Confidence Intervals for the Population Mean


 Edwin Webb
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1 Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios. So if X is distributed σ σ approximately ormally, P µ 2 < X < µ if we rearrage it so µ is σ σ i the middle the P x 2 < µ < x The iterval σ σ x 2, x + 2 has a probability of 0.95 of capturig the mea. Defiitio: If X is the sample mea of a radom sample of size from a populatio with variace 2 1 α 100% cofidece iterval for µ is give by σ, a ( ) σ X Z, X + Z α/2 α/2 σ Where Z α /2 is the Z value from the ormal table with area α /2 to its right. If σ, the populatio stadard deviatio is ukow, it ca be replaced by s, the sample stadard deviatio with o serious loss of accuracy for large sample cases. If we use α = 0.05, we report that we are 95% cofidet that the populatio mea will be withi our iterval. Why are we able to say we are 95% cofidet? We kow (from the Empirical Rule) that about 95% of all possible sample meas will lie withi two stadard errors of the actual populatio mea. We hope that our sample mea is oe of these, because if it is, the our cofidece iterval will cotai the populatio mea, ad our estimate will be correct. If ot, the our iterval will be icorrect. But this oly happes 5% of the time. The term "95% cofidece" meas that if we took repeated samples, ad foud a cofidece iterval for each sample, 95% of those cofidece itervals would actually cotai the populatio mea; 5% of them would ot. Whether our ow cofidece iterval cotais the populatio mea, we will ever kow! The Empirical Rule Theorem v.s. A Cofidece Iterval The Empirical Rule Theorem ad A Cofidece Iterval for the Mea are used to aswer two differet research questios. 1
2 Cofidece Itervals Math 283 The Empirical Rule Theorem is used to aswer the questio "Most of the values for the variable fall betwee what two values?" This is a rage of values used to discuss what we kow about the idividuals i our sample or populatio. A Cofidece Iterval is used to aswer the questio "What is the mea of the populatio?" This is a rage of values used to give reasoable values for the populatio mea. The average zic cocetratio recovered from a sample of zic measuremets i 36 differet locatios i a river is foud to be 2.6 grams per milliliter. Fid the 95% ad 99% cofidece itervals for the mea zic cocetratio i the river. Assume the populatio stadard deviatio is 0.3. A importat property of plastic clays is the percet of shrikage o dryig. For a certai type of plastic clay 45 test specimes showed a average shrikage of 18.4% with a stadard deviatio of 1.2. Estimate the mea percet shrikage for this type of clay with a 90% cofidece iterval. Aother way to thik about the cofidece iterval is: x ± MOE where MOE is the σ margi of error, MOE = Zα /2. Notice the width or precisio of our cofidece iterval depeds o cofidece level 1 α, sample size, ad stadard deviatio of the populatio. The accuracy of our sample mea depeds o the sample size,, the stadard deviatio of the populatio, σ, ad bias. 2
3 Cofidece Itervals Math 283 Decisio Makig with a Cofidece Iterval The owers of Geeral Light are plaig to advertise their light bulbs i the Suday editio of the ewspaper. I the ad, they wat to report "the mea lifetime of their light bulbs." To determie the mea lifetime of their light bulbs, they took a radom sample of 40 light bulbs. For their sample, the bulbs lasted o average, hours with a stadard deviatio of 58 hours. 1. Costruct a 95% cofidece iterval for the mea lifetime of light bulbs. 2. Should Geeral Light advertise that the mea lifetime of their light bulbs is 350 hours? Why or why ot? 3. Should Geeral Light advertise that the mea lifetime of their light bulbs is 310 hours? Why or why ot? Determiig Sample Size Whe our objective is to estimate the populatio mea, µ, we should do the followig to determie our sample size: 1. Determie the largest margi of error you are willig to accept ad a cofidece level. 2. Obtai or estimate the populatio stadard deviatio. σ 3. Fid the sample size,, that makes the followig true: Your MOE = Zα /2. 4. Check the sample size agaist your budget. If ecessary, retur to step 1. You are plaig a survey of startig salaries for liberal arts major graduates from you college. From a pilot study you estimate that the stadard deviatio is about $9000. What sample size do you eed to have a margi of error equal to $400 with 95% cofidece? 3
4 Cofidece Itervals Math 283 Cautios: Data must come from a SRS No correct method from data haphazardly collected with bias of ukow size. The sample mea is ot resistat to outliers. So look at your data carefully before determiig a CI. If is small ad populatio is ot ormal, the true cofidece level may be differet from what you used. o As log as 30, CLT applies o If 15, it is ok uless there are extreme outliers i.e. quite strog skewess. Must kow σ. The Case of the Ukow σ If X is the sample mea of a radom sample of size where X, 1 X are from a ormal distributio the the radom variable X µ t = s/ has a probability distributio called the tdistributio with degrees of freedom 1. Properties of the tdistributio (or Studet s tdistributio) Bell shaped with the mea zero. ν The variace where ν is the degrees of freedom. ν 2 The limitig distributio of the t is the stadard ormal distributio as goes to ifiity. See Table attached. A ( α ) 1 100% Cofidece Iterval for µ whe σ is ukow Let x ad s be the sample mea ad stadard deviatio of a radom sample of size from a ormally distributed populatio the the cofidece iterval is give by s X t, X + t α/2 α/2 where t α /2is the value from the tdistributio with degrees of freedom 1 ad α /2 is the upper tail probability. Note, this iterval is fairly robust to oormal data. If the data is ot too skewed, the t procedure is useful whe 15 < 40. Whe 40, the t procedure ca be used eve for skewed data. s 4
5 Cofidece Itervals Math 283 The cotets of 7 similar cotaiers of sulfuric acid are 9.8, 10.2, 10.4, 9.8, 10.0, 10.2, ad 9.6 liters. Fid a 95% cofidece iterval for the mea of all such cotaiers, assumig the data are from a ormal distributio. A radom sample of 12 graduates of a certai secretarial school typed a average of 79.3 words per miute with a sample stadard deviatio of 7.8 words per miute. Assumig a ormal distributio for the umber of words typed per miute, fid a 99% cofidece iterval for the mea umber of words per miute for all graduates. Cofidece Iterval for the Populatio Proportio If X, 1 X are idepedet observatios from a populatio with probability of success, the the radom variable X = X is distributed biomial with E ( X ) = p ad ( ) ( 1 ) i= 1 V X = p p. We showed that distributio as goes to ifiity. i Z = X p p ( 1 p) approaches the stadard ormal So the samplig distributio of pˆ = X / is approximately ormal with µ ˆp = p ad ( 1 p) p σ pˆ =. 5
6 Cofidece Itervals Math 283 A ( α ) 1 100% cofidece iterval for p, the populatio proportio is give by ( 1 ) ( 1 ) ˆ ˆ ˆ ˆ pˆ Z p p ˆ /2, p Z p p α + α/2 Where Z α /2 is the Z value from the ormal table with area α /2 to its right. A survey of 1280 studet loa borrowers foud that 448 had loas totalig more tha $20,000 for their udergraduate educatio. Give a 95% cofidece iterval for the proportio of all studet loa borrowers who have loas of $20,000 or more for their uder graduate degree. Determiig Sample Size Whe our objective is to estimate the populatio proportio, p, we should do the followig to determie our sample size: 1. Determie the largest margi of error you are willig to accept ad a cofidece level. 2. Determie p from previous study or use p = Fid the sample size,, that makes the followig true: p( 1 p) Your MOE = Zα /2. 4. Check the sample size agaist your budget. If ecessary, retur to step 1. You are plaig a evaluatio of a alcohol awareess program at your college that will take place six moths after the program. How large a sample should you take if you wat the margi of error for 95% to be about 0.1? 6
7 Cofidece Itervals Math 283 Studet s t Distributio Upper tail probability d.f
8 Cofidece Itervals Math The average weight of 40 radomly selected miivas was 4150 pouds. a. Fid ad iterpret a 98% cofidece iterval for the mea weight of all miivas. The stadard deviatio is kow to be 480 pouds. b. What could we do to reduce the width of this iterval? c. What are the advatages/disadvatages of your aswers i b? 2. The weight of grapefruit follows a ormal distributio. A radom sample of 12 ew hybrid grapefruit had a mea weight of 1.7 pouds with a stadard deviatio of 0.24 pouds. Fid a 95% cofidece iterval for the mea weight of the populatio of ew hybrid grapefruits. 3. A researcher wishes to estimate, withi $25, the true average amout of postage that parets of college studets sped each year. If she wishes to be 90% cofidet, how large a sample is ecessary? The stadard deviatio is kow to be $ A survey by Brides magazie foud that 8 out of 10 brides are plaig to take the surame of their ew husbad. How large a sample is eeded to estimate the true proportio to withi 3% with 98% cofidece? 5. A researcher wishes to estimate the proportio of adult females uder 5 feet tall. He wats to be 90% cofidet that his estimate is withi 5% of the true proportio. What sample size should he use? 6. I a survey of 200 workers, 169 said they were iterrupted three or more times a hour by phoe messages, faxes, etc. Fid ad iterpret a 90% cofidece iterval of the populatio of proportio of workers who are iterrupted three or more times a hour. 7. A sample of 17 states had these cigarette taxes (i cets): 112, 120, 98, 55, 71, 35, 99, 124, 64, 150, 150, 55, 100, 132, 35, 70, 93. Fid a 98% cofidece iterval for the mea cigarette tax i all 50 states. What assumptio is ecessary? 8
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