Basic formula for confidence intervals. Formulas for estimating population variance Normal Uniform Proportion

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1 Basic formula for the Chi-square test (Observed - Expected ) Expected Basic formula for cofidece itervals sˆ x ± Z ' Sample size adjustmet for fiite populatio (N * ) (N + - 1) Formulas for estimatig populatio variace Normal Uiform Proportio s b - a 6 s (b - a) 1 s p*q Poits o a Normal Distributio Oe Tail Two Tail 50 % % % % % % Poits o the Chi-square Distributio df Revised: September 8, 1999 / rtaylor@colorado.edu / Uiversity of Colorado - Boulder

2 z s e 110,000-0, Ł 6 ł You hypothesize that the average age of graduate studets o the Boulder Campus is 8 years, ad wat to test this hypothesis at the 5 percet level of sigificace. You take a sample of 100 graduate studets ad fid X _ 9.5 s^ 49 Would you reject or fail to reject? Why? Guess I ll start reviewig t-tests as the way to aswer this questio. t X _ - µ s^ Sice the t-value is above 1.96, you would reject the hypothesis, ad accept the alterative that the mea age is somethig other tha 8. (A aswer I had here which solved the problem a differet way ad said you would fail to reject the hypothesis was icorrect... ad I ca t figure out where those umbers came from.) You thik about 60 percet of the populatio are i favor of a ballot issue, but you would like to have a more precise estimate. How big of a sample would you have to take to be 95 percet sure your estimate was correct withi 3 percetage poits? 1.96 (.6)(.4) Compare ad cotrast probability ad o-probability samplig. Iclude a discussio of at least two specific methods for each (a total of four methods), ad the advatages ad disadvatages of each method you discuss.

3 z σ e 90,000 10, You would like to estimate the proportio of the populatio i favor of a ballot issue. How big of a sample would you have to take to be 95 percet sure your estimate was correct withi 3 percetage poits? 1.96 (.5)(.5) Describe a samplig distributio. Suppose you wated to determie the average umber of childre per household. You would like to be 95 percet sure your aswer is correct withi oe-half a child. You iitially assume that the umber of childre are uiformly distributed betwee 0 ad 6. How big of a sample should you take? Perhaps a trick questio, depedig o how far I got i the lecture. All of our previous problems assumed a ormal distributio i the populatio, ad we rage estimated the stadard deviatio of the populatio usig but here we are 6 assumig a uiform distributio i the populatio ad our estimate of the variace rage is. 1 z s e 1.96 (6-0) Suppose you took a sample of size 400 to determie the proportio of people i Boulder who work i Boulder. You foud this to be 35 percet. What is your 90 percet cofidece iterval o the proportio? X _ Z s or (.35)(.65) 400 or You wat to take a sample i Boulder to determie the average icome, ad wat to be 90 percet sure your estimate is correct withi $500. You estimate that most icomes rage from $0,000 to $110,000. What sample size should you take?

4 X _ Z s or or 3.65 She really could t be sure. Actually, with a sample mea of 3 which is right i the middle, ad the cofidece iterval beig balaced both ways, the true populatio mea could be i either directio. What if the sample mea was.5 or what bet would you make about the mea of the populatio? Bart calculated that he eeded to take a sample of 00 i order to estimate a populatio mea with 95% cofidece ad a estimatio error of 5 uits. The iterviews cost $15 each, ad he oly has $500 to sped. Specifically, what ca he do? He ca oly afford 166 iterviews. Based o that, you could calculate the level of cofidece ad/or the level of precisio which would give you 166 for a sample size. You have to icrease cofidece or icrease error (reduce precisio), or both. You would simply take the sample size formula z s e ad solve for the e or z which gives you the required. Suppose a researcher wats to estimate the mea aual expeditures for shampoo. She wats to be 90 percet sure that her estimate is correct withi $1. She believes that the rage of purchases is pretty well betwee $5 ad $65. What sample size should she take? z s 1.64 e A automobile dealership plas to coduct a survey to determie what proportio of ew car buyers cotiue to have their cars serviced at the dealership after the warraty period eds. It estimates that 30 percet of customers do so. It wats the results of its survey to be accurate withi 5 percetage poits, ad wats to be 95 percet cofidet of the results. What sample size is ecessary? 1.96 (.3)(.7) You wat to take a sample i Dever to determie the average icome, ad wat to be 95 percet sure your estimate is correct withi $1000. You estimate that most icomes rage from $10,000 to $90,000. What sample size should you take?

5 1.96 (.6)(.4) I this case, you wat to use the most coservative estimate of the proportio - the oe givig you the largest sample size. That would be.6 (ot the estimated.7 but the value closest to.5). Remember that for a proportio, s is PQ. Explai Systematic samplig, ad clearly explai why it is a form of Cluster samplig (which meas you also have to explai Cluster samplig). Compare ad cotrast Cluster ad Stratified samplig, clearly idicatig the purpose of each, how you sample i each, ad the advatages/disadvatages or beefits/limitatios of each. Paul is iterested i doig a sample of political opiios, ad is particularly iterested i the proportio of people who pla to vote for limitatios o smokig i public places. He wats to be 95% sure his aswer is correct withi 3 percetage poits. He does t have ay idea how may will say yes (or o). How big of a sample should he take? 1.96 (.5)(.5) Here you use.5 as the most coservative estimate of P. Lou wats to take a sample of studets to determie how much disposable icome they have each moth to sped for icidetals ad etertaimet. She believes that most studets have betwee $50 ad $350 to sped. She wats to be 90% sure her estimate is correct withi $5. How big of a sample should she take? z s 1.64 e Now I m dow to the poit of copyig solutios from above ad substitutig a differet set of umbers. At this poit, we have basically doe all variatios of the problems... ow it s just practice... or turig the formula back aroud ad computig cofidece itervals. Beth has take a sample of 50 people ad measured their attitudes about premarital sex. O oe scale the mea of the sample was 3.0, ad the variace of resposes was 16. What is her 99% cofidece iterval o the mea attitude i the populatio? If this is a 5 poit scale, with 1 beig i favor of ad 5 beig agaist, ca she be very sure that the average perso is agaist it? Why or why ot? (Drawig a picture of the distributio will help you.)

6 Marketig Research Sample Exam Questios Samplig Methods You wat to take a sample to determie the average icome, ad wat to be 90 percet sure your estimate is correct withi $500. You estimate that most icomes i the relevat populatio rage from $30,000 to $60,000. What sample size would you take? To determie sample size, you must have the level of cofidece, the degree of precisio, ad the variace of the populatio. You have to guess at the variace usig the rage. s rage 6 (60,000-30,000) 6 5,000 The you ca use the formula for sample size to fid z s e Suppose that you also kew that the populatio size was 500. What sample size would you take? The modificatio here is that with a fiite populatio, you do t have to take as large a sample. Usig the adjustmet factor we fid (ad I ca t remember the book s otatio, so I m usig as the sample size usig the ormal formula ad as the revised sample size - the book may have used the reverse otatio) ' N N (500) (69) ( ) 176 You wat to take a sample to estimate the mea age of a populatio. You expect most everyoe to be betwee the ages of 19 to 43. You would like to be 90 percet sure that your estimate is correct withi oe-half year. How big of a sample should you take. z s 1.64 e You would like to estimate the proportio of the populatio i favor of a ballot issue. You thik the aswer is aroud 70 percet ad are absolutely sure it is o greater tha 75 ad o less tha 60. How big of a sample would you have to take to be 95 percet sure your estimate was correct withi 5 percetage poits?

Math 124: Lecture for Week 10 of 17

Math 124: Lecture for Week 10 of 17 What we will do toight 1 Lecture for of 17 David Meredith Departmet of Mathematics Sa Fracisco State Uiversity 2 3 4 April 8, 2008 5 6 II Take the midterm. At the ed aswer the followig questio: To be revealed