Interval Estimation & Sampel Size

Transcription

1 Iterval Estimatio & Sampel Size Week 02 Preseted by : W. Rofiato

2 Iterval Estimatio Iterval Estimatio of a Populatio Mea: - Large-Sample Case - Small-Sample Case Determiig the Sample Size Iterval Estimatio of a Populatio Proportio [ ] [ ] [ ]

3 Samplig Error The absolute value of the differece betwee a ubiased poit estimate ad the populatio parameter it estimates is called the samplig error. For the case of a sample mea estimatig a populatio mea, the samplig error is Samplig Error = Probability Statemets About the Samplig Error Kowledge of the samplig distributio of eables us to make probability statemets about the samplig error eve though the populatio mea is ot kow. A probability statemet about the samplig error is a precisio statemet.

4 Probability Statemets About the Samplig Error Precisio Statemet There is a 1 - probability that the value of a sample mea will provide a samplig error of or less. z / 2 Samplig distributio of 1 - of all /2 /2 values

5 Iterval Estimate of a Populatio Mea: Kow Margi of Error With Kow z /2 where: is the sample mea 1 - is the cofidece coefficiet z /2 is the z value providig a area of /2 i the upper tail of the stadard ormal probability distributio is the populatio stadard deviatio is the sample size

6 Iterval Estimate for Populatio Mea (Kow σ) Lloyd s Departmet Store wats to determie a 95% cofidece iterval o the average amout spet by all of its customers (populatio mea). Lloyd took a sample of 100 customers ad foud that the average amout spet is $82 (this is sample mea or ). Populatio stadard deviatio (σ ) is $20. ± z /2 σ = 82 ± = 82 ± 3.92 We are 95% sure that the average amout spet by all Lloyd customers is betwee $78.08 ad $85.92.

7 Yes Kow Case Ca the populatio stadard deviatio be assumed kow? No Use the sample stadard deviatio s to estimate s Use z /2 Ukow Case t Use /2 s

8 t Distributio Stadard ormal distributio t distributio (20 degrees of freedom) t distributio (10 degrees of freedom) 0 z, t

9 I. E. of a Populatio Mea: Ukow Iterval Estimate t / 2 s where 1 - = the cofidece coefficiet t /2 = the t value providig a area of /2 i the upper tail of a t distributio with - 1 degrees of freedom s = the sample stadard deviatio

10 Eample: Apartmet Rets Iterval Estimatio of a Populatio Mea: Ukow A reporter for a studet ewspaper is writig a article o the cost of off-campus housig. A sample of 10 oe-bedroom uits withi a half-mile of campus resulted i a sample mea of $550 per moth ad a sample stadard deviatio of $60. Let us provide a 95% cofidece iterval estimate of the mea ret per moth for the populatio of oe-bedroom uits withi a half-mile of campus. We ll assume this populatio to be ormally distributed.

11 Eample: Apartmet Rets t Value At 95% cofidece, 1 - =.95, =.05, ad /2 =.025. t.025 is based o - 1 = 10-1 = 9 degrees of freedom. I the t distributio table we see that t.025 = Degrees Area i Upper Tail of Freedom

12 Eample: Apartmet Rets Iterval Estimatio of a Populatio Mea: Ukow t or $ to $ We are 95% cofidet that the mea ret per moth for the populatio of oe-bedroom uits withi a half-mile of campus is betwee $ ad $ s 60 10

13 Sample Size for a Iterval Estimate of a Populatio Mea Let E = the maimum samplig error metioed i the precisio statemet. E is the amout added to ad subtracted from the poit estimate to obtai a iterval estimate. E is ofte referred to as the margi of error. We have Solvig for we have E z /2 z ( / 2 ) E 2 2 2

14 Eample: BetaMart, Ic. Suppose that Betamart maagemet team wats a estimate of the populatio mea such that there is a.95 probability that the samplig error is $500 or less. How large a sample size is eeded to meet the required precisio whe is 4,500? At 95% cofidece, z.025 = ad = 4,500. Solvig for we have (1.96) (4, 500) 2 2 (500) We eed to sample 312 to reach a desired precisio of + $500 at 95% cofidece.

15 Iterval Estimatio of a Populatio Proportio Iterval Estimate p z / 2 p( 1 p) where: 1 - is the cofidece coefficiet z /2 is the z value providig a area of /2 i the upper tail of the stadard ormal probability distributio p is the sample proportio

16 Eample: Political Sciece, Ic. Iterval Estimatio of a Populatio Proportio Political Sciece, Ic. (PSI) specializes i voter polls ad surveys desiged to keep political office seekers iformed of their positio i a race. Usig telephoe surveys, iterviewers ask registered voters who they would vote for if the electio were held that day. I a recet electio campaig, PSI foud that 220 registered voters, out of 500 cotacted, favored a particular cadidate. PSI wats to develop a 95% cofidece iterval estimate for the proportio of the populatio of registered voters that favors the cadidate.

17 Eample: Political Sciece, Ic. Iterval Estimate of a Populatio Proportio p p z / 2 p( 1 p) where: = 500, = 220/500 =.44, z /2 = ( 1. 44) 500 PSI is 95% cofidet that the proportio of all voters that favors the cadidate is betwee.3965 ad.4835.

18 Sample Size for a Iterval Estimate of a Populatio Proportio Let E = the maimum samplig error metioed i the precisio statemet. We have Solvig for we have E z / 2 p( 1 p) ( z ) p( 1 p) / 2 2 E 2

19 Eample: Political Sciece, Ic. Sample Size for Iterval Estimate of a Populatio Proportio Suppose that PSI would like a 0.99 probability that the sample proportio is withi ±.03 of the populatio proportio. How large a sample size is eeded to meet the required precisio? At 99% cofidece, z.005 = ( z/ 2 ) p( 1 p) ( ) (. 44)(. 56) 2 2 E (. 03) Note: We used.44 as the best estimate of p i the above epressio. If o iformatio is available about p, the.5 is ofte assumed because it provides the highest possible sample size. If we had used p =.5, the recommeded would have bee 1843.

20