AdditionalExercises03

AdditionalExercises03 - Q , 1‘. 9000 m z: SCH: .m;...

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Unformatted text preview: Q , 1‘. 9000 m z: SCH: .m; E’Zsli I? goo :PQ‘WQQWGU) gaff: 1",“; ‘Jfi \IT'Q j; NM‘JIE'L’ w m «f i? mi moo (‘QQCP‘JMQ Mfr-1:146 31600 = "l {a 1? F“ Hm I} 5...”, ’9‘ [mm 56vpr 7139. 1.“; qufimM 5 “WK-“WK Mi’bplfl‘rfl? 5,1: :17. . WM I (:9 human Wm ’21 Mm“ :C'Z'LJCMLC “JFK—71)¢’l @S’L ’ H 10::- C53 “c? T‘ 3"” W R I“ x ,_ (1610(Ian 6.33 Tr??? ‘3' Ln) m 1@ gum \m C )(T 3‘ < )r “Idlés 1. r NF; ’1__ "-5) W ":— 3' T J7 5mm. Minam— wriwiflo M Ma merqueL, 1 Emrflfhfil) @g‘a (a) a" 3} Jami": , 5—1.; O'UQQ’L Owfl-Ol ‘Zfisvlwa W Eng/LE ~14“. 6'1 1.51.: or; the cjnqg'gLuol ‘25-{4' -‘: 11mth _ ‘fl-DD'L}i(1*31{)(‘fl‘fiuai)/© = (lfllflflglfrmflflg (k3mz 1*} '5' gym I L1, 31;)(fl-flauflg} NEIL ——— Q’MM calm/ML WWW; mgfibé ,h M m: HEM“ , Wk 49>! : Cw LHW'Q I-dl'l‘ m , m :4 M a @‘r—HMWW kffim m" LW‘J) @‘5’51- Qt) Fig-#21? H-n’fl <1r F—q (19) Ezflfigfl PM." (c) grip-:31; H1: I‘Wflh’ " MM (5563”) @M‘ P“: meflt Omhhrgh M ?$_-:, WWML g M r]- H’Dl' Erner Pf. 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OI‘IX (I 'Dggo'm ,_ = '1. 3,: yd: 23k; ’@ [fig-Wm C; 43 639 Chapter 6: Introduction to Inference Here are the Degree of Reading Power (DRP) scores for a sample of 44 third~grade students: www.mm—WW 40 26 39 14 42 18 25 43 46 2? 19 47 19 26 35 34 15 44 40 38 31 46 52 25 35 35 33 29 34 41 49 28 52 47 35 48 22. 33 41 51 27 14 54 45 Suppose that the standard deviation of the population of DR? scores is known to be cr = 11. Give a 95% confidence interval for the population mean score. The Confidence Intervals applet lets you simulate large numbers of confidence intervals quickly. Select 95% confidence and then sample 50 intervals. Record the number of intervals that cover the true value (this appears in the “hit” box in the applet). Press the reset button and repeat 30 times. Make a sternplot of the results and find the mean. Describe the results. If you repeated this experiment very many times, what would you expect the average number of hits to be? Repeat the previous exercise [or 80% confidence You are planning a survey of starting salaries for recent liberal arts major graduates from your college. From a pilot study you estimate that the standard deviation is about $8000. What sample size do you need to have a margin of error equal to $500 with 95% confidence? Suppose that in the setting of the previous exercise you are willing to settle _" for a margin of error of $1000. Will the required sample size be larger or smaller? Verify your answer by performing the calculations. How large a sample of one~bedroom apartments in Exercise 6.2 would be - needed to estimate the mean ,u within $3320 with 90% confidence? Researchers planning a study of the reading ability of third-grade children want to obtain a 95% confidence interval for the population mean score 011' a reading test, with margin of error no greater than 3 points. They carry 1 out a small pilot study to estimate the variability of test scores. The sample standard deviation is s 2 12 points in the pilot study, so in preliminary ' calculations the researchers take the population standard deviation to be .5: 0' I 12. '- {a} The study budget will allow as many as 100 students. Calculate the margin of error of the 95% confidence interval for the population based on n = 100. (3.1:) There are many other demands on the research budget. If all of these demands were met, there would be funds to measure only 10 child 'ii. What is the margin of error of the confidence interval based on a = '-.' measurements? . i :ores is 3' lulation'?‘ of _ nple SO' Ll€(tl'1l3;' cl repeat, cribe t would you I. 2 would ice? " .de chit ._ :an scor' hey ca _ The 8 ll -: ‘limin ' than to --- ulate the? ulation i'_. all of th r 10 child _-- ed on n 6.20 6.21 6.23 Section 6.1 Exercises 433 {c} Find the smallest value of n that would satisfy the goal of a 95% confidence interval with margin of error 3 or less. Is this sample size within the limits of the budget? How large a sample of Julie’s potassium levels in Exercise 6.1] would be needed to estimate her mean ,u within :006 with 95% confidence? In Exercises 6.? and (3.10, we compared confidence intervals based on corn yields from 35 and 50 small plots of ground. How large a sample is required to estimate the mean yield within :6 bushels per acre with 90% confidence? To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The scale readings are normally distributed with unknown mean (this mean is 10 grams if the scale has no bias). The standard deviation of the scale readings is known to be 0.0002 gram. / ta} The weight is weighed five times. The mean result is 10.0023 grams. Give a 98% confidence interval for the mean of repeated measurements of the weight. {b} How many measurements must be averaged to get a margin of error of i0.0001 with 98% confidence? The Gallup Poll asked 1571 adults what they considered to be the most serious problem facing the nation’s public schools; 30% said drugs. This sample percent is an estimate of the percent of all adults who think that drugs are the schools’ most serious problem. The news article reporting the poll result adds, “The poll has a margin of errorethe measure of its statistical accuracy—wotC three percentage points in either direction; aside from this imprecision inherent in using a sample to represent the whole, such practical factors as the wording of questions can affect how closely a poll reflects the opinion of the public in general.” The Gallup Poll uses a complex multistage sample design, but the sample percent has approximately a normal distribution. Moreover, it is standard practice to announce the margin of error for a 95% confidence interval unless a different confidence level is stated. {a} The announced poll result was 30% :r 3%, Can we be certain that the true population percent falls in this interval? (bl Explain to someone who knows no statistics what the announced result 30% i 3% means. (c) This confidence interval has the same form we have met earlier: estimate : {ca-estimate {Actually 0- is estimated from the data, but we ignore this for now.) What is the standard deviation trauma. of the estimated percent? {at} Does the announced margin of error include errors due to practical problems such as undercoverage and nonresponse? 434 6.24 69215 6.26 6.27 Chapter 6: Introduction to infereu When the statistic that estimates an unknown parameter has a normal distribution, a confidence interval for the parameter has the form estimate i 3," new”. In a complex sample survey design, the appropriate unbiased estimate of the population mean and the standard deviation of this estimate may require elaborate computations. But when the estimate is known to have a normal distribution and its standard deviation is given, we can calculate a confidence interval for it from complex sample designs without knowing the formulas that led to the numbers given. A report based on the Current Population Survey estimates the 1999 median annual earnings of households as $40,816 and also estimates that the standard deviation of this estimate is $19]. The Current Population Survey uses an elaborate multistage sampling design to select a sample of about 50,000 households. The sampling distribution of the estimated median income is approximately normal. Give a 95% confidence inteival for the 1999 median annual earnings of households. The previous problem reports data on the median household income for the entire United States. In a detailed report based on the same sample survey. you find that the estimated median income for four-person families ' in Michigan is $65,467. Is the margin of error for this estimate with 95% confidence greater or less than the margin of error for the national median, W'hv? As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That's the meaning of 95% confidence. If we use several such intervals, however, our confidence that all give correct results is less than 95%. Suppose we are interested in confidence intervals for the median household incomes for three states. We compute a 95% interval for each of the three, based on independent samples in the three states. (a) What is the probability that all three intervals cover the true median incomes? This probability (expressed as a percent) is our overall confidence level for the three simultaneous statements. (b) What is the probability that at least two of the three intervals cover the I true median incomes? The Bowl Championship Series (BCS) was designed to select the top two teams in college football for a final championship game. The teams are selected by a complicated formula. In 2001, the University of Miami Hurricanes and the University of Nebraska Cornhuskers played for the championship. However, many football fans thought that Nebraska should not have played in the game because it was rated only fourth in both major opinion polls. Third-ranked University of Colorado fans were particularly come for 1: sample I 3011 lamili with 95% final medi' in tervaL cover the use several _ tits is less '- edian for each 6 median verall tls cover the -._' e top {B teams if Miami for the she should both major Lrticularly 6.28 6.29 6.30 6.2 435 6.2 Tests of Significance upset because Colorado soundly beat the Cornhuskers late in the season. A new CNN/USA Todanyallup poll reports that a majority of fans would prefer a national championship playoff as an alternative to the BCS. The news media polled a random sample of 1019 adults 18 years of age or older. A summary of the results states that 54% prefer the play-off, and the margin of error is 3% for 95% confidence. (a) Give the 95% confidence interval. (b) Do you think that a newspaper headline stating that a majority of fans prefer a play-off is justified by the results of this study? Explain your answer. An advertisement in the student newspaper asks you to consider working for a telemarketing company. The ad states, "Earn between $500 and $1000 per week.” Do you think that the ad is describing a confidence interval? Explain your answer. A New York Times poll on women’s issues interviewed 1025 women and 472 men randomly selected from the United States, excluding Alaska and Hawaii. The poll found that 47% of the women said they do not get enough time for themselves. (a) The poll announced a margin of error of t3 percentage points for 95% confidence in conclusions about women. Explain to someone who knows no statistics why we can't just say that 47% of all adult women do not get enough time for themselves. (h) Then explain clearly what “95% confidence" means. (c) The margin of error for results concerning men was :4 percentage points. Why is this larger than the margin of error for women? A radio talk show invites listeners to enter a dispute about a proposed pay increase for city council members. “What yearly pay do you think council members should get? Call us with your number.” In all, 958 people call. The mean pay they suggest is 2? : $9740 per year, and the standard deviation of the responses is s = $1125. For a large sample such as this, s is very close to the unknown population a. The station calculates the 95% confidence interval for the mean pay ,u that all citizens would propose for council members to be $9669 to $9811. Is this result trustworthy? Explain your answer. Tests of Significance Confidence intervals are one of the two most common types of formal statisti- cal inference. They are appropriate when our goal is to estimate a population parameter. The second common type of inference is directed at a quite differ- ent goal: to assess the evidence provided by the data in favor of some claim about the population. lnferenc : ally Ho ‘ -_ :er difte r in eith computed. at least evident: ling distri" e data a _ 3 unkno I- 'd deviati' e compu 'vj' of stand I? ll incl sure bone ' 351's. To be - 1 “phanto centimeter. lites too lo I... assign the _ me whethe ment, The l nuch less A, ss time; 0 )- orm takes a time. t in a vane ur product sample of : 6.32 6.33 63.34 Section 6.2 Exercises 453 In each of the following situations, a significance test for a population mean u is called for. State the null hypothesis HO and the alternative hypothesis HCl in each case. {a} Experiments on learning in animals sometimes measure how long it takes a mouse to find its way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a noise as stimulus. {it} The examinations in a large history class are scaled after grading so that the mean score is 50. A self—confident teaching assistant thinks that his students have a higher mean score than the class as a whole. His students this semester can be considered a sample from the population of all students he might teach, so he compares their mean score with 50. {c} The Census Bureau reports that households spend an average of 31% of their total spending on housing. A homebuilders association in Cleveland wonders if the national finding applies in their area. They interview a sample of 40 households in the Cleveland metropolitan area to learn what percent of their spending goes toward housing. In each of the following situations, state an appropriate null hypothesis H0 and alternative hypothesis Hg. Be sure to identify the parameters that you use to state the hypotheses. (We have not yet learned how to test these hypotheses.) (a) A sociologist asks a large sample of high school students which academic subject they like best. She suspects that a higher percent of males than of females will name mathematics as their favorite subject. {is} An education researcher randomly divides sixthgrade students into two groups for physical education class. He teaches both groups basketball skills, using the same methods of instruction in both classes. He encourages Group A with compliments and other positive behavior but acts cool and neutral toward Group B. He hopes to show that positive teacher attitudes result in a higher mean score on a test of basketball skills than do neutral attitudes. (e) An economist believes that among employed young adults there is a positive correlation between income and the percent of disposable income that is saved. To test this, she gathers income and savings data from a sample of employed persons in her city aged 25 to 34. Translate each of the Following research questions into appropriate H0 and Hg. (23.) Census Bureau data show that the mean household income in the area served by a shopping mall is $62,500 per year. A market research firm questions shoppers at the mall to find out whether the mean household income of mall shoppers is higher than that of the general population. vInfE -.:',} : of 5 who ifferent' gnifican In; gnifican I oduction' your 3. Can he purity in Califo ; Out srn . me by la -1 ‘ongmthe ' ky areas.3 Here is th points, I", wrease (r25 us no stati-* ad the stati a calcium."- I y men. :ebrifior 1e conclusi d pressure I iis conclus‘ 6.4:} oil-21 6.43 6.44 6.45 Section 6.2 Exercises 455 especially the P~value, as if you were speaking to a doctor who knows no statistics. A social psychologist reports that “ethnocentrisrn was significantly higher (P < 0.05) among church attenders than among nonattenders." Explain what this means in language understandable to someone who knows no statistics. Do not use the word "significance" in your answer. A study examined the effect of exercise on how students perform on their final exam in statistics. The P—value was given as 0.87. (a) State null and alternative hypotheses that could have been used for this study. (Note that there is more than one correct answer.) (b) Do you reject the null hypothesis? (c) What is your conclusion? {d} What other facts about the study would you like to know for a proper interpretation of the results? The financial aid office of a university asks a sample of students about their employment and earnings. The report says that "for academic year earnings, a significant difference ( P 2 0.038) was found between the sexes, with men earning more on the average. No difference (P r 0.476) was found between the earnings of black and white students.”5 Explain both of these conclusions, for the effects of sex and of race on mean earnings, in language understandable to someone who knows no statistics. Statistics can help decide the authorship of literary works. Sonnets by a certain Elizabethan poet are known to contain an average of ,u 2 6.9 new words (words not used in the poet's other works). The standard deviation of the number of new words is o = 2.7. Now a manuscript with 5 new sonnets has come to light, and scholars are debating whether it is the poet's work. The new sonnets contain an average off : 1 1.2 words not used in the poet’s known works. We expect poems by another author to contain more new words, so to see if we have evidence that the new sonnets are not by our poet we test H0: ,LL 1: 6.9 Ha: ,u > 6.9 Give the z test statistic and its P~value. What do you conclude about the authorship of the new poems? The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Scores range from 0 to 200. The mean score for U.S. college audenmisabout115,andthesunuwrddewanonisabout30.Ateachm"who suspects that older students have better attitudes toward school gives the SSHA to 20 students who are at least 30 years of age. Their mean score is 35 = 135.2. 468 6.74 6.75 6.76 6.77 Chapter 6: Introduction to free of colds. This difference is statistically significant (P =2 0.03) in fat -' the vitamin C group. Can we conclude that vitamin C has a strong client preventing colds? Explain your answer. Every user of statistics should understand the distinction between stati significance and practical importance. A sufficiently large sample will declare very small effects statistically significant. Let us suppose that S mathematics (SATM) scores in the absence of coaching vary normally ‘ mean pi = 475 and 0' = 100. Suppose further that coaching may cha --.t but does not change on An increase in the SATM score from 475 to 478 ' no importance in seeking admission to college, but this unimportant :;E can be statistically veiy significant. To see this, calculate the P-value for i. test of H0: it : 475 HQ: ,u > 475 in each of the following situations: {a} A coaching service coaches 100 students; their SATM scores avera I“ : 478. . (b) By the next year, the service has coached 1000 students; their SATM scores average a? z 478. (c) An advertising campaign brings the number of students coached to 10,000; their average score is still I r 478. Give a 99% confidence interval for the mean SATM score it after coach):- in each part of the previous exercise. For large samples, the confidence ‘ interval says, "Yes, the mean score is higher after coaching, but only by 5.. small amount.” I a As in the previous exercises, suppose that SATM scores vary normally i; or : 100. One hundred students go through a rigorous training program. designed to raise their SATM scores by improving their mathematics Carry out a test of H0? it : 475 Ha: ,u > 475 in each of the following situations: {a} The students’ average score is 2? 2 491.4. Is this result significant at: 5% level? : {b} The average score is f : 491.5. Is this result significant at the 5%! The difference between the two outcomes in (a) and (b) is of no i : tance. Beware attempts to treat a: : 0.05 as sacred. A local television station announces a question for a call~in opinion poll g. the six o’clock news and then gives the response on the eleven o’clock 11 Today’s question concerns a proposed increase in funds for student lea ~- Of the 2372 calls received, 1921 oppose the increase. The station, follow ...
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This note was uploaded on 11/20/2009 for the course ECON 5000 taught by Professor Smith during the Summer '09 term at York University.

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AdditionalExercises03 - Q , 1‘. 9000 m z: SCH: .m;...

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