INFERENCE ABOUT A POPULATION PROPORTION

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1 CHAPTER 19 INFERENCE ABOUT A POPULATION PROPORTION OVERVIEW I this chapter, we cosider iferece about a populatio proportio p based o the sample proportio cout of successes i the sample p ˆ = cout of observatios i the sample obtaied from a SRS of size, where X is the umber of successes (occurreces of the evet of iterest) i the sample. To use the methods of this chapter for iferece, the followig assumptios eed to be satisfied. The data are a SRS from the populatio of iterest. The populatio is much larger tha the sample. The sample size is sufficietly large. Guidelies for sample sizes are give. I this case, we ca treat ˆp as havig a distributio that is approximately Normal with mea μ = p ad stadard deviatio σ = p(1 p) /. A approximate level C cofidece iterval for p is p ˆ ± z * pˆ(1 pˆ) where z * is the critical value for the stadard Normal desity curve with area C betwee z * ad z *. pˆ(1 pˆ) The stadard error of p ˆ is give by. The margi of error associated with the cofidece iterval described above is z * pˆ(1 pˆ). Use this iterval oly whe the couts of successes ad failures i the sample are both at least

2 186 Chapter 19 The cofidece iterval procedure described above is ofte quite iaccurate uless the sample size is large. A more accurate cofidece iterval for smaller samples is the plus four cofidece iterval. To get this iterval, add four imagiary observatios - two successes ad two failures - to your sample. The, with these ew values for the umber of failures ad successes, use the previous formula for the approximate level C cofidece iterval. Use the plus four cofidece iterval whe the cofidece level C is at least 90% ad the sample size is at least 10 (with ay combiatio of successes ad failures). The sample size required to obtai a cofidece iterval of approximate margi of error m for a proportio is = * z m 2 p *(1 p*) where p * is a guessed value for the populatio proportio ad z * is the critical value for the stadard Normal desity curve with area C betwee z * ad z *. To guaratee that the margi of error of the cofidece iterval is less tha or equal to m o matter what the value of the populatio proportio may be, use a guessed value of p * = ½. Tests of the hypothesis H 0 : p = p 0 are based o the z statistic z = pˆ p 0 p0(1 p0) with P-values calculated from the stadard Normal distributio. Use this test whe p 0 10 ad (1 p 0 ) 10. GUIDED SOLUTIONS Exercise 19.1 KEY CONCEPTS: Parameters ad statistics, proportios (a) To what group does the study refer? Populatio = Parameter p = (b) A statistic is a umber computed from a sample. What is the size of the sample ad how may i the sample said they prayed at least oce i a while? From these umbers compute cout of successes i the sample p ˆ = = cout of observatios i the sample

3 Iferece about a Populatio Proportio 187 Exercise 19.4 KEY CONCEPTS: Whe to use the cofidece iterval procedure for iferece about a proportio Recall the assumptios eeded to safely use the methods of this chapter to compute a cofidece iterval: The data are a SRS from the populatio of iterest. The populatio is much larger tha the sample. For a cofidece iterval, is large eough that both the cout of successes p ˆ ad the cout of failures (1 p ˆ ) are 15 or more. These are the coditios we must check. Are all the coditios met? Exercise 19.8 KEY CONCEPTS - large sample cofidece iterval, plus four cofidece iterval for a proportio We are iterested i estimatig with 95% cofidece the proportio of America tees with a MySpace profile that post their picture. (a) The large-sample cofidece iterval will be give by p ˆ ± z * pˆ(1 pˆ), where pˆ is the sample proportio of successes, is the sample size, ad z * is the critical value for the stadard Normal desity curve with area.9500 betwee z * ad z *. I this problem, = ˆp = ad for 95% cofidece, usig Table C, z * = Use these to costruct a 95% large-sample cofidece iterval for the proportio p of all tees with profiles who iclude photos of themselves:

4 188 Chapter 19 (b) The plus four estimate of p is give by p = 95% cofidece iterval for p is give by p ± z * First, compute the estimate: p ~ = umber of successes i the sample + 2. The plus-four + 4 p ( 1 p ) + 4, where ad z* are uchaged from part (a). Costruct the plus four 95% cofidece iterval for p: Fially, compare the two cofidece itervals for p you costructed. Are the margis of error almost the same? What is the differece betwee these cofidece itervals? Exercise KEY CONCEPTS: Sample size, margi of error The sample size required to obtai a cofidece iterval of approximate margi of error m for a proportio is z * 2 = p m * (1 p * ) where p * is a guessed value for the populatio proportio ad z * is the critical value of the stadard Normal distributio for the desired level of cofidece. To apply this formula here we must determie m = desired margi of error = p * = a guessed value for the populatio proportio = z * = critical value eeded for a 90% cofidece iterval = From the statemet of the exercise, what are these values? Oce you have determied them, use the formula to compute the required sample size. = z * m 2 p * (1 p * ) =

5 Iferece about a Populatio Proportio 189 Exercise KEY CONCEPTS: Whe to use the z test for a proportio Recall that the (large sample) z test for a proportio is appropriate if (i) the sample ca be cosidered a SRS (ii) the populatio we re samplig from is much larger tha the sample (iii) both p 0 10 ad (1 p 0 ) 10. These are the coditios we must check i (a) ad (b). (a) (b) Exercise KEY CONCEPTS: Cofidece iterval for a proportio; the four-step process. The four step process follows. State. What is the practical questio that requires estimatig a parameter? Pla. Idetify the parameter, choose a level of cofidece, ad select the type of cofidece iterval that fits your situatio. Solve. Carry out the test i two phases: 1. Check the coditios for the iterval you pla to use. 2. Calculate the cofidece iterval. Coclude. Retur to the practical questio to describe your results i this settig. To apply the steps to this problem, here are some suggestios. You ca use Example 19.5 i the text as a guide. State. What is the populatio beig studied i this problem? What do the researchers hope to estimate?

6 190 Chapter 19 Pla. What parameter are the researchers iterested i estimatig? What is the level of cofidece to be used here? I this sectio, two differet cofidece iterval forms were cosidered. Which oe is recommeded? Write the formula here: Solve. First, check coditios: Ca we cosider the sample to be a SRS from the populatio? Is the populatio much larger tha the sample? Is the sample large eough? Your aswer here will deped upo the cofidece iterval method you chose i Pla, above. Our sample size is = 117. Of these, 68 use a seatbelt. Compute the estimate correspodig to the cofidece iterval method selected i Pla, above. What is the critical value eeded for a 95% cofidece iterval? z* = Compute the appropriate 95% cofidece iterval: Coclude. State ay coclusios i the cotext of this problem.

7 Iferece about a Populatio Proportio 191 Exercise KEY CONCEPTS: Testig hypotheses about a proportio; the four-step process. The four step process for tests of sigificace follows. State. What is the practical questio that requires a statistical test? Pla. Idetify the parameter, state ull ad alterative hypotheses, ad choose the type of test that fits your situatio. Solve. Carry out the test i three phases: (1) Check the coditios for the test you pla to use, (2) Calculate the test statistic, (3) Fid the P-value. Coclude. Retur to the practical questio to describe your results i this settig. To apply the steps to this problem, here are some suggestios. Use Example 19.7 i the text as a guide. State. State the problem. Pla. What is the parameter of iterest? What does the researcher suspect, or what is he/she tryig to show? Write the ull ad alterative hypotheses of iterest: What type of test should be used? Solve. First check that the appropriate coditios for iferece are satisfied. Is the sample a SRS, or ca it be treated as such? Is the sample size much smaller tha the populatio size? Are both p 0 10 ad (1 p 0 ) 10?

8 192 Chapter 19 Compute the sample proportio of female Hispaic drivers i Bosto who wear seatbelts. ˆ p = Calculate the test statistic: pˆ p0 z = = p0(1 p0) Fid the P-value: P-value = Coclude. I the cotext of this problem, what do you coclude? COMPLETE SOLUTIONS Exercise 19.1 (a) The populatio is presumably all college studets. The parameter p is the proportio of all college studets who pray at least oce i a while. (b) The statistic is ˆ p, the proportio i the sample who said that they prayed at least oce i a while. Exercise 19.4 ˆ p = 107/127 = Though it is ot explicitly stated, there may be little reaso to believe that we ca t treat this sample as a SRS from the populatio of iterest. The populatio of adult heterosexuals is extremely large compared with the sample size of 2673 adult heterosexuals. However, the umber of successes is ˆ p = = We do t have at least 15 successes i the sample. We ca t use the large-sample cofidece iterval to estimate the proportio p who share these two risk factors.

9 Iferece about a Populatio Proportio 193 Exercise 19.8 (a) The sample proportio of successes is p ˆ = 385 = Usig Table C, the critical value eeded 487 for a 95% cofidece iterval is z * = Hece, a 95% large-sample cofidece iterval for the proportio of tees with MySpace profiles that posted photos of themselves is give by p ˆ ± z * pˆ(1 pˆ) ±1.96 or ± or to ( ) 487 umber of successes i the sample + 2 (b) The plus four estimate of p is p = + 4 The correspodig plus four cofidece iterval for p is ( ) p ± z * p 1 p ( ) ± or ± or to = = The margis of error with these itervals agree to at least four decimal places. The plus four estimate pulls the ordiary sample proportio toward 0.50, so the iterval i (b) is shifted slightly. Exercise We start with the guess that p* = For 90% cofidece we use z* = The sample size we eed for a margi of error m = 0.04 is thus = z * m p*(1 p*) = 0.75(1 0.75) = We roud up to = 318. Thus, a sample of size 318 is eeded to estimate the proportio of Americas with at least oe Italia gradparet who ca taste PTC to withi ± 0.04 with 90% cofidece. Exercise (a) We see that p 0 = (10)(0.5) = 5 < 10, so the ormal approximatio to the biomial should ot be used i this case. (b) We see that p 0 = (200)(0.99) = ad (1 p0 ) = (200)(1 0.99) = (200)(0.01) = 2 < 10. The ormal approximatio to the biomial should ot be used i this case.

10 194 Chapter 19 Exercise State. Of all Hispaic female drivers i Bosto, what proportio use seatbelts? Pla. Let p deote the ukow proportio of all Hispaic female drivers i Bosto who use seatbelts. We will costruct a 95% cofidece iterval for this proportio. We should use the plus four cofidece iterval p ( 1 p ) p ± z *, where umber of successes i the sample + 2 p =. This is a more accurate cofidece iterval tha the more traditioal large-sample cofidece iterval also described i the text. Solve. First, we check whether coditios ecessary for use of this method are met. Depedig o how the 117 Hispaic female drivers i our sample were chose, it might be reasoable to treat this as a SRS of all Hispaic female drivers i Bosto. (It s easy, however, to believe that this is ot a SRS: Suppose, for example, that the sample cosists oly of motorists that were pulled over by a police officer for a movig violatio. It s doubtful that violatig motorists represet all motorists.) We will use the 95% cofidece level, which is larger tha the required 90% cofidece level. Fially, our sample of 117 Hispaic female drivers i Bosto is larger tha the required 10. All coditios are satisfied. For our sample, ~ p = = The required critical value is z* = Hece, a 95% cofidece iterval for the proportio p of Hispaic female drivers i Bosto that use seatbelts is p ± z * ( ) p 1 p ( ) ± ± or to Coclude. We estimate with 95% cofidece that betwee about 49% ad 67% of all Hispaic female drivers i Bosto use seatbelts. Exercise State. We would like to kow if more tha 50% of Hispaic female drivers i Bosto wear seatbelts. Pla. Let p be the proportio of all Hispaic female drivers i Bosto who wear seatbelts. The researcher woders whether this proportio is larger tha 0.5. We wat to test the hypotheses H 0 : p = 0.5 H a : p > 0.5 We ll use the large-sample sigificace test (z test) for a proportio.

11 Iferece about a Populatio Proportio 195 Solve. The sample is assumed to be a radom sample of all Hispaic female drivers i Bosto. The sample of = 117 drivers is reasoably large, but is obviously much smaller tha the populatio of all Hispaic female drivers i Bosto. Now, p 0 = (117)(0.5) = ad (1 p 0 ) = 117(1 0.5) = , so the coditios for iferece are met. Ivestigators observed a radom sample of 117 Hispaic female drivers ad foud that 68 of these drivers were wearig seatbelts. I our sample, the proportio of Hispaic female drivers wearig seatbelts was The computed test statistic is ˆ p = 68/117 = z = ˆ p p 0 ( ) p 0 1 p 0 = ( ) 117 = =1.76 The P-value is the area uder the stadard Normal desity to the right of z = 1.76, which is = Coclude. There is reasoably strog evidece that more tha half of all Hispaic female drivers i Bosto wear seatbelts. Note: You might woder why i Problem above, 50% was cotaied i a 95% cofidece iterval for p, while i Problem we reject 50% as a plausible value for p at the 5% level of sigificace. The most importat reaso is that ifereces made from cofidece itervals such as the oe used i Problem above coicide with two-sided tests of sigificace. Ideed, if i Problem we had a two-sided alterative hypotheses H a : μ.50, the correspodig P-value would have bee about.078, ad we would ot have rejected p =.50 as plausible.