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2 Stat 250 Guderso Lecture Notes 5: Learig about a Populatio Proportio Part 1: Distributio for a Sample Proportio To be a statisticia is great!! You ever have to be "absolutely sure" of somethig. Beig "reasoably certai" is eough! - - Pavel E. Guarisma, North Carolia State Uiversity Recall: Parameters, Statistics, ad Statistical Iferece Some distictios to keep i mid: Populatio versus Sample Parameter versus Statistic Populatio proportio p versus sample proportio pˆ Populatio mea µ versus sample mea X Sice we hardly ever kow the true populatio parameter value, we take a sample ad use the sample statistic to estimate the parameter. Whe we do this, the sample statistic may ot be equal to the populatio parameter, i fact, it could chage every time we take a ew sample. Will the observed sample statistic value be a reasoable estimate? If our sample is a RANDOM SAMPLE, the we will be able to say somethig about the accuracy of the estimatio process. Statistical Iferece: the use of sample data to make judgmets or decisios about populatios. The two most commo statistical iferece procedures are cofidece iterval estimatio ad hypothesis testig. Cofidece Iterval Estimatio: A cofidece iterval is a rage of values that the researcher is fairly cofidet will cover the true, ukow value of the populatio parameter. I other words, we use a cofidece iterval to estimate the value of a populatio parameter. We have already ecoutered the idea of a margi of error ad usig it to form a cofidece iterval for a populatio proportio. Hypothesis Testig: Hypothesis testig uses sample data to attempt to reject a hypothesis about the populatio. Usually researchers wat to reject the otio that chace aloe ca explai the sample results. Hypothesis testig is applied to populatio parameters by specifyig a ull value for the parameter a value that would idicate that othig of iterest is happeig. Hypothesis testig proceeds by obtaiig a sample, computig a sample statistic, ad assessig how ulikely the sample statistic would be if the ull parameter value were correct. I most cases, the researchers are tryig to show that the ull value is ot correct. Achievig statistical sigificace is

3 equivalet to rejectig the idea that the observed results are plausible if the ull value is correct.

4 A Overview of Samplig Distributios The value of a statistic from a radom sample will vary from sample to sample. So a statistic is a radom variable ad it will have a probability distributio. This probability distributio is called the samplig distributio of the statistic. Defiitio: The distributio of all possible values of a statistic for repeated samples of the same size from a populatio is called the samplig distributio of the statistic. We will study the samplig distributio of various statistics, may of which will have approximately ormal distributios. The geeral structure of the samplig distributio is the same for each of the five scearios. The samplig distributio results, alog with the ideas of probability ad radom sample, play a vital role i the iferece methods that we cotiue studyig throughout the remaider of the course. Samplig Distributios for Oe Sample Proportio May resposes of iterest produce couts rather tha measuremets - - sex (male, female), political preferece (republica, democrat), approve of ew proposal (yes, o). We wat to lear about a populatio proportio ad we will do so usig the iformatio provided from a sample from the populatio. Example: Do you work more tha 40 hours per week? A poll was coducted by The Heldrich Ceter for Workforce Developmet (at Rutgers Uiversity). A probability sample of 1000 workers resulted i 460 (for 46%) statig they work more tha 40 hours per week. Populatio = Parameter = Sample = Statistic = Ca ayoe say how close this observed sample proportio pˆ is to the true populatio proportio p? If we were to take aother radom sample of the same size = 1000, would we get the same value for the sample proportio pˆ? So what are the possible values for the sample proportio pˆ if we took may radom samples of the same size from this populatio? What would the distributio of the possible pˆ values look like? What is the samplig distributio of pˆ?

5 Aside: Ca you Visualize It? Cosider takig your oe radom sample of size ad computig your oe pˆ value. (As i the previous example, our oe pˆ = 460/1000 = 0.46.) Suppose we did take aother radom sample of the same size, we would get aother value of pˆ, say. Now repeat that process over ad over; takig oe radom sample after aother; resultig i oe pˆ value after aother. Example picture showig the possible values whe = pˆ values Observatios: How would thigs chage if the sample size were eve larger, say =? Suppose our first sample proportio tured out to be pˆ =. Now imagie agai repeatig that process over ad over; takig oe radom sample after aother; resultig i may pˆ possible values. Example picture showig the possible values whe = pˆ values Observatios:

6 Let s take a closer look at the sample proportio pˆ. The sample proportio is foud by takig the umber of successes i the sample ad dividig by the sample size. So the cout variable X X of the umber of successes is directly related to the proportio of successes as p ˆ =. Earlier we studied the distributio of our first statistic, the cout statistic X (the umber of successes i idepedet trials whe the probability of a success was p). We leared about its exact distributio called the Biomial Distributio. We also leared whe the sample size was large, the distributio of X could be approximated with a ormal distributio. Normal Approximatio to the Biomial Distributio If X is a biomial radom variable based o trials with success probability p, ad is large, N p, p(1 p). the the radom variable X is also approximately ( ) Coditios: The approximatio works well whe both p ad (1 p) are at least 10. So ay probability questio about a sample proportio could be coverted to a probability questio about a sample cout, ad vice- versa. If is small, we would eed to covert the questio to a cout ad use the biomial distributio to work it out. If is large, we could covert the questio to a cout ad use the ormal approximatio for a cout, OR use a related ormal approximatio for a sample proportio (for large ). The Stat 250 formula card summarizes this related ormal approximatio as follows: Let s put this result to work i our ext Try It! Problem.

7 Try It! Do Americas really vote whe they say they do? To aswer this questio, a telephoe poll was take two days a electio. From the 800 adults polled, 56% reported that they had voted. However, it was later reported i the press that, i fact, oly 39% of America adults had voted. Suppose the 39% rate reported by the press is the correct populatio proportio. Also assume the resposes of the 800 adults polled ca be viewed as a radom sample. a. Sketch the samplig distributio of pˆ for a radom sample of size = 800 adults. b. What is the approximate probability that a sample proportio who voted would be 56% or larger for a radom sample of 800 adults? c. Does it seem that the poll result of 56% simply reflects a sample that, by chace, voted with greater frequecy tha the geeral populatio? More o the Stadard Deviatio of pˆ The stadard deviatio of pˆ is give by: s.d.( pˆ ) = p ( 1 p) This quatity would give us a idea about how far apart a sample proportio pˆ ad the true populatio proportio p are likely to be. We ca iterpret this stadard deviatio as approximately the average distace of the possible pˆ values (for repeated samples of the same size ) from the true populatio proportio p.

8 I practice whe we take a radom sample from a large populatio, we oly kow the sample proportio. We geerally would ot kow the true populatio proportio p. So we could ot compute the stadard deviatio of pˆ. However we ca use the sample proportio i the formula to have a estimate of the stadard deviatio, which is called the stadard error of pˆ. The stadard error of pˆ is give by: s.e.( pˆ ) = p ˆ (1 pˆ ) This quatity is a estimate of the stadard deviatio of pˆ. So we ca iterpret this stadard error as estimatig, approximately, the average distace of the possible pˆ values (for repeated samples of the same size ) from the true populatio proportio p. Moreover, we ca use this stadard error to create a rage of values that we are very cofidet will cotai the true proportio p, amely, pˆ ± (a few)s.e.( pˆ ). This is the basis for cofidece iterval for the true proportio p, discussed ext Try It! Love at first sight? I a radom sample of = 500 adults, 300 stated they believe i love at first sight. a. Estimate the populatio proportio of adults that believe i love at first sight. b. Fid the correspodig stadard error of for the estimate i part a ad use this stadard error to provide a iterval estimate for the populatio proportio p, with 95% cofidece.

9 Stat 250 Guderso Lecture Notes 5: Learig about a Populatio Proportio Part 2: Estimatig Proportios with Cofidece Big Idea of Cofidece Itervals: Use sample data to estimate a populatio parameter. Recall some of the laguage ad otatio associated with the estimatio process. Populatio ad Populatio Parameter Sample ad Sample Statistic (sample estimate or poit estimate) The sample estimate provides our best guess as to what is the value of the populatio parameter, but it is ot 100% accurate. The value of the sample estimate will vary from oe sample to the ext. The values ofte vary aroud the populatio parameter ad the stadard deviatio give a idea about how far the sample estimates ted to be from the true populatio proportio o average. The stadard error of the sample estimate provides a idea of how far away it would ted to vary from the parameter value (o average). The geeral format for a cofidece iterval estimate is give by: Sample estimate ± (a few) stadard errors The few or umber of stadard errors we go out each way from the sample estimate will deped o how cofidet we wat to be. The how cofidet we wat to be is referred to as the cofidece level. This level reflects how cofidet we are i the procedure. Most of the itervals that are made will cotai the truth about the populatio, but occasioally a iterval will be produced that does ot cotai the true parameter value. Each iterval either cotais the populatio parameter or it does t. The cofidece level is the percetage of the time we expect the procedure to produce a iterval that does cotai the populatio parameter. Cofidece Iterval for a Populatio Proportio p pˆ Goal: we wat to lear about a populatio proportio. How? We take a radom sample from the populatio ad estimate p with the resultig sample proportio pˆ. Let s first recall how those may possible values for the sample proportio would vary, that is, the samplig distributio of the statistic pˆ.

10 Samplig Distributio of pˆ : If the sample size is large ad p 10 ad ( 1 p) 10, the pˆ is approximately p(1 p) N p,. Desity N(, ) 1. Cosider the followig iterval or rage of values ad show it o the picture. p( 1 p) p( 1 p) p( 1 p) p ± 2 p 2, p What is the probability that a (yet to be computed) sample proportio pˆ will be i this iterval (withi 2 stadard deviatios from the true proportio p)? 3. Take a possible sample proportio pˆ ad cosider the iterval ( ) ( ) p( 1 p) p 1 p p 1 p pˆ ± 2 pˆ 2, pˆ + 2 Show this rage o the ormal distributio picture above. 4. Did your first iterval aroud your first pˆ cotai the true proportio p? Was it a good iterval? 5. Repeat steps 3 ad 4 for other possible values of pˆ.

11 Big Idea: Cosider all possible radom samples of the same large size. Each possible radom sample provides a possible sample proportio value. If we made a histogram of all of these possible pˆ values it would look like the ormal distributio o the previous page. About 95% of the possible sample proportio pˆ values will be i the iterval p( 1 p) p ± 2 ; ad for each oe of these sample proportio pˆ values, the iterval p( 1 p) p ± 2 will cotai the populatio proportio p. Thus about 95% of the itervals p( 1 p) pˆ ± 2 will cotai the populatio proportio p. Thus, a iitial 95% cofidece iterval for the true proportio p is give by: p( 1 p) pˆ ± 2 The Dilemma: Whe we take our oe radom sample, we ca compute the sample proportio pˆ p( 1 p), but we ca t costruct the iterval pˆ ± 2 because we do t kow the value of p. The Solutio: Replace the value of p i the stadard deviatio with the estimate, that is use pˆ ( 1 pˆ ) called. pˆ A approximate 95% cofidece iterval (CI) for the populatio proportio p is: pˆ ± 2 pˆ(1 pˆ) Note: The ± part of the iterval 2 p ˆ(1 pˆ ) is called the 95% margi of error. Note: The approximate is due to the multiplier of 2 beig used. We will lear about other multipliers, icludig the exact 95% multiplier value later.

12 Try It! Gettig Alog with Parets I a Gallup Youth Survey = 501 radomly selected America teeagers were asked about how well they get alog with their parets. Oe survey result was that 54% of the sample said they get alog VERY WELL with their parets. a. The sample proportio was foud to be Give the stadard error for the sample proportio ad use it to complete the setece that iterprets the stadard error i terms of a average distace. We would estimate the average distace betwee the possible values (from repeated samples) ad to be about b. Compute a 95% cofidece iterval for the populatio proportio of teeagers that get alog very well with their parets. c. Fill i the blaks for the typical iterpretatio of the cofidece iterval i part b: Based o this sample, with 95% cofidece, we would estimate that somewhere betwee ad of all America teeagers thik they get alog very well with their parets. d. Ca we say the probability that the above (already observed) iterval (, ) cotais the populatio proportio p is 0.95? That is, ca we say P( p ) = 0.95? e. Ca we say that 95% of the time the populatio proportio p will be i the iterval computed i part b?

13 Just what does the 95% cofidece level mea? Iterpretatio The phrase cofidece level is used to describe the likeliess or chace that a yet- to- be costructed iterval will actually cotai the true populatio value. However, we have to be careful about how to iterpret this level of cofidece if we have already completed our iterval. The populatio proportio p is ot a radom quatity, it does ot vary - oce we have looked (computed) the actual iterval, we caot talk about probability or chace for this particular iterval aymore. The 95% cofidece level applies to the procedure, ot to a idividual iterval; it applies before you look ad ot after you look at your data ad compute your iterval. Try It! Gettig Alog with Parets I the previous Try It! you computed a 95% cofidece iterval for the populatio proportio of teeagers that get alog very well with their parets i part (b). This was based o a radom sample of = 501 America teeagers. You iterpreted the iterval i part (c). Write a setece or two that iterprets the cofidece level. The iterval we foud was computed with a method which if repeated over ad over... Try It! Completig a Graduate Degree A researcher has take a radom sample of = 100 recet college graduates ad recorded whether or ot the studet completed their degree i 5 years or less. Based o these data, a 95% cofidece iterval for the populatio proportio of all college studets that complete their degree i 5 years or less is computed to be (0.62, 0.80). a. How may of the 100 sampled college graduates completed their degree i 5 years or less? b. Which of the followig statemets gives a valid iterpretatio of this 95% cofidece level? Circle all that are valid. i. There is about a 95% chace that the populatio proportio of studets who have completed their degree i 5 years or less is betwee 0.62 ad ii. If the samplig procedure were repeated may times, the approximately 95% of the resultig cofidece itervals would cotai the populatio proportio of studets who have completed their degree i 5 years or less. iii. The probability that the populatio proportio p falls betwee 0.62 ad 0.80 is 0.95 for repeated samples of the same size from the same populatio. What about that Multiplier of 2?

14 The exact multiplier of the stadard error for a 95% cofidece level would be 1.96, which was rouded to the value of 2. Where does the 1.96 come from? Use the stadard ormal distributio, the N(0, 1) distributio at the right ad Table A.1. Researchers may ot always wat to use a 95% cofidece level. Other commo levels are 90%, 98% ad 99%. Usig the same idea for cofirmig the value of 1.96, fid the correct multiplier if the cofidece level were 90%. The geeric expressio for this multiplier whe you are workig with a stadard ormal distributio is give by z*. Here are a few other multipliers for a populatio proportio cofidece iterval. Cofidece Level 90% 95% 98% 99% Multiplier z* (or about 2) Now, the easiest way to fid multipliers is to actually look ahead a bit ad make use of Table A.2. Look at the df row marked Ifiite degrees of freedom ad you will fid the z* values for may commo cofidece levels. Check it out! From Utts, Jessica M. ad Robert F. Heckard. Mid o Statistics, Fourth Editio Used with permissio. Whe the cofidece level icreases, the value of the multiplier icreases. So the width of the cofidece iterval also icreases. I order to be more cofidet i the procedure (have a procedure with a higher probability of producig a iterval that will cotai the populatio value, we have to sacrifice ad have a wider iterval. The formula for a cofidece iterval for a populatio proportio p is summarized ext.

15 Cofidece Iterval for a Populatio Proportio p: where pˆ is the sample proportio ad z * is the appropriate multiplier pˆ 1 pˆ ad s.e.( pˆ ) = is the stadard error of the sample proportio. Coditios: 1. The sample is a radomly selected sample from the populatio. However, available data ca be used to make ifereces about a much larger group if the data ca be cosidered to be represetative with regard to the questio(s) of iterest. 2. The sample size is large eough so that the ormal curve approximatio holds p 10 ad ( 1 p) 10 Try It! A 90% CI for p A radom sample of = 501 America teeagers resulted i 54% statig they get alog very well with their parets. The stadard error for this estimate was foud to be 2.2%. The 95% cofidece iterval for the populatio proportio of teeagers that get alog very well with their parets wet from 49.6% to 58.4%. The correspodig 90% cofidece iterval would go from 50.4% to 57.6%, which is ideed arrower (but still cetered aroud the estimate of 54%). The Coservative Approach From the geeral form of the cofidece iterval, the margi of error is give as: pˆ ( 1 pˆ ) Margi of error = z* s.e.( pˆ ) = z* For ay fixed sample size, this margi of error will be the largest whe pˆ = ½ = 0.5. Thik about the fuctio pˆ (1 pˆ ). So usig ½ for pˆ i the above margi of error expressio we have: z * ( ) pˆ(1 pˆ) = z * * 2 ( ) z = 2 p ˆ ± z s.e.( pˆ) By usig this margi of error for computig a cofidece iterval, we are beig coservative. The resultig iterval may be a little wider tha eeded, but it will ot err o beig too arrow. This leads to a correspodig coservative cofidece iterval for a populatio proportio. Coservative Cofidece Iterval for a Populatio Proportio p z * pˆ ± 2 where pˆ is the sample proportio ad z * is the appropriate multiplier. *

16 Earlier we saw the margi of error for a proportio was give as 1. This is actually a 95% coservative margi of error. What happes to the coservative margi of error i the box above whe you use z * = 2 for 95% cofidece? Choosig a Sample Size for a Survey The choice of a sample size is importat i plaig a survey. Ofte a sample size is selected (usig the coservative approach) that such that it will produce a desired margi of error for a give level of cofidece. Let s take a look at the coservative margi of error more closely. (Coservative) Margi of Error = m = * z Solvig this expressio for the sample size we have: = 2m If this does is ot a whole umber, we would roud up to the ext largest iteger. Try It! Coke versus Pepsi A poll was coducted i Caada to estimate p, the proportio of Caadia college studets who prefer Coke over Pepsi. Based o the sampled results, a 95% coservative cofidece iterval for p was foud to be (0.62, 0.70). a. What is the margi of error for this iterval? 2 z * 2 b. What sample size would be ecessary i order to get a coservative 95% cofidece iterval for p with a margi of error of 0.03 (that is, a iterval with a width of 0.06)? c. Suppose that the same poll was repeated i the Uited States (whose populatio is 10 times larger tha Caada), but four times the umber of people were iterviewed. The resultig 95% coservative cofidece iterval for p will be: twice as wide as the Caadia iterval 1/2 as wide as the Caadia iterval 1/4 as wide as the Caadia iterval 1/10 as wide as the Caadia iterval the same width as the Caadia iterval Usig Cofidece Itervals to Guide Decisios Thik about it: A value that is ot i a cofidece iterval ca be rejected as a likely value of the populatio proportio. A value that is i a cofidece iterval is a acceptable possibility for the value of a populatio proportio.

17 Try It! Coke versus Pepsi Recall the poll coducted i Caada to estimate p, the proportio of Caadia college studets who prefer Coke over Pepsi. Based o the sampled results, a 95% coservative cofidece iterval for p was foud to be (0.62, 0.70). Do you thik it is reasoable to coclude that a majority of Caadia college studets prefer Coke over Pepsi? Explai. Additioal Notes A place to jot dow questios you may have ad ask durig office hours, take a few extra otes, write out a extra problem or summary completed i lecture, create your ow summary about these cocepts.

18 Stat 250 Guderso Lecture Notes 5: Learig about a Populatio Proportio Part 3: Testig about a Populatio Proportio We make decisios i the dark of data. - - Stu Huter Overview of Testig Theories We have examied statistical methods for estimatig the populatio proportio based o the sample proportio usig a cofidece iterval estimate. Now we tur to methods for testig theories about the populatio proportio. The hypothesis testig method uses data from a sample to judge whether or ot a statemet about a populatio is reasoable or ot. We wat to test theories about a populatio proportio ad we will do so usig the iformatio provided from a sample from the populatio. Basic Steps i Ay Hypothesis Test Step 1: Determie the ull ad alterative hypotheses. Step 2: Verify ecessary data coditios, ad if met, summarize the data ito a appropriate test statistic. Step 3: Assumig the ull hypothesis is true, fid the p- value. Step 4: Decide whether or ot the result is statistically sigificat based o the p- value. Step 5: Report the coclusio i the cotext of the situatio. Formulatig Hypothesis Statemets May questios i research ca be expressed as which of two statemets might be correct for a populatio. These two statemets are called the ull ad the alterative hypotheses. The ull hypothesis is ofte deoted by H 0, ad is a statemet that there is o effect, o differece, that othig has chage or othig is happeig. The ull hypothesis is usually referred to as the status quo. The alterative hypothesis is ofte deoted by H a, ad is a statemet that there is a relatioship, there is a differece, that somethig has chaged or somethig is happeig. Usually the researcher hopes the data will be strog eough to reject the ull hypothesis ad support the ew theory i the alterative hypothesis. It is importat to remember that the ull ad alterative hypotheses are statemets about a populatio parameter (ot about the results i the sample). Fially, there will ofte be a directio of extreme that is idicated by the alterative hypothesis. To see these ideas, let's try writig out some hypotheses to be put to the test.

19 Try It! Statig the Hypotheses ad defiig the parameter of iterest 1. About 10% of the huma populatio is left- haded. Suppose that a researcher speculates that artists are more likely to be left- haded tha are other people i the geeral populatio. H 0 : H a : let = parameter = writte descriptio Directio: 2. Suppose that a pharmaceutical compay wats to be able to claim that for its ewest medicatio the proportio of patiets who experiece side effects is less tha 20%. H 0 : H a : let = parameter = writte descriptio Directio: 3. The US Cesus reports that 48% of households have o childre. A radom sample of 500 households will be take to assess if the populatio proportio has chaged from the Cesus value of H 0 : H a : let = parameter = writte descriptio Directio: Notes: 1. Whe the alterative hypothesis specifies a sigle directio, the test is called a oe- sided or oe- tailed hypothesis test. I practice, most hypothesis tests are oe- sided tests because ivestigators usually have a particular directio i mid whe they cosider a questio. 2. Whe the alterative hypothesis icludes values i both directio from a specific stadard, the test is called a two- sided or two- tailed hypothesis test. 3. A geeric ull hypothesis could be expressed as H 0 : populatio parameter = ull value, where the ull value is the specific umber the parameter equals if the ull hypothesis is true. I all of the examples above, the populatio parameter is p, the populatio proportio. Example 1 above has the ull value of 10% or 0.10.

20 The Logic of Hypothesis Testig: What if the Null is True? Thik about a jury trial H 0 : The defedat is H a : The defedat is We assume that the ull hypothesis is true util the sample data coclusively demostrate otherwise. We assess whether or ot the observed data are cosistet with the ull hypothesis (allowig reasoable variability). If the data are ulikely whe the ull hypothesis is true, we would reject the ull hypothesis ad support the alterative theory. The Big Questio we ask: If the ull hypothesis is true about the populatio, what is the probability of observig sample data like that observed (or more extreme)? Reachig Coclusios about the Two Hypotheses We will be decidig betwee the two hypotheses usig data. The data is assumed to be a radom sample from the populatio uder study. The data will be summarized via a. I may cases the test statistic is a stadardized statistic that measures the distace betwee the sample statistic ad the ull value i stadard error uits. Test Statistic = Sample Statistic Null Value (Null) Stadard Error I fact, our first test statistic will be a z- score ad we are already familiar with what makes a z- value uusual or large. With the test statistic computed, we quatify the compatibility of the result with the ull hypothesis with a probability value called the p- value. The p- value is computed by assumig the ull hypothesis is true ad the determiig the probability of a result as extreme (or more extreme) as the observed test statistic i the directio of the alterative hypothesis. Notes: (1) The p- value is a probability, so it must be betwee 0 ad 1. It is really a coditioal probability the probability of seeig a test statistic as extreme or more extreme tha observed give (or coditioal o) the ull hypothesis is true. (2) The p- value is ot the probability that the ull hypothesis is true. The the p- value, the stroger the evidece is AGAINST H 0 (ad i favor of H a ). Commo Covetio: Reject H 0 if the p- value is. This borderlie value is called the ad deoted by. Whe the p- value is α, we say the result is. Commo levels of sigificace are:

21 Two Possible Results: The p- value is α, so we reject H 0 ad say the results are statistically sigificat at the level α. We would the write a real- world coclusio to explai what rejectig H 0 traslates to i the cotext of the problem at had. The p- value is > α, so we fail to reject H 0 ad say the results are ot statistically sigificat at the level α. We would the write a real- world coclusio to explai what failig to reject H 0 traslates to i the cotext of the problem at had. Be careful: we say fail to reject H 0 ad ot accept H 0 because the data do ot prove the ull hypothesis is true, rather the data were ot covicig eough to support the alterative hypothesis. Testig Hypotheses About a Populatio Proportio I the cotext of testig about the value of a populatio proportio p, the possible hypotheses statemets are: 1. H 0 : versus H a : 2. H 0 : versus H a : 3. H 0 : versus H a : Where does p 0 come from? Sometimes the ull hypothesis is writte as H 0 :p = p 0 as we compute the p- value assumig the ull hypothesis is true, that is, we take the populatio proportio to be the ull value p 0. The sample data will provide us with a estimate of the populatio proportio p, amely the sample proportio pˆ. For a large sample size, the distributio for the sample proportio will be: If we have a ormal distributio for a variable, the we ca stadardize that variable to compute probabilities, as log as you have the mea ad stadard deviatio for that statistic. I testig, we assume that the ull hypothesis is true, that the populatio proportio p = p 0. So the stadardized z- statistic for a sample proportio i testig is: z = If the ull hypothesis is true, this z- test statistic will have approximately a.

22 The stadard ormal distributio will be used to compute the p- value for the test.

23 Try It! Left- haded Artists About 10% of the huma populatio is left- haded. Suppose that a researcher speculates that artists are more likely to be left- haded tha are other people i the geeral populatio. The researcher surveys a radom sample of 150 artists ad fids that 18 of them are left- haded. Perform the test usig a 5% sigificace level. Step 1: Determie the ull ad alterative hypotheses. H 0 : H a : where the parameter represets Note: The directio of extreme is Step 2: Verify ecessary data coditios, ad if met, summarize the data ito a appropriate test statistic. The data are assumed to be a radom sample. Check if p 0 10 ad (1 p 0 ) 10. Observed test statistic: z = pˆ p p (1 p ) Step 3: Assumig the ull hypothesis is true, fid the p- value. The p- value is the probability of gettig a test statistic as extreme or more extreme tha the observed test statistic value, assumig the ull hypothesis is true. Sice we have a oe- sided test to the right, toward the larger values p- value = probability of gettig a z test statistic as large or larger tha observed, assumig the ull hypothesis is true. = Step 4: Decide if the result is statistically sigificat based o the p- value. Step 5: Report the coclusio i the cotext of the situatio.

24 Aside: The researcher chooses the level of sigificace α before the study is coducted. I our Left- Haded Artists example we had H 0 : p = 0.10 versus H a : p > If oly 12 LH artists i our sample, we would have pˆ = 0.08, we would certaily ot reject H 0 With our 18 LH artists i our sample, our pˆ = 0.12, z=0.82, p- value=0.206 ad our decisio was to fail to reject H 0 What if we had 20 LH artists i our sample, our pˆ = 0.133, our z=1.36, p- value= ad our decisio would be What if we had 22 LH artists i our sample, our pˆ = 0.147, our z=1.91, p- value= ad our decisio would be What if we had 24 LH artists i our sample, our pˆ = 0.16, our z=2.45, p- value=0.007, ad our decisio would be Selectig the level of sigificace is like drawig a lie i the sad separatig whe you will reject H 0 ad whe there the evidece would be strog eough to reject H 0. This first test was a oe- sided test to the right. How is the p- value foud for the other directios of extreme? The table below provides a ice summary. From Utts, Jessica M. ad Robert F. Heckard. Mid o Statistics, Fourth Editio Used with permissio.

25 Try It! Households without Childre The US Cesus reports that 48% of households have o childre. A radom sample of 500 households was take to assess if the populatio proportio has chaged from the Cesus value of Of the 500 households, 220 had o childre. Use a 10% sigificace level. Step 1: Determie the ull ad alterative hypotheses. H 0 : p = 0.48 H a : p 0.48 where the parameter p represets the populatio proportio of all households today that have o childre. Note: The directio of extreme is two- sided. Step 2: Verify ecessary data coditios, ad if met, summarize the data ito a appropriate test statistic. The data are assumed to be a radom sample. Check if p 0 10 ad (1 p 0 ) 10. pˆ p0 Observed test statistic: z = p (1 p ) 0 0 Step 3: Assumig the ull hypothesis is true, fid the p- value. The p- value is the probability of gettig a test statistic as extreme or more extreme tha the observed test statistic value, assumig the ull hypothesis is true. Sice we have a two- sided test, both large ad small values are extreme. Sketch the area that correspods to the p- value: Compute the p- value: Step 4: Decide if the result is statistically sigificat based o the p- value. Step 5: Report the coclusio i the cotext of the situatio.

26 What if is small? Goal: we still wat to lear about a populatio proportio p. We take a radom sample of size where is small (i.e. p 0 < 10 or (1 p 0 ) < 10). If the sample size is small, we have to go back to the exact distributio for a cout X, called the biomial distributio. If X has the biomial distributio Bi(, p), the k k! P( X = k) = p (1 p) for k = 0,1,2,..., where = k k k!( k)! ad the mea of X = µ = p ad the stadard deviatio of X = p( 1 p) We will use the biomial probability formula for computig the exact p- value. Testig Hypotheses about a Populatio Proportio p whe is small With a small sample size, we will do a Biomial test. Small- Sample Biomial Test for the populatio proportio p To test the hypothesis H 0 : p = p 0 we compute the cout test statistic X = the umber of successes i the sample of size which has the Bi(, p 0 ) distributio whe H 0 is true. This Bi(, p 0 ), distributio is used to compute the p- value for the test. Try It! New Treatmet A group of 10 subjects with a disease are treated with a ew treatmet. Of the 10 subjects, 9 showed improvemet. Test the claim that a majority of people usig this treatmet improved usig a 5% sigificace level. Let p be the true populatio proportio of people who improve with this treatmet. State the hypotheses: H 0 : H a : The observed test statistic value is just p- value = At the 5% sigificace level, we would ad coclude:

27 Let s revisit the flow chart for workig o problems that deal with a populatio proportio. Whe the sample size is large we ca use the How to deal with questios large sample ormal approximatio for about proportios computig probabilities about a sample proportio, for testig hypotheses about a Proportios Respose Variable is Categorical populatio proportio (based o the resultig with 2 outcomes: S, F sample proportio), ad for computig a cofidece iterval estimate for the value of a populatio proportio (agai usig the sample proportio as the poit estimate). Whe the sample size is small we use the biomial distributio to compute probabilities about a sample proportio or to test hypotheses about a populatio proportio (based o the resultig sample cout of successes). We did ot discuss the small sample cofidece iterval for a populatio proportio usig the biomial distributio. is small Covert to cout. Use X ~ Bi(,p) Use p-hat p-hat ~ N(p, p(1-p)/) Sample Size, Statistical Sigificace, ad Practical Importace The size of the sample affects our ability to make firm coclusios based o that sample. With a small sample, we may ot be able to coclude aythig. With large samples, we are more likely to fid statistically sigificat results eve though the actual size of the effect is very small ad perhaps uimportat. The phrase statistically sigificat oly meas that the data are strog eough to reject the ull hypothesis. The p- value tells us about the statistical sigificace of the effect, but it does ot tell us about the size of the effect. Cosider testig H 0 : p = 0.5 versus H a : p > 0.5 at α = Case 1: 52 successes i a sample of size = 100 à pˆ = 0.52 Test Statistic: z = ( ) / [0.5(1-0.5)/100] = 0.4 p- value = P(Z 0.4) = So we would fail to reject H 0. A icrease of oly 0.02 beyod 0.50 seems icosequetial (ot sigificat). Case 2: 520 successes i a sample of size = 1000 à pˆ = 0.52 Test Statistic: z = ( ) / [0.5(1-0.5)/1000] = 1.26 p- value = P(Z 1.26) = So we would agai fail to reject H 0. Here a icrease of 0.02 beyod 0.50 is approachig sigificace. is large p 10 ad (1-p) 10 Covert to cout Use X ~ N(p, p(1-p)) Case 3: 5200 successes i a sample of size = 10,000 à pˆ = 0.52 Test Statistic: z = ( ) / [0.5(1-0.5)/10,000] = 4.0 p- value = P(Z 4) = So we would certaily reject H 0. Here a icrease of 0.02 beyod 0.50 is very sigificat! Small samples make it very difficult to demostrate much of aythig. Huge sample sizes ca make a practically uimportat differece statistically sigificat. Key: determie appropriate sample sizes so fidigs that are practically importat become statistically sigificat. or

28 What Ca Go Wrog: Two Types of Errors We have bee discussig a statistical techique for makig a decisio betwee two competig theories about a populatio. We base the decisio o the results of a radom sample from that populatio. There is the possibility of makig a mistake. I fact there are two types of error that we could make i hypothesis testig. Type 1 error = rejectig H 0 whe H 0 is true Type 2 error = failig to reject H 0 whe H a is true I statistics we have otatio to represet the probabilities that a testig procedure will make these two types of errors. P(Type 1 error) = P(Type 2 error) = There is aother probability that is of iterest to researchers if there really is somethig goig o (if the alterative theory is really true), what is the probability that we will be able to detect it (be able to reject H 0 )? This probability is called the power of the test ad is related to the probability of makig a Type 2 error. Power = P(rejectig H 0 whe H a is true) = 1 P(failig to reject H 0 whe H a is true) = 1 P(Type 2 error) = 1 β. So we ca thik of power as the probability of advocatig the ew theory give the ew theory is true. Researchers are geerally iterested i havig a test with high power. Oe dilemma is that the best way to icrease power is to icrease sample size (see commets below) ad that ca be expesive. Commets: 1. I practice we wat to protect the status quo so we are most cocered with. 2. Most tests we describe have the Geerally, for a fixed sample size, Ideally we wat the probabilities of makig a mistake to be small, we wat the power of the test to be large. However, these probabilities are properties of the procedure (the proportio of times the mistake would occur if the procedure were repeated may times) ad ot applicable to the decisio oce it is made. 5. Some factors that ifluece the power of the test Sample size: larger sample size leads to higher power. Sigificace level: larger α leads to higher power. Actual parameter value: a true value that falls further from the ull value (i the directio of the alterative hypothesis) leads to higher power (however this is ot somethig that the researcher ca cotrol or chage).

29 Simple Example H 0 : Basket has 9 Red ad 1 White H a : Basket has 4 Red ad 6 White Data: 1 ball selected at radom from the basket. What is the most reasoable Decisio Rule? Reject the ull hypothesis if the ball is With this rule, what are the chaces of makig a mistake? P(Type 1 error) = P(Type 2 error) = What is the power of the test? Suppose a ball is ow selected from the basket ad it is observed ad foud to be WHITE. What is the decisio? You just made a decisio, could a mistake have bee made? If so, which type? What is the probability that this type of mistake was made? Note: The Decisio Rule stated i the simple example resembles the rejectio regio approach to hypothesis testig. We will focus primarily o the p- value approach that is used i reportig results i jourals.

30 Additioal Notes A place to jot dow questios you may have ad ask durig office hours, take a few extra otes, write out a extra problem or summary completed i lecture, create your ow summary about these cocepts.