Math 124: Lecture for Week 10 of 17

Size: px

Start display at page:

Download "Math 124: Lecture for Week 10 of 17"

Christian Webster
5 years ago
Views:

1 What we will do toight 1 Lecture for of 17 David Meredith Departmet of Mathematics Sa Fracisco State Uiversity April 8, II Take the midterm. At the ed aswer the followig questio: To be revealed i class.

2 Determie poit estimates i simple cases Retur i 15 miutes Suppose we wat to kow somethig about a populatio Average height Proportio with type B blood Percetage of flash bulbs that will work Take a sample ad measure it. The result is the poit estimate for the desired parameter. Explai the idea of iterval estimatio Determie poit estimates i simple cases The sample will ot measure the populatio exactly. What ca we lear about a populatio from the sample? I additio to the poit estimate, we will be able to costruct a iterval or rage aroud the poit estimate that we are cofidet cotais the actual populatio parameter. The poit estimate says: I thik the populatio parameter is about here. The iterval estimate says: I am cofidet that the populatio parameter is betwee these two limits. Suppose you wat to estimate the proportio of a populatio that has some characteristic The average height of me i the US The average SAT scores of eterig freshme at America uiversities The average class size at SF State The first step is to take a radom sample from your populatio, the largest ad best desiged that you ca afford The you measure it ad get a poit estimate for the desired parameter

3 Oce you have your poit estimate, what else ca you say? SOMETIMES you ca calculate a cofidece iterval, a pair of limits that you are 95% cofidet cotais the true populatio parameter Sample statistics, for the sample size you are usig, have to be ormally distributed either the factor has to be ormally distributed i the populatio or the sample size has to be larger tha 30. You have to kow the populatio stadard deviatio (coditio to be relaxed later) If m is the measured average of your sample of size, ad if σ is the populatio stadard deviatio for the same characteristic, the the 95% cofidece iterval is (m 2 σ, m + 2 σ ) Suppose you measured 50 me ad foud that their average height was 66.9", ad you kew that the stadard deviatio for male heights was 3.0". Sice the sample size if greater tha 30, you ca calculate the 95% cofidece iterval: ( , ) = (66.1, 67.7 ) We are 95% cofidet that the true average height lies i this iterval. The expressio 2 σ = = 0.8 is called the margi of error i the cofidece iterval. Questio 1: Fid the margi of error ad the 95% cofidece iterval for heights if the sample size is = 100. Did icreasig the sample size make the cofidece iterval bigger or smaller? What does "95% cofidet" mea? Why ot just say: "The probability that the true populatio mea lies i the iterval (65.8, 68.0 ) is 95%"? Because we ca t talk about the probability that the populatio has a certai value. Either the true populatio mea lies i the iterval (65.8, 68.0 ) or it does ot. There is othig probable about it. If the populatio mea was 66.9", the 95% of sample averages (for samples of size 30) would fall i the iterval. Whe we have kowledge of a sample istead of a populatio, we talk of cofidece istead of probabilities. Suppose we wated a margi of error < 0.33". How big a sample would we eed? 2 3 < < ( 2 3 ) 2 < < Use a sample size of = 331 or larger to get a margi of error smaller tha 0.33".

4 If m is the desired upper limit for the margi of error the: Suppose we wat a differet cofidece level. To fid cofidece level 90%, first fid z 90 such that m > actual margi of error m > 2 σ ( ) 2σ 2 > m P( z 90 < Z < z 90 ) = 0.90 P(Z < z 90 ) = 0.95 The 90% margi of error is z 90 σ. z 90 = Questio 2: What sample size is eeded to get a margi of error smaller tha 0.25" Questio 3: Fid the 90% cofidece iterval for heights if the sample size is = 100. Is this smaller or larger tha the 95% cofidece iterval for the same sample size? Usig the same methods, we could calculate: z 99 = z 95 = 1.96 But we always simplify to the more memorable z 95 = 2 Just remember, the cofidece iterval for cofidece level p is σ m ± z p z p σ is the margi of error for cofidece level p ad sample size ad populatio stadard deviatio σ If the sample size is large, say more tha 30 (ad certaily if it is more tha 100), you ca calculate the cofidece iterval usig the sample stadard deviatio s i place of the (perhaps ukow) populatio stadard deviatio σ. For smaller samples (provided the characteristic we wat to study is ormally distributed i the populatio) we will lear later how to replace σ with the sample stadard deviatio s ad replace z p with a slightly larger factor t p to get a margi of error ad cofidece iterval.

5 Determie poit estimates i simple cases Fid cofidece itervals for the populatio proportio Suppose you wat to estimate the proportio of a populatio that has some characteristic The proportio of me over 6 tall The proportio of wome who are married The proportio of ewbors who are female The first step is to take a radom sample from your populatio, the largest ad best desiged that you ca afford The you measure it ad get a poit estimate for the desired parameter Oce you have your poit estimate, you ca calculate a cofidece iterval for the populatio proportio The methods we will discuss assume that ˆp 10 ad (1 ˆp) 10 where is the sample size ad ˆp is the measured proportio i the sample. The methods are exactly the same as for estimatig the populatio mea, except that σ is replaced by ˆp(1 ˆp) Fid cofidece itervals for the populatio proportio Fid cofidece itervals for the populatio proportio O March 31, a poll of 730 Democratic voters i Pesylvaia foud that 47% favored Hillary Clito (ad 42% favored Barack Obama: /presidet/pa/pesylvaia_ democratic_primary-240.html). What is the 95% cofidece iterval for the proportio of voters favorig Clito? 0.47(1 0.47) The margi of error is m = 2 = or 4%. The cofidece iterval is 0.47 ± m = (0.43, 0.51). Based o the poll, we ca be 95% cofidet that betwee 43% ad 51% of Pesylvaia Democrats favor Clito. Questio 4: Aother poll of 406 Pesylvaia Democrats foud that 49% favored Clito. Based o this poll, fid the 95% margi of error ad the 95% cofidece iterval for the proportio of Pesylvaia Democrats favorig Clito.

6 The margi of error is less tha m = We must solve the followig equatio for : A typical disclaimer that is ofte give with poll results is: "this poll is accurate withi plus or mius 3 percetage poits". A more accurate statemet would be: "you ca be 95% cofidet that this poll is accurate withi plus or mius 3 percetage poits". How may people do you have to poll to achieve this level of cofidece? m > 2 ˆp(1 ˆp) But this equatio icludes the ukow ˆp. Before solvig it, we set ˆp = m > 2 m > 1 > 1 m 2 = = For a survey to be accurate to ±3% (at the 95% cofidece level), you must survey at least 1112 people. Questio 5: How may people must be surveyed to get a result accurate to ±2% at the 95% cofidece level?

CHAPTER 8 Estimating with Confidence

CHAPTER 8 Estimating with Confidence CHAPTER 8 Estimatig with Cofidece 8.2 Estimatig a Populatio Proportio The Practice of Statistics, 5th Editio Stares, Tabor, Yates, Moore Bedford Freema Worth Publishers Estimatig a Populatio Proportio