. Fid the sample proportio p, ad calculate the value of the z statistic uder Step 2 i Table 2-1 to complete the secod step of the hypothesis test. (You should fid that p = 27/25 =.18 ad z = ) The third step is to defie the rejectio regio, decide whether or ot to reject H, ad obtai the p-value of the test. Uder Step 3 i Table 2-1, draw a graph to display the rejectio regio, write the rejectio regio algebraically, idicate whether or ot to reject H, ad write the p-value of the test. (Figure 2-3 displays the rejectio regio graphically where z.25 = 1.96, ad this rejectio ca be writte algebraically as z or z 1.96, or as z 1.96; you should fid that H is rejected ad that p-value =.185 which is obtaied by averagig the etries for 2.35 ad 2.36 i Table A.2 ad doublig this result.) Complete the fourth step of the hypothesis test by summarizig the results uder Step 4 i Table 2-1. Recall that i our earlier illustratio cocerig the lighter, we did ot reject the H, thus cocludig that the populatio proportio was equal to the hypothesized value. I the illustratio of Table 2-1 cocerig uderweight packages, we reject the H, thus cocludig that the populatio proportio was ot equal to the hypothesized value. Rejectig the H ad cocludig that λ is ot equal to the hypothesized value leaves us with the questio of whether λ is less tha or greater tha the hypothesized value. 142

6 To aswer this questio i the hypothesis test of Table 2-1 cocerig the uderweight packages, we fid that the observed test statistic z = is i the half of the rejectio regio correspodig to positive values of z; this is because the sample proportio p = 27/25 =.18 was greater tha the hypothesized value.7. It is for this reaso we say that the results of the test seem to suggest that the proportio of uderweight packages is greater tha.7. Cosequetly, we should iclude this iformatio as part of our statemet of results. The statemet of results i Table 2-1 should look as follows: Sice z = ad z.25 = 1.96, we have sufficiet evidece to reject H. We coclude that the proportio of uderweight packages is differet from.7 (p-value =.185). The data suggest that the proportio of uderweight packages is greater tha.7. I our itroductio to hypothesis testig, we have focused solely o tests cocerig a populatio proportio λ. Shortly, however, we shall cosider hypothesis tests cocerig a populatio mea µ. 143

7 Self-Test Problem 2-1. A cadidate i a upcomig electio for goveror would like to see if there is ay evidece that the percetage of registered voters i the state itedig to vote for her is differet from the 4% that a poll from two weeks earlier suggested. She chooses a.1 sigificace level to perform a hypothesis test. I a simple radom sample of 5 registered voters, 165 say they will vote for her. (a) Complete the four steps of the hypothesis test by completig the table titled Hypothesis Test for Self-Test Problem 2-1. (b) Verify that the sample size is sufficietly large for the z statistic to be appropriate. (c) Cosiderig the results of the hypothesis test, decide which of the Type I or Type II errors is possible, ad describe this error. (d) Decide whether H would have bee rejected or would ot have bee rejected with each of the followig sigificace levels: (i) α =.5, (ii) α =.1. (e) What would be a appropriate graphical display for the data used i this hypothesis test? Self-Test Problem 2-2. Suppose a.5 sigificace level is chose for a hypothesis test to see if there is ay evidece that the mea yearly icome per household i a particular coutry is differet from $2,. I a simple radom sample of 45 households, the mea yearly icome per household is foud to be x = $18,4. (a) Complete the first step of the hypothesis test by statig H ad H 1, ad by idetifyig the chose sigificace level. (b) What prevets us from calculatig a z test statistic based o x i the same way we calculate the z test statistic based o p? 144

8 Aswers to Self-Test Problems 2-1 (a) Step 1: H : λ =.4, H 1 : λ.4, α =.1 Step 2: p = 165/5 =.33 ad z = Step 3: The rejectio is z or z (which ca be writte as z 1.96). H is rejected; p-value <.2. Step 4: Sice z = ad z.5 = 2.576, we have sufficiet evidece to reject H. We coclude that the proportio of registered voters i the state itedig to vote for the cadidate is differet from.4 (p-value <.2). The data suggest that the proportio is less tha.4. (b) 5(.4) = 2 ad 5(1.4) = 3 are both larger tha 5 implyig is sufficietly large. (c) Sice H is rejected, a Type I error is possible, which is cocludig that λ.4 whe really λ =.4. (d) H would have bee rejected with α =.5 ad with α =.1. (e) a bar chart or pie chart 2-2 (a) H : µ = 2,, H 1 : µ 2,, α =.5 (b) b) We do ot kow σ x = σ / = σ / 45. Summary The hypothesis which is assumed to be true at the outset of a hypothesis test is called the ull hypothesis, abbreviated H, ad is geerally a statemet of equality. The hypothesis for which sufficiet evidece is required before it will be believed is called the alterative hypothesis (or sometimes also called the research hypothesis), abbreviated H 1, ad is geerally a statemet of iequality. The first step i a hypothesis test is statig H ad H 1, ad choosig a sigificace level α. The hypotheses i a test to see if there is evidece that a populatio proportio is differet from a hypothesized value ca be writte as H : λ = λ vs. H 1 : λ λ. A test statistic i hypothesis testig is a statistic which is used to decide whether to believe the H or to believe the H 1. With a ull hypothesis H : λ = λ, the z statistic z = λ p λ ( 1 λ ) ca be used shall if the simple radom sample size is large eough so that λ 5 ad (1 λ ) 5. The secod step of a hypothesis test is to collect data ad calculate the value of the test statistic. A rejectio regio i hypothesis testig, sometimes also called a critical regio, is a set of test statistic values which lead to rejectig the ull hypothesis i favor of the alterative hypothesis. I geeral, the rejectio for the hypothesis test H : λ = λ vs. H 1 : λ λ with sigificace level α will be defied by z z α/2 or z z α/2 (i.e., z z α/2 ). Sice the H is assumed to be true uless sufficiet evidece is foud agaist it, the custom i statistical termiology has bee to state the decisio i terms of the H. If we decide that there is sufficiet evidece agaist the H, we say that we reject the H, ad sayig that we reject H is equivalet to sayig that we accept H 1. However, if we decide that there is ot sufficiet evidece agaist the H, it has become customary i statistical termiology to say that we do ot reject H istead of sayig that we accept the H ; this is to emphasize that we decided to believe H because there is ot sufficiet evidece agaist H ad ot because there is evidece to support H. The probability value, usually just called the p-value, of a hypothesis test is the probability of obtaiig a test statistic value more supportive of the alterative hypothesis tha the test statistic value actually observed, uder the assumptio H is true. The third step of a hypothesis test is to defie the rejectio regio, decide whether or ot to reject H, ad obtai the p-value of the test. The fourth step of a hypothesis test cosists of statig results ad performig ay further aalysis which may be required. A hypothesis test should be summarized i a few seteces which make clear the results to the reader. It is suggested that the summary of results iclude the observed test statistic value, the tabled value 145

9 which defies the rejectio regio, the coclusio, ad the p-value. Whe the ull hypothesis H : λ = λ is rejected, the statemet of the results of the test should also iclude a idicatio of whether the populatio proportio appears to be less tha or greater tha the hypothesized proportio. 146