Chapter 20. Inference About a Population Proportion

Transcription

1 Chapter 20. Iferece About a Populatio Proportio 1 Chapter 20. Iferece About a Populatio Proportio The Sample Proportio ˆp Defiitio. The statistic that estimates the parameter p, a proportio of a populatio that has some property, is the sample proportio ˆp = umber of successes i the sample total umber of idividuals i the sample. The sample proportio ˆp is called p-hat. S Proportioed Stooges. A radom sample of size 20 of Three Stooges films icludes 9 films which have Curly i the role of third stooge. What is the statistic which this sample yields ad what parameter of the populatio does this estimate?

2 Chapter 20. Iferece About a Populatio Proportio 2 The Samplig Distributio of ˆp Defiitio. Draw a SRS of size from a large populatio that cotais proportio p of successes. Let ˆp be the sample proportio of successes, umber of successes i the sample ˆp =. The As the sample size icreases, the samplig distributio of ˆp becomes approximately ormal. The mea of the samplig distributio is p. The stadard deviatio of the samplig distributio is p(1 p). Note. The coditio for iferece about a proportio iclude: We ca regard our data as a simple radom sample (SRS) from the populatio. This is, as usual, the most importat coditio. The sample size is large eough to esure that the distributio of ˆp is close to ormal. We will see that

3 Chapter 20. Iferece About a Populatio Proportio 3 differet iferece procedures require differet aswers to the questio how large is large eough? Example. Exercise 20.3 page 495. Large-Sample Cofidece Itervals for a Proportio Note. The large-sample cofidece iterval for a populatio proportio test cosists of the followig steps: Draw a SRS of size from a large populatio that cotais a ukow proportio p of successes. A approximate level C cofidece iterval for p is ˆp ± z ˆp(1 ˆp) where z is the critical value for the stadard ormal desity curve with area C betwee z ad z. Use this iterval oly whe the umbers of successes ad failures i the sample are both at least 15. Example. Exercise page 509.

4 Chapter 20. Iferece About a Populatio Proportio 4 Accurate Cofidece Itervals for a Proportio Note. The plus four cofidece iterval for a proportio cosists of the followig steps: Draw a SRS of size from a large populatio that cotais a ukow proportio p of successes. To get the plus four cofidece iterval for p, add four imagiary observatios, two successes ad two failures. The use the largesample cofidece iterval with the ew sample size ( + 4) ad cout of successes (actual cout +2). Use the iterval whe the cofidece level is at least 90% ad the sample size is at least 10. (Dr. Bob says: Hmmmm... ) Choosig the Sample Size Note. The level C cofidece iterval for a populatio proportio p will have margi of error approximately equal to a specified value m whe the sample size is ( ) z 2 = p (1 p ) m where p is a guessed value for the sample proportio. The margi of error will be less tha or equal to m if you take the guess p to be 0.5.

5 Chapter 20. Iferece About a Populatio Proportio 5 Example S Curly Proportios. A stoogeologist wats to kow the proportio of Three Stooges fas who claim that their favorite third stooge is Curly. She wats to be C = 95% cofidet that the margi of error of her estimate is withi m = 3% of the populatio proportio. How large a sample must she take? Solutio. We have C = 95% ad m = 3% = The critical value that correspods to C = 95% = 0.95 is z = To be coservative, we take p = 0.5 ad we get from the above formula: ( ) z 2 ( ) = p (1 p ) = (0.5)(1 0.5) = m 0.03 We roud up to get that the desired sample should be size = 1068.

6 Chapter 20. Iferece About a Populatio Proportio 6 Sigificace Tests for a Proportio Note. The sigificace test for a proportio cosists of the followig steps: Draw a SRS of size from a large populatio that cotais a ukow proportio p of successes. To test the hypothesis H 0 : p = p 0, compute the z statistic z = ˆp p 0 p 0 (1 p 0 ) I terms of a variable Z havig the stadard ormal distributio, the approximate O-value for a test of H 0 agaist H a : p > p 0 is P(Z z) H a : p < p 0 is P(Z z) H a : p p 0 is 2P(Z z ) Use this test whe the sample size is so large that both p 0 ad (1 p 0 ) are 10 or more..

7 Chapter 20. Iferece About a Populatio Proportio 7 Example S Testig Shemp. The presidet of the Three Stooges Fa Club thiks that Shemp is uderappreciated ad that at least 20% of Three Stooges fas would cosider Shemp the fuiest stooge. He seds a questioaire to a radom sample of 100 of the members of the Fa Club ad all of the questioaires are retured. Oly 15 of the fas replied that Shemp was their favorite Stooge. Perform the relevat hypothesis test for the presidet. Solutio. We have p 0 = 0.20, = 100, ad ˆp = 15/100 = We compute the z statistic: z = ˆp p 0 p 0 (1 p 0 ) = (1 0.20) 100 = We have H 0 : p = p 0 = 0.20 ad H a : p 0.2. From Table A we fid that P(Z z) = P(Z 1.25) = = We would reject the ull hypothesis if P(Z z) were ear 0. It is ot, so we fail to reject the ull hypothesis ad the sample is ot supportive of H a ad so ot supportive of the Fa Club presidet s opiio. Example. Exercise page 511. rbg