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Ch11LastProblems - 11.2 Comparing Two Means (at Fill in the...

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Unformatted text preview: 11.2 Comparing Two Means (at Fill in the values in this sununarv table: Group treannenr n x s 1 lDX ? ? ? 2 Untreated ? ? ? (b) What degrees of freedoln would you use in the conservative two—sample t proce- dures to compare the two treatments? 11.52 TREATING SCRAPIE, || Exercise 115] contains the results ofa study to determine whether IDX is an effective treatment of scrapie. {a} Is there good evidence that hamsters treated with IDX live longer on the average? {tr} Give a 95% confidence interval for the mean amount by which IDX prolongs life. 11.53 TEACHING READING An educator believes that new reacting activities in the classroom will help elementary school pupils improve their reading ability. She an‘angcs for a third- grade class of 21 students to follow these activities for an 8—week period. A control classroom of 23 third graders follows the same curriculum without the activities. At the end of the 3 weeks, all students are given the Degree of Reading Power (DRP) test which measures the aspects of reading ability that the treatment is designed to improve. Here are. the (lat‘ :22 Treatment Control 24 43 58 T] 43 42 4% 55 26 62 4‘) 6] 44 67 4‘) 37 33 41 19 54 5% 56 59 32 62 20 85 46 10 17 '34 5? 33 46 43 60 53 42 3? 42 '3? 55 28 48 {a} Examine the data with a graph. Are there strong outliers or skewness that could prevent use of the t procedures? (b) Is there good evidence that the new activities improve the mean DRP score? Carry out a test and report your conclusions. (c: Although this study is an experiment, its design is not ideal because it had to be done in a school without disrupting classes. W'hat aspect of good experimental design is missing? 11.54 WEIGHT LOSS PROGRAM In a study of the effectiveness ofa weight loss program, 47 subjects who were at least 20% overweight took part in the program for 10 weeks. Private weighings determined each subject’s weight at the beginning of the program and 6 months after the program’s end. rf' he matched pairs t test was used to assess the significance of the average weight loss. The paper reporting the study said, “The sub— jects lost a significant amount ofweight over time, H46) 2 4.68, f) < Hi.” It is com- mon to report the results of statistical] tests in this abbreviated style.23 {a} “7hr was the matched pairs 1‘. test appropriate? (bl Explain to someone who knows no statistics but is interested in weight—loss pro- grams what the practical conclusion is. 659 571} Chapter 11 Inference for Distributions (c) The paper follows the tradition of reporting significance only at fixed levels such as a : 0.01. In fact, the results are more significant than “p < .01” suggests. Use Table C to say more about the P—valuc of the t test. 11.55 COMPARING TWO DRUGS Makers of generic drugs must show that they do not differ significantly from the “reference” drug that they imitate. One aspect in which drugs might differ is their extent of absorption in the blood. Table 1 1.6 gives data taken from 20 heal thy nonsmoking male subjects for one pair of drugs. This is a matched pairs design. Subjects 1 to 10 received the generic drug first, and Subjects 11 to 20 received the reference drug First. In all cases, a washout period separated the [we drugs so that the first had disappeared from the blood before the subject took the second The subject numbers in the table were assigned at random to decide the order of the drugs for each subject. (a) Do a data analysis of the differences between the absorption measures for the generic and reference drugs. Is there any reason not to apply t procedures? (b) Give a 90% confidence interval for the mean difference in gains between treatment and control. TABLE 11.3 Absorption extent for two versions of a drug Refererte Generic Reference Generic Sumen dmg one Bowed dmg dmg 15 4108 1755 4 234e 2738 3 2526 1138 16 186" 2302 9 2729 1613 ' 6 1022 1284 13 3852 2254 I 10 2256 3052 12 1833 1310 j 5 938 1287 8 2463 2120 7 1339 1930 18 2059 1851 14 1262 1964 20 1709 1878 = 11 1438 2.549 17 1829 1682 1 17.35 .3340 2 2594 2613 t 19 1020 3050 Source: Data from Lianng Yuli, “A biopharmaceutieal example for undergraduate students," unpublished manuscript. 11.55 EUAEHING AND SAT SCORES Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without pay— ing for coaching generally raise their scores. A random sample ofstudcnts who l’Ole' the SA’ 1 ' twice found 427 who were coached and 2,733 who were uncoached.24 Starting with their Verbal scores on the first and second tries, we have these summary statistics: Trgl 1th Gain Mean Std.dev. Mean Std.dev. Mean Stddev. Coached 500 92 Uncoached 506 101 527 101 21 52 11.2 Comparinng Means Let’s first ask if students who are coached increased their scores significantly. {at You cordd use the information given to carry out either a two-sample t test corn— paring Try 1 with Try 2 for coached students or a matched pairs t test using Gain. Which is the correct test? Why? {1)} Carry out the proper test. What do you conclude? {'c) Give a 99% confidence interval for the mean gain of all students who are coached. 11. 5? EUAEHING AND SAT SCORES, || What we really want to know is whether coached stu— dents improve more than uncoached students, and whether any advantage is large enough to be worth paying for. Use the information in the previous problem to answer these questions. {a} Is there good evidence that coached students gained more on the average than uncoaehed students? {b} How much more do coached students gain on the average? Give a 99% confi- dence interval. {U} Based on your work, what is your opinion: do you think coaching courses are worth paying for? 11.58 COACHING AND SAT SCORES: CRITIQUE The data yOu used in the previous two exer— cises came from a random sample of students who took the SAT twice. rt'he response rate was 63%, which is pretty good for nongovernment surveys, so let’s accept that the respondents do represent all students who took the exam twice. Nonetheless, we can’t be sure that coaching actually caused the coached students to gain more than the uucoach ed students. Explain briefly but clearly why this is so. 11.59 STU DENTS' SELZLEDNEEPT Here is SAS output for a study of the sclf~concept of seventh— grade students. The variable SC is the score on the Pierswl—Iarris Self Concept Scale. The analysis was done to see if male and female students differ in mean self—concept SCOICLZs TTEST PROCEDURE Variable: SC SEX N Mean Std Dev Std Error F 31 55.5..612903 12.69611743 2.28029001 M 47 57.92.4'89362 12.26488410 1.78901722 Variances T DF Prob> | T ] Uneqaal —0.8276 62-8 0.4110 Equal —0.8336 76.0 0.4071 Write a sentence or two Summarizing the comparison of females and males, as if you were preparing a report for publication. fi?2 Chapter 11 Inference for Distributions The remaining exercises concern the power ofthe two-sample t test, an optioncii topic. ifyou have read Section 10.4 and the discussion of the power of the one—sampie t test on pages 639—640, Exercise 11.6-1- guides you in finding the power ofthe two—sample at. 11.60 111 Example 11.10 (page 650), a small study of black men suggested that a calcium supplement can reduce blood pressure. Now we are planning a larger clinical trial of this effect. We plan to use 100 subjects in each of the two groups. Are these sample sizes large enough to make it very likely that the study will give strong evidence (a : 0.01) of the effect of calcium it in fact calcium lowers blood pressure by 5 millimeters more than a placebo? rlb answer this question, we will compute the power of the tu-D—smnple t test of against the specific alternative M1 — 1L3 : 5. Based 011 the pilot study reported in Example 1 1.10, we take 8, the larger of the two obscured s—values, as a rough estimate of both the population {7’5 and future sample 3’s. (a) that is the approximate value of the a : 0,01 critical value if“ for the two—sample 1. statistic when :11 : u2 : 100? (1)) Step 1: Write the mile for rejecting 1'1“ in terms of f1 — 352. The test rejects HLJ when Take both 31 and 32 to be 8, and ul and rig to be 100. bind the number 0 such that the test rejects H‘J when f1 —- E2 2 c. to) Step 2: The power is the probability of rejecting 11[) when the altemative is true. Su ose that p. — u, = 5 and that both or and 0‘1 are 8. The owcr we seek is the )rob— PP 1 _ 1 L P l ability that E1 — E2 2 c under these assumptions Ca1culate the power. (d) Describe a Type 1 and a Type I] error in this experiment. Much is more serious? 11.61 A bank asks you to compare two ways to increase the use of its credit cards. 1’1 a n A would offer customers a cash—back rebate based on their total amount charged. Plan B would reduce the interest rate charged on card balances. 'l 'he response variable is the total amount a customer charges during the test period. You decide to offer each cat—Plan A and Plan B to a separate SRS of the bank’s credit card customers. In the past, the mean amount charged in a six—month period has been about $1100, with a standard deviation of $400. Will a twosamplc t test based on SRSs of 350 customers in each group detect a difference ot$100 in the mean amounts charged under the two plans? (a) State H“ and H”, and write the formula for the test statistic. (b) Give the CE = 0.03 critica1 value For the test when n] : :11 = 3'50. (c) Calculate the power of the test with or 0.05, using $400 as a rough cstiu'iatc of all standard deviations. td] Describe a '1"vpe 1 error and a Type 11 error in this setting. Which is of more con- cern to the bank? .F 11.2 Comparing Two Means 573 CHAPTER REVIEW This chapter presents t tests and confidence intervals for inference about the mean of a single population and for comparing the means of two populations. The oneisample t procedures do inference about one mean and the two—sample t procedures compare two means Matched pairs studies use one—sample proce— dures because you first create a single sample by taking the differences in the responses within each pair. 'lhese t procedures are among the most common methods of statistical inference. 'Ihe figure below helps you decide when to use them. Before you use any inference method, think about the design of the study and examine the data for outliers and other problems. The t Procedures for Means STUDY DESIGN? One Sample Matched Fairo, Two Sample DATA ANALYSiS: CAN WE U5E t2? Strong Skewnaos? Outliers)? Small Sample? CHOOSE t PROCEDURE The t procedures require that the data be random samples and that the dis- tribution of the population or populations be nornlai. One reason for the wide use of t procedures is that they are not very strongly affected by lack of nor— mality. If you can’t regard your data as a random sample, however, the results of inference may be oflittle value. Chapter It] concentrated on the reasoning of confidence intervals and tests. Understanding the reasoning is essential for wise use of the t and other inference methods. The discussion in this chapter paid more attention to prac- tical aspects of using the methods. We saw that there are several versions of the tworsainple t, for example. Which one you use depends largely on whether or not you use statistical software. Before you use any inference method, think abOut the design of the study and examine the data for outliers and other prob— lems The chapter exercises are important in this and later chapters. You must now recognize probiem settings and decide which of the methods presented in the chapter fits In this chapter, you must recognize oncesample studies, matched pairs studies, and two~sarnplc studies. Here are the most important skills you should have after reading this chapter. A. RECOGNITION 1. Recognize when a problem requires inference about a mean or comparing two means. Dfioign? T551: or" Confidence Intervai? g l PRACTICAL CONCLUSION? 574 Chapter 11 CHAPTER 11 REVIEW EXERCISES Inference for Distributions 2. Recognize from the design of a study whether one-sample, matched pairs, or two—sample procedures are needed. B. ONE-SAMPLE t PROCEDURES 1. Use the t procedure to obtain a confidence interval at a stated level of con— fidence for the mean [.L of a population. 2. Carry out a t test for the hypothesis that a population mean ,u has a speci— fied value against either a one-sided or a two—sided alternative. Use Table C 03‘ t critical values to approximate the P—value or carry out a fixed or test. 3. Recognize when the t procedures are appropriate in practice, in particular that they are quite robust against lack of normality but are influenced by outliers. 4. Also recognize when the design of the studyP outliers, or a small sample from a skewed distribution make the t procedures risky. 5. Recognize matched pairs data and use the t procedures to obtain confi— dence intervals and to perform tests of significance for such data. C. TWO-SAM PLE t PROCEDURES 1. Give a confidence interval for the difference between two means. Use the two—sample 1‘ statistic with conservative degrees of freedom if you do not have statistical software. Use the "Fl-83,89 or software if you have it. A 2. Test the hypothesis that two populations have equal means against either a one-sided or a two-sided alternative. Use the twousample t test with conserva- tive degrees of freedom if you do not have statistical software. Use the TI-83i’89 or software if you have it. 3. Recognize when the two—sample tprocedures are appropriate in practice. 11.52 EXPENSWE ADS Consumers who think a product’s advertising is expensive often also think the product must be of high quality. Can other information undermine this effect? To find out, marketing researchers did an experiment. The subjects were 90 women from the clerical and administrative staff of a large organization. All subjects read an ad dlat described a fictional line of food products called “Five Chefs.” The ad also described the major TV commercials that would soon be shown, an unusual expense for this type of product, The 45 women in the control group read nothing else. The 45 in the “undermine group” also read a news story headlined “No link between Advertising Spending and New Product Quality." All the subjects then rated the quality of Five Chefs products on a seven—point scale. The study report ._said, “Thc mean quality ratings were significantly lower in the undermine treatment (Xfll = 4.56) than in the control treatment (XC = 5.05; t = 2.64, p < .01).”26 (a) ls the matched pairs t test or the two-sample t test the right test in this setting? “div? (1)) W’ hat degrees of Freedom would you use for the t statistic you chose in (a)? (e) The distribution of individual responses is not normal, because there is only a seven-point scale. Why is it nonetheless proper to use a t test? 11.53 SHARKS Great white sharks are hig and hungry. Here are the lengths in feet of44 great whilemx 18.7 12.3 18.6 16.4 15.7 18.3 14.6 15,8 14.9 17,6 12.1 16.4 16.7 17.8 16.2 12,6 17.8 13.8 12.2 15.2 14.7 12.4 13.2 15.8 14.3 16.6 9.4 18.2 13.2 13.6 15.3 16.1 13.5 19.1 16.2 22.8 16.8 13.6 13.2 15.7 19.7 18.7 13,2 16.8 {3) Examine these data for shape, center, spread, and outliers. 'I‘he distribution is rea— sonably normal except for one outlier in each direction. Because these are not extreme and preserve the symmetry of the distribution, use of the t procedures is safe with 44 observations. (b) Give a 95% confidence interval for the mean length of great white sharks. Based on this interval, is there significant evidence at the 5% level to reject the claim “Great white sharks average 20 feet in length"? (c) it isn’t clear exactly what parameter p. you estimated in What information do you need to say what ,u. is? 11.54 INDEPENDENT SAMPLES VERSUS PAIRED SAMPLES Deciding whether to perform a matched pairs t test or a two-sample t test can be tricky.28 Your decision should be based on the design that produced the data. Which procedure would you choose in each of the following Situations? (a) To compare the average weight gain of pigs fed two different rations, nine pairs of pigs were used. The pigs in each pair were littemiates. (b) To test the effects ofa new fertilizer, 100 plots are treated with the new fertilizer, and 100 plots are treated with another fertilizer. (c) A sample of college teachers is taken. We wish to compare the average salaries of male and female teachers. (d) A new fertilizer is tested on 100 plots. Each plot is divided in half. Fertilizer A is applied to one half and B to the other. to) Consumers Union wants to compare two types of calculators. They get 100 vol- unteers and ask them to carry out a series of 50 routine calculations (such as figuring discounts, sales tax, totaling 3 bill, etc.). Each calculation is done on each type of cal— culator, and the time required for each calculation is recorded. 11.55 KIEKINE A HELIUM-EILLED FDDTBAll On a calm, clear Saturday in 1993, the Auburn 'l‘igers were faced with fourth down deep in their own territory. "1111 eir opposition, the Mississippi State Bulldogs, looked for good field position following a punt. The foot— ball was snapped, kicked, and eyed in disbelief as it sailed an estimated 71 yards Chapter Review 575 675 Chapter 11 Inference for Distributions through the air. Shocked, the Mississippi State coaches cried foul and the football was immediately seized by the officials. The football was later tested to see ifit had been filled with helium, as many thought that this might explain its unusually long flight. No helium was found in that football, but the possible benefits of filling a football with gas lighter than air would be kicked around both science and sports communities in the weeks to come. Many devised their own experiments to see if helium—filled balls traveled farther than footballs fi1led with air. The Columbus Dispatch conducted one such study. "two identical footballs, one air—filled and one helium—tilled, were used outdoors on a windless day at Ohio State Universith athletic complex. The kicker was a novice punter and was not informed which football contained the helium. Each football was kicked 39 times and the two footballs were alternated with each kick. Table 11.7 provides the data from this exper— uncut. TABLE Distance traveled (in yards) by two kicked footballs, one filled with helium and one filled with air Toal aw Hehum i tea as HeHum ] 1661 Ah‘ Hefium j 16a as HeHum 1 25 25 5 11 25 12 21 31 31 5 31 22 26 2 23 16 i 12 19 28 1 22 27 34 F 32 26 32 3 18 25 g 13 27 28 j 23 22 39 E 33 28 30 4 16 14 § 14 25 31 24 29 32 g 34 32 29 5 35 23 i 15 3 22 25 28 14 ; 35 28 30 6 15 29 t 16 26 29 26 29 28 j 36 25 29 3 26 25 i 17 20 23 1 27 22 30 i 37 31 29 8 24 26 E 18 22 26 f 28 31 27 38 28 3o 9 24 22 i 19 33 35 a 29 25 33 = 39 28 26 10 28 26 g 20 29 24 f 30 20 11 Source: Data from the EESEF‘. story “Kicking a l-Iclium-ll‘illed Football.” Based on the summary statistics, the researcher concluded that there is “not much dif— ference” in the results for the two footballs, {a} Perform an appropriate statistical test of this statement. (b) Conduct your test from (a) with any outliers in the data set removed. Compare the two results. (c) 'I'he researcher also stated: “The kicker changed footballs on each kick, guaran- leeing that his leg would play no favorites if he tired. However, it appears he improved with practice.” Perform an appropriate statistical analysis to address this claim. 11.55 LEARNING TD SGWE A MAZE Table 11.2 (page 629) contains the times required to complete a maze for 21 subjects wearing scented and unscented masles. Example 1 1‘1 used the matched pairs t test to show that the scent makes no significant difference in the time. Now we ask whether the re is a learning effect, so that subjects complete the maze faster on their second trial. All of the odd-numbered subjects in rFable 11.2 first worked the maze wearing the unscented mask. E1-‘e11—nun‘1bered subjects wore the scented mask first. 'I 'he numbers were assigned at randonL {a} we will compare the unscented times for “unscented first” subjects with the unscented times for the “scented first” subjects. Explain why this comparison requires two—sample procedures. (b]I We suspect that on the average subjects are slower when the unscented time is their first trial. Make a back—to—back sternplot of unscented times for “scented first” and “unscented first” subjects. Find the mean unscented times for these two groups. Do the data appear to support our suspicion? Do the data have features that prevent use of the t procedures? {c} Do the data give statistically significant support to our suspicion? State hypothe~ ses, carry out a test, and report your conclusion. 11.57 EDMPARING WELFARE PROGRAMS A major study of alternative welfare programs randomly assigned women on welfare to one of two programs, called “MN” and “Options.” WIN was the existing program. The new Options program gave more incentives to work. An important question was how Inuch more {on the average) women in Options eamed than those in WIN. Here is Minitab output for earnings in dollars over a 3—year period?” TWOSAMPLE '1‘ FOR ‘OPT’ VS ‘WIN’ N MEAN STDEV SE MEAN OPT 1362 7638 289 7.8309 WIN 1395 6595 247 6.6132 95 PCT C: FOR MU OPT — MU WIN: (1022.90, 1063.10) (a) Give a 99% confidence interval for the amount by which the mean earnings of Options participants exceeded the mean earnings oleN subjects (Minitab will give a 99% confidence interval if you instruct it to do so. Here we have only the basic out- put, which includes the 95% confidence interval.) (b) The distribution of incomes is strongly skewed to the right but includes no extreme outliers because all the subjects were on welfare. What fact about these data allows us to use t procedures despite the strong skewness? 11.68 EACH DAY [AM GETTING BETFER IN MMH A “subliminal” message is below our thresh- old ot awareness but may nonetheless influence us. Can subliminal messages help stue dents learn math? A group of students who had failed the mathematics part of the City University of New York Skills Assessment Test agreed to participate in a study to find out. All received a daily subliminal message, Flashed on a screen too rapidly to be con— sciously read. The treatment group of 10 students (chosen at random) was exposed to Hliiacll day I am getting better in math.” The control group of 8 students was exposed to a neutral message, “People are walking on the s eet.” All students participated in a summer program designed to raise their math skilis, and all took the assessment test again at the end of the program. liable 11.8 gives data on the subjects’ scores before and after the program. (a) Is there good evidence that the treatment brought abouta greater improvement in math scores than the neutral message? State hypotheses, carry out a test, and state your conclusion. Is your result significant at the 5% level? At the lU% level? Chapter Review 5?? 1) 678 Chapter 11 Inference for Distributions TABLE 11.3 Mathematics skills scores before and after a subliminal message Treatment Group Controi Group Pro-test Post~te5t Pro-test Post—test 18 24 18 29 18 25 24 29 2] 33 20 24 18 29 18 26 18 ' 33 24 38 20 36 22 27 23 34 15 22 23 36 19 31 21 34 17 27' Source: Data provided by Warren Page, New York City Technical College, from a study done by John Hudcsman. (b) Give a 90% confidence interval for the mean difference in gains between treat— ment and control. a 11.59 STRESS RMDNE PHS AND FRIENDS Stress is a fact of everyday life. Researchers explored how the presence of others can affect certain stress indicators when a person performs a stressful task. In this study, the researchers asked 45 women to perform mental arithmetic in the presence of their pet dog (P), a good female friend (F), or alone (C, for control). To record the participants’ stress levels during the task and rest periods, the experimenters measured maximum heart rate (beatsfininute). The researchers were interested in esplor— ing whether the fact that a human friend could evaluate the subject’s performance at arith— metic while a dog could not would affect the participant’s stress level.3U Condition Max.heartrate I Condition Max.heartrate I Condition Max.heartrate c 115 P 128 a P 72 e 110 P 122 P 72 e 113 I P 103 P 74 c 103 P 128 P 68 c 114 P 131 P 61 c 112 P 118 P 82 c 115 P 83 a P 72 c 96 P 127 l P 78 c 107 P 132 p P 92 e 103 I P 103 P 127 c 95 P 126 i P 87 e 113 , P 116 i P 73 c 120 P 110 l P 74 e 96 P 11.3 P 76 c 34 P 120 P 70 Use these data to examine the researchers’ question of interest If you find statistically significant difference between two of the groups, estimate the size of that difference. 11.70 You look up a census report that gives the poPulations of all 92 counties in the state of Indiana. Is it proper to apply the one—sample t method to these data to give a 95% confidence interval for the mean pOpulation of an Indiana county? Explain your answer. 11.?1 Exercise 1.28 (page 35) gives 29 measurements of the density of the earth, made in 1798 by Henry Cavendish. Display the data graphically to check for skewness and outliers. Then give an estimate for the density of the earth from Cavendish’s data and a margin of error for your estimate. 11.72 CHOLESTEROL 1N DUES High levels of cholesterol in the blood are not healthy in either humans or dogs. Because a diet rich in saturated fats raises the cholesterol level. it is plausible that dogs owned as pets have higher cholesterol levels than dogs owned by a veterinary research clinier “Normal” levels of cholesterol based on the clinic’s dogs would then be misleading. A clinic compared healthy dogs it owned with healthy pets brought to the clinic to be neutered. The summary statistics for blood cholesterol lcxL cls (milligrams per deciliter of blood) appear below.31 Group :1 i 5 Pets 26 193 68 Clinic 23 174 44 (a) ls there strong evidence that pets have higher mean cholesterol level than clinic dogs? State the H U and H d and carry out an appropriate test. Give the P—value and state your conclusion. (1)) Give a 95% confidence interval for the difference in mean cholesterol levels between pets and clinic dogs. (c) Give a 95% confidence interval for the mean cholesterol level in pets. (d) What conditions must be satisfied to justify the procedures you used in (a), (b), and (e)? Assuming that the cholesterol measurements have no outliers and are not strongly skewed, what is the chief threat to the validity of the results of this study? 11.73 ACTIVE VERSUS PASSIVE LEARNING A study of computer—assisted learning exalrlincd the learning of “Blissymbols” by children. Blissymbols are pictographs (think of Egyptian hieroglyphs) that are sometimes used to help leaminguimpaired children connnunicate. The researcher designed two computer lessons that taught the same content using the same examples. One lesson required the children to interact with the material, while in the other the children controlled only the pace of the lesson. Call these two styles “Active” and “Passive.’1 After the lesson, the computer presented a quiz that asked the Children to identify 56 Blissymbols. Here are the numbers of cor— rect identifications by the 24 children in the Active group:32 29 28 24 31 24 35 21 24 44 28 15 24 27 23 20 22 23 21 17 21 21 20 28 16 Chapter Review 679 EBB Chapterll lnferenceforDistrihutions The 24 children in the Passive group had these counts of correct identifications: 16 14 17 15 26 I? 12 25 2] 20 18 21 20 16 18 15 26 15 13 17 21 19 15 12 (a) Is there good evidencc that active learning is superior to passive learning? Give appropriate statistical justification for your answer. (13) Give a 90% confidence interval for the mean number of Blissymbols identified correctly in a large population of children after the Active computer lesson. ...
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