Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.The Diversityof Samplesfrom theSame PopulationChapter 19Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.2Thought Question 1:a. If randomly sample 10 people, will exactly four(40%) disagree with law? Surprised if only two insample disagreed? How about if none disagreed?b. If randomly sample 1000 people, will exactly 400(40%) disagree with law? Surprised if only 200 insample disagreed? How about if none disagreed?c. Explain how long-run relative-frequencyinterpretation of probability and gambler’s fallacyhelped you answer parts a and b.40% of large population disagree with new law.In parts a and b, think about role of sample size.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.3Thought Question 2:a. Recalling Empirical Rule for bell-shaped curves, inwhat range would you expect 95% of women’sweights to fall?b. If randomly sampled 10 women at university, howclose do you think their average weight would beto 135 pounds? If sampled 1000 women, wouldyou expect average weight to be closer to 135pounds than for the sample of only 10 women?Mean weight of all women at large university is 135 pounds with a standard deviation of 10 pounds.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.4Thought Question 3:Survey of 1000 randomly selected individualshas a margin of error of about 3%, so resultsaccurate to within ± 3% most of the time.Suppose 25% of adults believe in reincarnation.If ten polls are taken, each asking a different randomsample of 1000 adults about belief in reincarnation,would you expect each poll to find exactly 25%of respondents expressing belief in reincarnation?If not, into what range would you expect theten sample proportions to reasonably fall?Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.519.1 Setting the StageWorking Backward from Samples to Populations• Start with question about population.• Collect a sample from the population, measure variable.• Answer question of interest for sample.• With statistics, determine how close such an answer,based on a sample, would tend to be from the actualanswer for the population.Understanding Dissimilarity among Samples• Suppose most samples are likely to provide an answerthat is within 10% of the population answer.• Then the population answer is expected to be within10% of whatever value the sample gave.• So, can make a good guess about the population value.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.619.2 What to Expect ofSample Proportions40% of population carry a certain geneDo Not Carry Gene = , Do Carry Gene = XA slice of the population:Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.7Sample 1: Proportion with gene = 12/25 = 0.48 = 48%Sample 2: Proportion with gene = 9/25 = 0.36 = 36%Sample 3: Proportion with gene = 10/25 = 0.40 = 40%Sample 4: Proportion with gene = 7/25 = 0.28 = 28%Possible Samples• Each sample gave a different answer.• Sample answer may or may not match population answer.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.81. There exists an actual population with fixedproportion who have a certain trait. OrThere exists a repeatable situation for which a certainoutcome is likely to occur with fixed probability.2. Random sample selected from population (soprobability of observing the trait is same for eachsample unit). OrSituation repeated numerous times, with outcomeeach time independent of all other times.3. Size of sample or number of repetitions is relativelylarge – large enough to see at least 5 of each of thetwo possible responses.Conditions for Rule for Sample ProportionsCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.9Example 1: Election PollsPollster wants to estimate proportion of voters whofavor a certain candidate. Voters are the populationunits, and favoring candidate is opinion of interest.Example 2: Television RatingsTV rating firm wants to estimate proportion ofhouseholds with television sets tuned to a certaintelevision program. Collection of all householdswith television sets makes up the population, andbeing tuned to program is trait of interest.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.10Example 3: Consumer PreferencesManufacturer of soft drinks wants to know whatproportion of consumers prefers new mixture ofingredients compared with old recipe. Populationconsists of all consumers, and response of interest ispreference of new formula over old one.Example 4: Testing ESPResearcher wants to know the probability thatpeople can successfully guess which of 5 symbols ison a hidden card. Each symbol is equally likely.Repeatable situation is a guess, and response ofinterest is successful guess.Is the probability of correct guess higher than 20%?Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.11If numerous samples or repetitions of the same size aretaken, the frequency curve made from proportions fromvarious samples will be approximately bell-shaped.Mean will be true proportion from the population.Standard deviation will be: (true proportion)(1 – true proportion)sample sizeDefining the Rule for Sample ProportionsCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.12Example 5: Using Rule for Sample ProportionsSuppose 40% of all voters in U.S. favor candidate X.Pollsters take a sample of 2400 people. What sampleproportion would be expected to favor candidate X?The sample proportion could be anything from a bell-shaped curve with mean 0.40 and standard deviation:(0.40)(1 – 0.40) = 0.01 2400• 68% chance sample proportion is between 39% and 41%• 95% chance sample proportion is between 38% and 42%• almost certain sample proportion is between 37% and 43%For our sample of 2400 people:Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.1319.3 What to Expect ofSample Means• Want to estimate average weight loss for all whoattend national weight-loss clinic for 10 weeks.• Unknown to us, population mean weight loss is8 pounds and standard deviation is 5 pounds.• If weight losses are approximately bell-shaped,95% of individual weight losses will fall between–2 (a gain of 2 pounds) and 18 pounds lost.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.14Results:Sample 1: Mean = 8.32 pounds, std dev = 4.74 poundsSample 2: Mean = 6.76 pounds, std dev = 4.73