I. Estimatig with Large Samples Probability & Statistis Poit Estimate of a parameter is a estimate of a populatio parameter give by a sigle umber. Use x (the sample mea) as a poit estimate for (the populatio mea) ad s (the sample stadard deviatio) as a poit estimate for (the populatio stadard deviatio). For large samples of size 30, s is a good estimate, for most pratial purposes. Error of estimate is the magitude of the differee betwee the poit estimate ad the true parameter value. Usig x as a poit estimate for, the error of estimate is the magitude of absolute value otatio x. The reliability of a estimate will be measured by the ofidee level,. x or i Theoretially, a be ay value betwee 0 ad, but usually is equal to a umber suh as 0.90, 0.95, 0.99. The value z is the umber suh that the area uder the stadard ormal urve fallig betwee z ad z is equal to. The value z is alled the ritial value for a ofidee level of. P z z z
Example (a) Is it true that the oditio P z0.99 z z0.99 0. is equivalet to the oditio P 0 z z0. 99 0. 99? Why? 99 (b) Use the iformatio of part a ad Table 5 of Appedix II to fid the value z. of 0. 99 Aswers. (a) It is true that the oditios are equivalet beause the stadard ormal urve is symmetrial about its mea 0. (b) To omplete the omputatio, we divide both sides of the equatio P 0 z z0. 99 0. 99 by, ad get the equivalet equatio 0.99 P 0 z z0. 99 0. 4950 We look up the area 0.4950 i Table 5 ad the fid the z value that produers that area. The value 0.4950 is ot i the table; however, the values 0.4949 ad 0.495 are i the table. Eve though 0.4950 is exatly halfway betwee the two values, the two values are so lose together we use the higher value 0.495. This gives us z.58. 0.99
Equatio () P z x z this uses the laguage of probability to give us a idea of the size of the error of estimate for the orrespodig ofidee level. The error of estimate (or absolute error) usig x as a poit estimate for is x. I most pratial problems, is ukow, so the error of estimate is also ukow. However, Equatio () allows us to ompute a error of tolerae E, whih serves as a boud error of estimate. Usig a % level of ofidee, we a say that the poit estimate x differs from the populatio mea by a maximal error tolerae of E z Sie s for large samples, we have s E z whe 30 where E is the maximal error tolerae o the error of estimate for a give ofidee level (i.e., x E with probability ); z is the ritial value for the ofidee level (see Table 8-); s is the sample stadard deviatio; is the sample size. For large samples 30 take from a distributio that is approximately moud-shaped ad symmetrial, ad for whih the populatio stadard deviatio is ukow, a ofidee iterval for the populatio mea is give by the followig: Cofidee Iterval for (Large Samples) x E x E where x = sample mea E z s s = sample stadard deviatio = ofidee level 0 z = ritial value for ofidee level (See Table 8- for frequetly used values.) = sample size 30
Example Walter usually meets Julia at the trak. He prefers to jog 3 miles. While Julia kept her reord, he also kept oe for his time required to jog 3 miles. For his 90 times, the mea was x. 50 miutes ad the stadard deviatio was s. 40 miutes. Let be the mea joggig time for the etire distributio of Walter s 3-mile ruig times over the past several years. How a we fid a 0.99 ofidee iterval for? (a) What is the value of z 0. 99? (See Table 8-) (b) Sie the sample size is large, what a use for? () What is the value of E? (d) What are the edpoits for a 0.99 ofidee iterval for? Aswers. (a) z 0.99. 58 (b) s. 40 () E z.40.58 0. 65 90 (d) The edpoits are give by x E.50 0.65.85 x E.50 0.65 3.5
Example 3 A large loa ompay speializes i makig automobile loas for used ars. The board of diretors wats to estimate the average amout loaed for ars durig the past year. The ompay takes a radom sample of 5 ustomer files for this period. The mea amout loaed for this sample of 5 loas is x $ 800 ad the stadard deviatio is s $ 750. Let be the mea of all ar loas made over the past year. Fid a 0.95 ofidee iterval for. Aswer. Sie 5 is a large sample, we take s 750. From Table 8-, we see that z. 96. The 0.95 s 750 E z.96 98 5 x E 800 98 $80 x E 800 98 $898 The iterval from $80 to $898 is a 0.95 ofidee iterval for. See Calulator Note P. 385.
Example 4 We have said that a sample size 30 or larger is a large sample. I this setio we idiated two importat reasos why our methods require large samples. What are these reasos? Aswers. Reaso : Reaso : Our methods require x to have approximately a ormal distributio. We kow from the etral limit theorem that this will be the ase for large samples. Uless we somehow kow, our methods require us to approximate with the sample stadard deviatio s. This approximatio will be good oly if the sample size is large.
II. Estimatig with Small Samples Studet s t distributio Used whe the sample size is small (less tha 30). t x s where x is the mea of a radom sample of measuremets, is the populatio mea of the x distributio, ad s is the sample stadard deviatio. Whe we use the t distributio, we will assume that the x distributio is ormal. Degrees of Freedom Table 6 i Appedix II gives the values of the variable t orrespodig to what is alled the umber of degrees of freedom d.f. = where d.f. stads for the degrees of freedom ad is the sample size beig used. The graph of a t distributio is always symmetrial about its mea. The mai differee betwee a t distributio ad the stadard ormal z distributio is that a t distributio has somewhat thiker tails.
Example 5 Use Table 6 i Appedix II to fid t, for a 0.90 ofidee level of a t distributio with sample size = 9. (a) (b) () (d) We fid the olum headed by =. This is the (first, seod, third, fourth, fifth, sixth) olum. The degrees of freedom are give by d.f. = =. Read dow the olum foud i part a util you reah the etry i the row headed by d.f. = 8. The value of t 0. 90 is for a sample size of 9. Fid t for a 0.95 ofidee level of a t distributio with sample size = 9. Aswers (a) = 0.90. This is the fourth olum. (b) d.f. = = 9 = 8 () t. 860 for sample size = 9. 0.90 0.95 (d) t. 306 for a sample size = 9.
C Cofidee Iterval for (Small Sample) x E x E where x = sample mea s E t = ofidee level (0 < < ) t = ritial value for ofidee level, ad degrees of freedom d.f. = take form the t distributio. = sample size (small samples, < 30) s = sample stadard deviatio Example 6 A ompay has a ew proess for maufaturig large artifiial sapphires. The produtio of eah gem is expesive, so the umber available for examiatio is limited. I a trial ru sapphires are produed. The mea weight for these gems is x 6.75arats, ad the sample stadard deviatio is s 0. 33arats. Let be the mea weight for the distributio of all sapphires produed by the ew proess. (a) What is d.f. for this settig? (b) Use Table 6 i Appedix II to fid t 0. 95. () Fid E. (d) Fid a 95% ofidee iterval for. (e) What assumptio about the distributio of all sapphires had to be made to obtai these aswers?
Aswers. (a) d.f. = where is the sample size. Sie =, d.f. = =. (b) Usig Table 6 with d.f. = ad = 0.95, we fid t. 0 s 0.33 () E t0.95 (.0) 0. (d) (e) x E x E 6.75 0. 6.75 0. 0.95 6.54 6.96 The populatio of artifiial sapphire weights is approximately ormal. See Calulator Note P. 398-399.
Summary: Cofidee Itervals for the Mea Large Sample Cases 30 ) If is ot kow, the a ofidee iterval for is s s x z x z ) If is kow, the a % ofidee iterval for is x z x z Small Sample Case < 30 If the populatio is approximately ormal ad is ot kow, the a % ofidee iterval for is s s x t x t Use d.f. = For Ay Sample Size If the populatio is ormal ad is kow, the for ay sample size (large or small) a % ofidee iterval for is x z x z
III. Estimatig p i the Biomial Distributio The biomial distributio is determied by the umber of trials ad the probability p of suess i a sigle trial. Let r be the umber of suesses out of trials i a biomial experimet. We will take the sample proportio of suesses pˆ (read p hat ) = r/ as our poit estimate for p, the populatio proportio of suesses. Poit estimate for p Poit estimate for q r pˆ qˆ pˆ If, p, ad q are suh that p > 5 ad q > 5 the a ofidee iterval for p is pˆ E p pˆ E r where pˆ pˆ pˆ E z z ritial value for ofidee level take from a ormal distributio (see Table 8-).
Example 7 A radom sample of 88 books purhased at a loal bookstore showed that 66 of the books were murder mysteries. Let p represet the proportio of books sold by this store that are murder mysteries. (a) What is a poit estimate for p? (b) Fid a 90% ofidee iterval for p. () What is the meaig of the ofidee level you just omputed? (d) To ompute the ofidee iterval, we used a ormal approximatio. Does this seem justified?
Aswers r 66 (a) p ˆ 0. 35 88 Probability & Statistis (b) E z pˆ pˆ 0.35 0.35.645 88 0.057 The ofidee iterval is pˆ E p pˆ E 0.35 0.057 p 0.35 0.057 0.9 p 0.4 () If we had omputed the iterval for may differet sets of 88 books, we would fid that about 90% of the itervals atually otaied p, the populatio proportio of mysteries. Cosequetly, we a be 90% ofidet that our iterval is oe of the oes that otais the ukow value p. (d) 88 p 0.35 q 0.65 Sie p 68.5 5 ad q. 5, the approximatio is justified. See Calulator Note P. 409-40.
Geeral Iterpretatio of Poll Results ) Whe a poll states the results of a survey, the proportio reported to respod i the desigated maer is pˆ, the sample estimate of the populatio proportio. ) The margi of error is the maximal error E of a 95% ofidee iterval for p. 3) A 95% ofidee iterval for the populatio proportio p is poll report pˆ -margi of error E < p < poll report pˆ +margi of error E How Poll Was Coduted The Wall Street Joural/NBC News poll was based o a atiowide telephoe iterviews of 508 adults oduted last Friday through Tuesday by the pollig orgaizatios of Peter Hart ad Robert Teeter. The sample was draw from 35 radomly seleted geographi poits i the otietal U.S. Eah regio was represeted i proportio to its populatio. Households were seleted by a method that gave all telephoe umbers, a equal hae of beig iluded. Oe adult, 8 years or older, was seleted from eah household by a proedure to provide the orret umber of male ad female respodets. Chaes are 9 of 0 that if all adults with telephoes i the U.S. had bee surveyed, the fidigs would differ from these poll results by o more tha.6 peretage poits i either diretio.
Example 8 (a) What ofidee level orrespods to the phrase haes are 9 of 0 that if? (b) The artile idiates that everyoe i the sample was asked the questio, Whih party, the Demorati Party or the Republia Party, do you thik would do a better job hadlig eduatio? Possible resposes were Demorats, either, both or Republias. The poll reported that 3% of the respodets said Demorats. Does 3% represet the sample statisti pˆ or the populatio parameter p for the proportio of adults respodig Demorat? () Cotiue readig the last paragraph of the artile. It goes o to state, if all adults with telephoes i the U.S. had bee surveyed, the fidigs would differ from these poll results by o more tha.6 peretage poits i either diretio. Use this iformatio together with parts a ad b to fid a 95% ofidee iterval for the proportio p of the speified populatio who would respod Demorat to the questio.
Aswers. (a) 9/0 = 0.95 A 95% ofidee iterval is beig disussed. (b) 3% represets a sample statisti pˆ beause 3% represets the peretage of the adults i the sample who respoded Demorats. () The value.6 peretage poits represets the margi of error. Sie the margi of error is equivalet to E, the maximal error of estimate for a 95% ofidee iterval, the ofidee iterval is 3%.6% p,3%.6% 9.4% p 34.6% The poll idiates that at the time of the poll, betwee 9.4% ad 34.6% of the speified populatio thik that Demorats would do a better job hadlig eduatio.
IV. Choosig the Sample Size Whe desigig statistial researh projets you eed to deide i advae o the ofidee level you wish to use ad selet the maximum error of estimate E you wat for your projet. Solvig E z for, we get the equatio z E Example 9 A large state uiversity has over 800 faulty members. The dea of faulty wats to estimate the average teahig experiee (i years) of the faulty members. A prelimiary radom sample of 60 faulty members yields a sample stadard deviatio of s = 3.4 years. The dea wats to be 99% ofidet that the sample mea x does ot differ from the populatio mea by more tha half a year. How large a sample should be used? (a) What value a we use to approximate? Why a we do this? (b) What is z 0. 99? (Hit: See Table 8-.) () (d) What is E for this problem? Use the formula for to fid the miimum sample size. Should your aswer be rouded up or dow to a whole umber? Aswers. (a) s = 3.4 years is a good approximatio beause a prelimiary sample of 60 is fairly large. (b) z 0.99. 58 () E = 0.5 year (d).583.4 z 307.8 E 0.5 Always roud up to the ext whole umber. Our fial aswer = 308 is the miimum size.
z p p E z 4 E Example 0 I Idiaapolis, the departmet of publi health wats to estimate the proportio of hildre (grades -8) who require orretive leses for their visio. A radom sample of hildre is take, ad r of these hildre are foud to require orretive leses. Let p be the true proportio of hildre requirig orretive leses. The health departmet wats to be 99% sure that the poit estimate pˆ r for p will be i error either way by less tha 0.03. (a) If o prelimiary study is made to estimate p, how large a sample should the health departmet use? (i) Whih formula (above) should we use? (ii) What is the value of E, ad what is the value of z i this problem? (iii) What is the value of? (b) A prelimiary radom sample of 00 hildre idiates that 3 require orretive leses. Usig the results of this prelimiary study, how large a sample should the health departmet use? (i) Whih formula should we use? (ii) What are the values of E ad z for this problem? (iii) What approximate value shall we use for p? (iv) What is the value of?
Aswers (a) z (i) beause we do ot have a estimate for p. 4 E (ii) E = 0.03 ad z. 58(see Table 8-) 0.99 z.58 (iii) 849 4 E 4 0.03 So without a prelimiary study to fid p, we will eed a sample size of at least = 849 hildre. (b) (i) p p z E beause we have a estimate of p from a prelimiary study. (ii) E = 0.03 ad z. 58 (iii) p 0. 3 0.99 z.58 E 0.03 Therefore, the sample size should be at least 30 hildre. (iv) p p 0.30.77 309. 83
V. Estimatig ad p p Probability & Statistis How a we tell if two populatios are differet? - Compare the differee i populatio meas or the differee i populatio proportios. What we eed to make a statistial estimate about the differee betwee two populatio parameters: a sample from eah populatio. The samples a be idepedet or depedet. Idepedet The way we take a sample from oe populatio is urelated to the seletio of sample data from the other populatio. These our very aturally whe we draw two radom samples, oe from the first populatio ad oe from the seod populatio. Depedet Samples are hose i a way that eah measuremet i oe sample a be aturally paired with a measuremet i the other sample. These our very aturally i before ad after situatios where the same objet or item is measured twie. All examples i this setio will ivolve idepedet radom samples. Example For eah experimet, ategorize the samplig as idepedet or depedet, ad explai your hoie. (a) I may medial experimets, a sample of subjets is radomly divided ito two groups. Oe group is give a speifi treatmet, ad the other group is give a plaebo. After a ertai period of time, both groups are measured for the same oditio. Do the measuremets from these two groups ostitute idepedet or depedet samples? (b) I a aoutability study, a group of studets i a Eglish ompositio ourse is give a pretest. After the ourse, the same studets are give a posttest over similar material. Are the two groups of sores idepedet or depedet?
Aswers. (a) (b) Sie the subjets are radomly assiged to the two treatmet groups, the resultig measuremets would form idepedet samples. Sie the pretest sores ad the posttest sores are from the same studets, the samples are depedet. Eah studet has both a pretest sore ad a posttest sore, so there is a atural pairig of data values. All examples i this setio will ivolve idepedet radom samples. Cofidee Iterval for (Large Samples) A ofidee iterval for is x x E x x E where x = sample mea for populatio s = sample stadard deviatio for populatio = sample size from populatio 30 x = sample mea for populatio s = sample stadard deviatio for populatio = sample size from populatio 30 s s E z z = ritial value for ofidee level (Table 8-) = ofidee level (0<<)
Cofidee Iterval for (Small Idepedet Samples) x x E x x E Where the samples are idepedet, ad the stadard deviatios are approximately equal. Populatio Populatio 30= sample size 30= sample size x = sample mea x = sample mea s = sample stadard deviatio s = sample stadard deviatio s s s. f. E ts = ofidee level, 0<< t = ritial value for ofidee level ad degrees of freedom d (See Table 6 i Appedix II)
Cofidee Iterval for p p (Large Samples) pˆ pˆ E p p pˆ p E ˆ where (usig the otatio of Theorem 8.3) pˆ E z qˆ ˆ ˆ pq = ofidee level, 0<< z = ritial value for ofidee level (Table 8-) = umber of trials i biomial experimet r umber of suesses i biomial experimet = umber of trials i biomial experimet r umber of suesses i biomial experimet r r pˆ ad qˆ r r pˆ ad qˆ We assume that all for quatities ˆp ˆq ˆp ˆq are greater 5.
Example (a) A study reported a 90% ofidee iterval for the differee of meas to be 0 0 For this iterval, what a you olude about the respetive values of ad? (b) A study reported a 95% ofidee iterval for the differee of proportios to be 0.3 p p 0. 6. From this iterval, what a you olude about the respetive values of p ad p? Aswers. (a) At a 905 level of ofidee, we a say that the differee is positive, so 0 ad. (b) At the 95% ofidee level, we see that the differee of proportios rages from egative to positive values. We aot tell from this iterval if p is greater tha p or p is less tha p. See Calulator Note P. 436