8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These kids of decisios are called tests of sigificace. Goal: To test the sigificace of the differece betwee the sample ad the stadard. Small differece: there is o reaso to coclude that the stadard does t hold. Large eough differece: If it ca t reasoably be attributed to chace, you ca coclude that the stadard o loger holds. 1 About 2% of bar swallows have white feathers i places where the plumage is ormally blue or red. The white feathers are caused by geetic mutatios. I 1986, the Russia uclear reactor at Cherobyl leaked radioactivity. Researchers cotiue to be cocered that the radiatio may have caused mutatios i the gees of humas ad aimals that were passed o to offsprig. I a sample of bar swallows captured aroud Cherobyl i 1991 ad 1996, about 14% had white feathers i places where the plumage is ormally blue or red. Researchers compared the proportio.14 i the sample of captured bar swallows to the stadard of.2. If the overall percetage was still oly 2%, it is ot reasoably likely to get 14% i their sample. So they came to the coclusio that there was a icreased probability of geetic mutatios i the Cherobyl area. Source: Los Ageles Times, October 9, 1997, page B2.
Iformal Sigificace Testig People ted to believe that peies are balaced. They geerally have o qualms about flippig a pey to make a fair decisio. Is it really the case that pey flippig is fair? What about spiig peies? The logic ivolved i decidig whether or ot to reject the stadard that spiig a pey results i heads 5% of the time makes use of the same logic as that ivolved i estimatig a proportio i Sectio 8.1. Spiig Peies Jey ad Maya s Spis Jey ad Maya woder if heads ad tails are equally likely whe a pey is spu. They spi peies times ad get 17 heads. Should they reject the stadard that peies fall heads 5% of the time eve if heads ad tails are equally likely? 17 = =.425 The value.425 falls i the reasoably likely iterval obtaied from p =.5. 5 Statistical Sigificace A sample is statistically sigificat if it is ot a reasoably likely outcome whe the proposed stadard is true. Jey ad Maya s result is ot statistically sigificat sice their sample proportio of.425 falls withi the reasoably likely iterval of p =.5. Spiig Peies Miguel ad Kevi s Spis Miguel ad Kevi also spu peies ad got 1 heads out of spis for a sample proportio of.25. Is this a statistically sigificat result? 1 = =.25 The value.25 falls outside the reasoably likely iterval obtaied from p =.5. This is a statistically sigificat result!
Spiig Peies Miguel ad Kevi s Spis Miguel ad Kevi also spu peies ad got 1 heads out of spis for a sample proportio of.25. Is this a statistically sigificat result? 1 = =.25 The value.25 falls outside the reasoably likely iterval obtaied from p =.5. Aother solutio is to calculate the 95% Cofidece Iterval usig =.25 ad verify that.5 is ot i it. ± 1.96 (1 ) = (.25)(1.25).25 ± 1.96 = = (.11581,.38419) Basic Notatio. p p Populatio proportio of successes (Ukow i geeral) Sample proportio of successes (What we recorded from our sample) Hypothesized value of the populatio proportio. (The Stadard) 9 Discussio: Statistical Sigificace A 1997 article reported that two-thirds of tees i grades 7 12 wat to study more about medical research. You woder if this proportio still holds today ad decide to test it. You take a radom sample of tees ad fid that oly 23 wat to study more about medical research. Source: CNN Iteractive Story Page, www.c.com/tech/974/22/teetech.poll/, April 22, 1997. a. What is the stadard (the hypothesized value, p, of the populatio proportio)? b. What is a alterate hypothesis? c. What is the sample proportio? d. Is the result statistically sigificat? That is, is there evidece leadig you to believe that the proportio today is differet from the proportio i 1997? Discussio: Statistical Sigificace a. What is the stadard (the hypothesized value, p, of the populatio proportio)? Aswer: p = 2/3 = 66.66% b. What is a alterate hypothesis? Aswer: That the proportio owadays is differet from that i 1997, that is that p is differet from 2/3. c. What is the sample proportio? Aswer: The sample proportio is = 23/ =.575
Discussio: Statistical Sigificace d. Is the result statistically sigificat? That is, is there evidece leadig you to believe that the proportio today is differet from the proportio i 1997? Oe possible solutio is to calculate the 95% Cofidece Iterval ad the check whether the value p = 2/3 is i the iterval or ot. Do it! Here is the stadard procedure. The Test Statistic To check if the sample proportio is statistically sigificat with respect to the stadard p we just eed to check if is a rare evet i the distributio geerated by p. We kow the distributio is approximately ormal with p(1 p) µ = p, ad σ = So we ca calculate the z-score of : µ p z = = σ p(1 p) Ad if we get z < 1.96 or z > 1.96 the is statistically sigificat. The umbers 1.96 ad 1.96 are called critical values. This value of z is called the Test Statistic 13 Other Critical Values The dividig poits are called critical values (deoted z*). Other z*- values commoly used as critical poits are * z 1.645 = 1.96 2.576 for a level of for a level of for a level of sigificace of α = 1% sigificace of α = 5% sigificace of α = 1% If the value of the test statistic is more extreme tha the critical values you have chose, you reject the stadard ad say that the result is statistically sigificat. A larger critical value makes it harder to reject the stadard. If you use z* = ±1.96, the to reject the stadard, the test statistic z must fall i the outer 5% of the stadard ormal distributio. If you use z* = ±1.645, the value of z must fall i oly the outer 1% of the distributio. Each critical value is associated with a correspodig percetage, α (alpha), called the level of sigificace. If a level of sigificace is t specified, it is usually safe to assume that α =.5 ad z* = ±1.96. Fid the test statistic from Jey ad Maya s data o spiig peies. (17 Heads out of spis) What do you coclude if α =.1? p Recall that the test statistic is: z = p(1 p) Sice we are testig if the probability of gettig heads is.5, the we have p =.5, =, ad sample proportio p hat of 17/ =.425. Thus z =.425.5.5(1.5) =.948683 The critical values associated to α =.1 are z* = ±1.645. Sice our test statistic falls i betwee, the the coclusio is that the sample is ot statistically sigificat.
Use your z-table (or your calculator preferably) to aswer these questios. a. What level of sigificace is associated with critical values of z* = ±2.576? Aswer: We eed to fid the area uder the curve betwee 2.576 ad 2.576 i the stadard ormal distributio. We ca do this usig: ormalcdf( 2.576, 2.576)=.9948 So the level of sigificace is equal to 1% 99% = 1%. b. What critical values are associated with a level of sigificace of 2%? Aswer: We eed to fid the z-values that correspod to the middle 98% of the area uder the stadard ormal distributio, that is the values that leave 1% at the begiig ad 1% at the ed. By symmetry we ca fid oly oe of them. We ca do this usig: ivnorm(.1)= 2.32634 Formal Laguage of Test Sigificace (Compoets of a Sigificace Test for a Proportio) 1. Give the ame of the test ad check the coditios for its use. For a sigificace test for a proportio, three coditios must be met. The sample is a simple radom sample from a biomial populatio. Both p ad (1 p ) are at least 1. The populatio size is at least 1 times the sample size. 17 Formal Laguage of Test Sigificace (Compoets of a Sigificace Test for a Proportio) 2. State the hypotheses, defiig ay symbols. Whe testig a proportio, the ull hypothesis H is H : The percetage of successes p i the populatio from which the sample came is equal to p. The alterate hypothesis, Ha, ca be of three forms: H a : The percetage of successes p i the populatio from which the sample came is ot equal to p. H a : The percetage of successes p i the populatio from which the sample came is greater tha p. H a : The percetage of successes p i the populatio from which the sample came is less tha p. Formal Laguage of Test Sigificace (Compoets of a Sigificace Test for a Proportio) 3. Compute the test statistic z ad compare it to the critical values z* (or fid the P-value as explaied later i this sectio). The test statistic is z = p p (1 p ) Compare the value of z to the predetermied critical values. Iclude a sketch that illustrates the situatio.
Formal Laguage of Test Sigificace (Compoets of a Sigificace Test for a Proportio) 4. Write a coclusio. There are two parts to statig a coclusio: Say whether you reject the ull hypothesis or do t reject the ull hypothesis, likig your reaso to the results of your computatios. Do the whole aalysis for Miguel ad Kevi s spiig of peies gettig 1 of them heads. Tell what your coclusio meas i the cotext of the situatio. Note: You should ever say that you accept the ull hypothesis, because all other values i the cofidece iterval for p could be plausible values, you caot assert that p is the right oe. 21