Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

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1 CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter (the margi of error) ad what type of variability would ot be expected to happe by radom chace (statistical sigificace). : iterval of values computed from sample data that is likely to iclude the true populatio value. We use our kowledge of the sample to estimate the populatio value. : how likely it is (the probability) that a cofidece iterval actually captures the truth we are seekig (the populatio parameter of iterest). Noted as C. How to calculate a cofidece iterval sample statistic ± multiplier stadard error margi of error The multiplier determies the amout of cofidece we have that the iterval will cotai the populatio parameter. It is determied by the cofidece level C. Cofidece Iterval for a Populatio Proportio, p Two situatios to estimate a populatio proportio 1. A populatio exists ad we are iterested i kowig what proportio has a certai trait, opiio or characteristic, respose to a treatmet ad so o. 2. A repeatable situatio exists, ad we are iterested i the log-ru probability of a specific outcome Necessary Coditios (to assume ormal samplig distributio for ˆp ): 1. Probability of success is fixed (p does ot chage) 2. Radom Sample selected from the populatio (represetative sample) 3. Sample size: is large eough such that a. ˆp 10 b. (1- ˆp ) 10 1

2 Cofidece Iterval (CI): Sample Statistics: the sample proportio, ˆp Stadard Error: the stadard error of the sample proportio, se..( pˆ ) = pˆ(1 pˆ) Multiplier: the z-score, z* such that the area is betwee z ad +z is equivalet to the cofidece level C. Example: For a 95% cofidece iterval, fid z such that P(-z Z z) =.95 I other words, fid z such that PZ ( z) = (100 C%) / 2 Recall that Z = pˆ p, so the for a cofidece level C pˆ(1 pˆ) pˆ p P( z* Z + z* ) = P z* + z* = C pˆ(1 pˆ) Rearrage ad solve for p ˆ(1 ˆ) ˆ(1 ˆ) ˆ * p P p z p p pˆ z* p p + C = Therefore, we write the cofidece iterval at level C for p as ˆ(1 ˆ) ˆ * p p± z p or ˆ(1 ˆ) ˆ(1 ˆ) ˆ * p p z p, pˆ z* p p + 2

3 Example: I a recet Gallup Poll (9/23-9/25) of 1547 radomly selected people, the approval ratig of Presidet Obama was 52% approval. Durig the same week, Fox News took a poll of 900 radomly selected people ad foud that Presidet Obama s approval was at 54%. Calculate a 95% cofidece iterval for the proportio of the populatio that approve of Obama s job performace based o the Gallup Poll. Coditios Satisfied: Sample Statistic: Stadard Error: Multiplier: Cofidece Iterval: Iterpretatio of CI: With % cofidece, the proportio of fall betwee ad based o the. Are the two polls i disagreemet? Which approval ratig is correct? 3

4 Questios: Do you expect the cofidece iterval at a specified level C to be the same for all samples (thik about Gallup vs. Fox)? Why? Cosider the followig applet: At a cofidece level of 95%, how may of the itervals calculated for 100 samples do you expect cotai the actual populatio parameter? What happes to the size (the rage of the upper ad lower limits) as we icrease the cofidece level? What happes whe we decrease the cofidece level? What do you expect to happe to the size of the cofidece iterval as we icrease the sample size? What if we decrease the sample size? Why does this happe? Cofidece itervals are radom quatities which vary from sample to sample ad they may miss the true populatio parameter. Cofidece level is the proportio of possible samples for which the cofidece iterval will capture the true parameter. 4

5 I-Class Activity: A recet report of a sample survey of 1517 Americas reported that With 95% cofidece betwee 35% ad 42% of all Americas feel that Walmart has a positive ifluece o the ecoomy. Explai to someoe who kows o statistics what the phrase 95% cofidece meas i this report. 5

6 I Class Activity: The Mars Compay (the makers of M&Ms) recetly fired the factory maager at the M&M factory. They are worried that he might have tampered with the equipmet before he left, specifically; they thik he might have messed with machie the mixig machie that allots differet proportios of each color of M&Ms per bag (there is a fixed percetage of red, yellow, orage, brow, gree ad blue radomly mixed i each bag). I this activity we'll ivestigate the proportio of M&M's that are blue. To get a estimate of this proportio, we eed to take a sample. The sample proportio is subject to chace variability ad we ca express this by providig a cofidece iterval. Each group gets a bag of M&Ms (a idepedet radom sample). Follow the istructios below. 1. Your istructor will provide you with a bag of plai M&M's. Carefully ope the bag ad cout out 40 cadies, i a way that does ot bias your results. (For istace, pour them out oe at a time util you have 40.) What is the sample size? 2. Defie the populatio of iterest ad the parameter p beig estimated. Explai briefly. 3. Cout the umber of blue M&M's that you obtaied. Defie the sample ad the sample statistic p ˆ. We foud blue M&M's i my sample. As a fractio of the sample size this gives ˆ p =. 4. Estimate the stadard error for the proportio of blue M&M's i your sample. 6

7 5. Fid the z-score that you would use for a 80% cofidece iterval. 6. Calculate a 80% cofidece iterval for the proportio of blue M&Ms i your sample. Does our study meet the coditios uder which ifereces about the populatio ca be made (is it a radom sample, is the sample size big eough to use a CI for iferece?) My 80% cofidece iterval is ±, which goes from a proportio of up to blue based o my sample. 7. Calculate a 95% cofidece iterval for the proportio of blue M&Ms i your sample. A 95% cofidece iterval for the proportio of blue M&M's based o my sample would go from up to. 8. Whe you have completed your 80% ad 95% itervals type them ito the spread sheet. Accordig to the Departmet of Cosumer Iformatio of Mars, Ic., M&M's plai cadies are radomly mixed to cotai 24% blue cadies Did the 80% cofidece iterval that you costructed cotai the p = 0.24 figure give by the compay? Did the 95% CI cotai the parameter of iterest? 7

8 Differet samples were draw by each studet ad they each costructed their ow cofidece iterval. List all of the 80% cofidece itervals here. Group umber ˆ p ˆ se(p) Begiig of Iterval Ed of Iterval 9. How may of these 80% cofidece itervals icluded the p = 0.24 figure supplied by the compay? How does this compare with what you would expect? Explai. 8

9 Usig the iformatio from questio 6 for your whole class, list all of the 95% cofidece itervals here. Group umber ˆ p ˆ se(p) Begiig of Iterval Ed of Iterval 10. How may of the 95% Cofidece Itervals that were made by the studets i class today iclude the compay figure of 0.24? Is this close to what you expected? Explai. 9

10 Studet Names 11. What if I bought a 12 oz. bag of M&Ms ad calculated a 95% cofidece iterval for the proportio of blue M&Ms i the bag ad foud the same proportio of blues as you did. Would my cofidece iterval be smaller or wider tha your cofidece iterval? Explai. 12. Cosider your 95% cofidece iterval for the proportio of blue M&M s that you foud i the previous questios. Write a paragraph explaiig what 95% cofidece meas to someoe who kows o statistics. Use the results geerated i class as part of your explaatio. 13. Should the Mars Compay be worried about tamperig with the mixig machie? 10

11 Cofidece Itervals vs. Z-statistics Example: Suppose I cout the umber of orage M&Ms out of a pack of 60 ad fid 10 orage. With 95% cofidece, what is the true proportio of orage M&Ms amog all M&Ms (the populatio)? Example: M&M tells me that the true proportio of orage M&Ms is 17%. What is the probability that I got 10 out of 60 orage M&Ms? Whe to calculate a z-score z = pˆ p p(1 p) Whe to fid a z*. 11

12 Cofidece Iterval for a Populatio Proportio, p sample mea ± multiplier stadard error Steps to fid a Cofidece Iterval margi of error 1. Necessary Coditios (to assume ormal samplig distributio for ˆp ): a. Radom Sample selected from the populatio (represetative sample) b. Sample size: is large eough such that i. ˆp 10 ii. (1- ˆp ) Sample Statistics: calculate the sample proportio, umber of successes p ˆ = total sample size, 3. Stadard Error: calculate the stadard error of the sample proportio, se..( pˆ ) = pˆ(1 pˆ) 4. Multiplier: fid the z-score, z* such that the area is betwee z* ad +z* is equivalet to the cofidece level C. I other words, a. fid z such that PZ ( < z*) = (100 C) / 2%, OR b. fid z such that PZ ( > z*) = (100 C)/2% 5. Calculate the cofidece iterval for cofidece level C (use positive z*) ˆ(1 ˆ) ˆ * p p± z p or ˆ(1 ˆ) ˆ(1 ˆ) ˆ * p p z p, pˆ z* p p + 12

13 Cofidece Iterval for a Populatio Mea, µ Cofidece Iterval (CI): sample mea ± multiplier stadard error Sample Statistic: the sample mea, x margi of error Stadard Error: the stadard error of the sample mea, se( x) = Multiplier: Not the z-score why? 1. We do t kow the true variace of the populatio ad must use the stadard deviatio of the sample as a approximatio. 2. Eve if the sample is from a bell-shaped populatio, the z-score is ot a accurate represetatio if we estimate the stadard deviatio from a small sample. s : a smooth, symmetric curve ad its exact shape depeds o the size of the sample : oe less tha the sample size (df = 1). The shape of the t- distributio relys o the degrees of freedom. Greater the degrees of freedom, the closer it is to the Normal Normal distributio T-distributio, df=5 T-distributio, df=

14 Fidig the t-score for a cofidece level C 1. Calculate the degrees of freedom, df = Use the t-table to fid the value associate with the cofidece level C. I other words, fid t such that PT ( t) = (100 C %)/2correspodig to df=-1 Example: We wat to calculate a cofidece iterval for the mea height of me (which is bell-shaped) based o a sample of me. Fid the followig t-values: 95% CI for =10, 20 90% CI for =10, 20 Cofidece Iterval for the Populatio Mea, µ s s x ± t 1 or x t 1, x + t 1 margi of error s Example: I am iterested i the mea salary of graduate studets i the Biology Departmet at NCSU. I take a radom sample of 20 studets ad discover that the sample mea is $18,000 with a sample stadard deviatio of $2,500. The distributio of the salaries is bell-shaped. Calculate the 95% cofidece iterval for the mea salary of NCSU Biology graduate studets. Coditios Satisfied: Sample Statistic: Stadard Error: Multiplier: 14

15 Cofidece Iterval: Iterpretatio of CI: Questios: Do you expect the cofidece iterval at a specified level C to be the same for all samples? Why? At a cofidece level of 95%, how may of the itervals calculated for 100 samples of size do you expect cotai the actual populatio parameter? What happes to the size (the rage of the upper ad lower limits) as we icrease the cofidece level? What happes whe we decrease the cofidece level? Sample size ad the margi of error Proportio: Margi of Error is ˆ(1 ˆ) z * p p s Mea: Margi of Error is t 1 How does t chage as the sample size icrease? How does the stadard error chage as the sample size icreases? How, the, will the margi of error chage (icrease/decrease) as the sample size icreases? 15

16 Cofidece Iterval for the Populatio Mea, µ sample mea ± multiplier stadard error margi of error Steps to fid a Cofidece Iterval for the Populatio Mea µ 1. Necessary Coditios: a. Radom Sample selected from the populatio (represetative sample) b. Sample size: is large eough such that i. is small, but the origial distributio is bell-shaped/symmetric ii. 30, if the origial distributio is ukow Note: Eve though we try to follow guidelies about the distributio of the data, the ifereces usig the t-statistic are robust, meaig they are ot as sesitive to outliers ad skew i data ifereces usig the z-statistic. 2. Sample Statistics: calculate the sample mea x 3. Stadard Error: calculate the stadard error of the sample mea, se..( x) = s 4. Multiplier: t -1 a. Calculate the degrees of freedom, df = -1 b. Fid the t-score with df = -1 such that the area is betwee t* ad +t* is equivalet to the cofidece level C. 5. Calculate the cofidece iterval for cofidece level C x t s ± 1 or 1 1 s x t, x + t s Statcruch>>Stat>>T statistics>>oe Sample>> Select Colum>>CI & Level 16

17 Notes o Cofidece Itervals Here are 50 cofidece itervals for the mea ( The red itervals are the oes that did ot capture the true populatio mea. Samplig Distributio of the Sample Mea 95% Cofidece Itervals for the Populatio Mea Why do we expect some of the itervals to ot capture the true mea? What do you otice about the meas associated with the red itervals? Where do they fall o the samplig distributio of the sample meas? If the samplig distributio of the sample meas is ormal, why do we use a t-distributio to calculate the cofidece iterval? 17

18 Iterpretatio of Cofidece Itervals Example 1: Commercial growers of orametal shrubs ofte wat to retard the growth of the shrubs so that they do ot become too big before sale. I a experimet, 10 radomly selected shrubs were selected ad treated with a growth retardat ad aother 10 shrubs were radomly selected as the cotrol ad ot treated. After 13 weeks, the heights of all the shrubs were measured (You may assume that the distributio of heights is symmetric). Here are the results: Shrubs x s Utreated Treated Costruct a 90% cofidece iterval for both the mea height of the treated shrubs ad the mea height of the utreated shrubs. Iterpret the results. Do the two cofidece itervals overlap? What are the possible coclusios ca you make about the effectiveess of the experimetal growth retardat? 18

19 Example 2: I a recet poll of 1001 adults atiowide, the AP reports that 41% of Americas believe that laws limitig gu owership do ot ifrige o the public s right to bear arms uder the secod amedmet of the U.S. Costitutio. I additio, they reported a margi of error of ±3.1%. How did they determie the margi of error? Calculate a 90% CI for the proportio of Americas who believe that gu laws ifrige o the public s Costitutioal rights. The ews is ow reportig that 43% of Americas believe that laws limitig gu owership ifrige o the public s right to bear arms. Gu-law activists are toutig this as a sig that public opiio is ow more i favor of stricter gu laws. Is this a accurate statemet? Is there eough evidece to prove the activists claim is correct? 19

20 Oe-miute paper: Watch the followig youtube.com clip of a perso explaiig the iterpretatio of the cofidece iterval (igore the fact that he uses a z-score istead of a t-score for the multiplier). Cosider the iterpretatio of the cofidece iterval at the ed of the video. What is wrog with his iterpretatio? What is the correct way to iterpret the cofidece iterval? 20

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