Estimation and Confidence Intervals: Additional Topics

Transcription

1 Chapte 8 Etimation and Confidence Inteval: Additional Topic Thi chapte imply follow the method in Chapte 7 fo foming confidence inteval The text i a bit dioganized hee o hopefully we can implify Etimation: Additional Topic Chapte Topic Population Mean, Dependent Sample- OMIT Example : Population Mean, Independent Sample Population Popotion OMIT Vaiance Same goup befoe v. afte teatment Goup v. independent Goup Popotion v. Popotion Vaiance of a nomal ditibution Statitic fo Bui ne and Economi c, 6e 007 Peaon Education, Inc. Chap 9-3 Figue 8.:

2 CHAPTER 8. ESTIMATION AND CONFIDENCE INTERVALS: ADDITIONAL TOPICS 8. Popotion of Succee Theameideacanbeuedtofind point and inteval etimate of popotion. Suppoe, fo example, that we have obevation, with uccee, o that Then the numbe of uccee ha ˆ = = (8.) () =; () =( ) (8.) while the popotion of uccee ha (ˆ) =; [] = ( ) (8.3) We can etimate the vaiance of the popotion by ( ) = (8.4) Then we imply ue the nomal appoximation to the binomial: = ˆ p ( ) (8.5) The 00( ) %confidence inteval fo ˆ when n i lage i: {ˆ p ( ) ˆ + p ( )} =( ) (8.6) Notice that thi expeion involve the unknown,oweaegoingtoeplace by it etimato f to give: { pˆ( ˆ)+ pˆ( ˆ)} =( ) (8.7) Keep in mind that thi appoximation hold only fo lage ample

3 8.. CONFIDENCE INTERVAL FOR THE DIFFERENCE BETWEEN MEANS Example of Coinfidence Inteval fo Population Popotion Aampleof00 vote ae aked about a municipal bond iue; 64 favou it. What i a 95 pecent confidence inteval fo the popotion who favou it in the entie municipal population? Hee =00, =64,andˆ = 64. So the 95 pecent confidence inteval i (55 73). (Can you get thee numbe?) Intepet thi inteval. 8. Confidence Inteval fo the Diffeence Between Mean It look like thee ae a whole bunch of cae to emembe hee, but actually we ae jut eapplying the ame pinciple ove and ove Fo intance, aume that we have two independent ample of ize and. = = Let the mean and vaiance be and 8.. Known Vaiance Let u aume that the ample ae both nomal, it hould be obviou afte ome thinking that the ( )% fo known vaiance i = ± (8.8) + ince the vaiable ae independent (the vaiance of the um i the um of the vaiance) [ ]= [ ]+ [ ] fo any

4 4CHAPTER 8. ESTIMATION AND CONFIDENCE INTERVALS: ADDITIONAL TOPICS 8.. Unknown Vaiance with Vaiance Peumed Diffeent If the vaiance ae unknown (and aumed diffeent) which i uually the cae we want to etimate by : = ± ( ) (8.9) + In my own wok I have ued = + wheea NCT ue a teibly complicate expeion: = h + i + ( ) ( ) Of coue a and get lage (tandad nomal) and it make no diffeence which we ue fo 8..3 Unknown Vaiance that ae Aumed to be Equal: = = Notice then that = + = + we can etimate by uing a pooled etimato ( ) = +( ) + and the etimate fo = + i = + which can be ubtituted in (8.9) with = +

5 8.3. CONFIDENCE INTERVALS FOR THE DIFFERENCES IN POPULATION PROPORTIONS5 8.3 Confidence Inteval fo the Diffeence in Population Popotion Thi hould be taightfowad a anothe example: Recall fom Chapte 7, conide two independent population.. Population : = numbe of uccee, = numbe in ample, o ˆ =. Recall [ ]= [ˆ ]= ( ) µ ˆ ( ( ) aymptotically. Population : : = numbe of uccee, = numbe in ample, o ˆ = [ˆ ]= [ˆ ]= ( ) µ ˆ ( ( ) aymptotically 3. Fom Diffeence of Sample Popotion: ˆ ˆ If and ae lage, i.e. ( ) 9 and ( ) 9 then: ˆ ˆ ( ( ) + ( ) ) 4. So the confidence inteval i ˆ ˆ ± ˆ ( ˆ ) + ˆ ( ˆ )

6 6CHAPTER 8. ESTIMATION AND CONFIDENCE INTERVALS: ADDITIONAL TOPICS 8.4 SampleSizeDetemination 8.4. Sample Size Detemination fo Population Mean fom known vaiance Let ( ) ampling eo bound (called magin of eo in text) in any confidence inteval fo ˆ ˆ ± ( ) ( ) i a function of the ample ize, o that we can fo a given ize fo calculate a ingle ample ize Fo intance, fo the confidence inteval of the population mean fom known vaiance we have ( ) = which we could eaange to olve fo Example ( ) = = ( ) Suppoe we have metal od in an indutial pove with a =8 and want a 99% confidence inteval that extend no futhe than 0.50 on eithe ide of ample mean ( (099) = 50)Whatitheequiedampleize? Anwe( = 005 =576): ( ) = (576) (8) (050) Sample Size Detemination fo Population Popotion In thi cae ( ) = ( ) Notice that ince ( ) involve the ample popotion (which obviouly depend on ) we cannot imply calculate thi diectly Intead we can et an uppe limit fo by auming a ample popotion of 0.5 (thi give the laget value of and hence i the mot conevative) o that 05(05) ( ) = 05 = =05 q

7 8.4. SAMPLE SIZE DETERMINATION 7 which lead to 05 = = 5 ( ) ( ) Suppoe a polling fim want to have no moe than 3% band aound the population popotion confidence inteval at the 95% level, how lage a polling ample hould be obtained Anwe ( = 05 =96) = 5 ( ) = people