Ebook Applied statistics - In business and economics (3E): Part 2

(BQ) Part 2 book Applied statistics - In business and economics has contents: Two-Sample hypothesis tests, analysis of variance, simple regression, multiple regression, time series analysis, nonparametric tests, quality management, simulation.

doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 390 Find more at www.downloadslide.com CHAPTER 10 Two-Sample Hypothesis Tests Chapter Contents 10.1 Two-Sample Tests 10.2 Comparing Two Means: Independent Samples 10.3 Confidence Interval for the Difference of Two Means, μ1 − μ2 10.4 Comparing Two Means: Paired Samples 10.5 Comparing Two Proportions 10.6 Confidence Interval for the Difference of Two Proportions, π1 − π2 10.7 Comparing Two Variances Chapter Learning Objectives When you finish this chapter you should be able to LO1 Recognize and perform a test for two means with known σ1 and σ2 LO2 Recognize and perform a test for two means with unknown σ1 and σ2 LO3 Recognize paired data and be able to perform a paired t test LO4 Explain the assumptions underlying the two-sample test of means LO5 Perform a test to compare two proportions using z LO6 Check whether normality may be assumed for two proportions LO7 Use Excel to find p-values for two-sample tests using z or t LO8 Carry out a test of two variances using the F distribution LO9 Construct a confidence interval for μ1 − μ2 or π1 − π2 390 doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 391 Find more at www.downloadslide.com The logic and applications of hypothesis testing that you learned in Chapter will continue here, but now we consider two-sample tests The two-sample test is used to make inferences about the two populations from which the samples were drawn The use of these techniques is widespread in science and engineering as well as social sciences Drug companies use sophisticated versions called clinical trials to determine the effectiveness of new drugs, agricultural science continually uses these methods to compare yields to improve productivity, and a wide variety of businesses use them to test or compare things 10.1 TWO-SAMPLE TESTS What Is a Two-Sample Test? Two-sample tests compare two sample estimates with each other, whereas one-sample tests compare a sample estimate with a nonsample benchmark (a claim or prior belief about a population parameter) Here are some actual two-sample tests from this chapter: Automotive A new bumper is installed on selected vehicles in a corporate fleet During a 1-year test period, 12 vehicles with the new bumper were involved in accidents, incurring mean damage of $1,101 with a standard deviation of $696 During the same year, vehicles with the old bumpers were involved in accidents, incurring mean damage of $1,766 with a standard deviation of $838 Did the new bumper significantly reduce damage? Did it reduce variation? Marketing At a matinee performance of X-Men Origins: Wolverine, a random sample of 25 concession purchases showed a mean of $7.29 with a standard deviation of $3.02 For the evening performance a random sample of 25 concession purchases showed a mean of $7.12 with a standard deviation of $2.14 Is there less variation in the evenings? Safety In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility During a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of yellow fire trucks made 135,035 runs and had accidents Is the difference in accident rates significant? 391 doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 392 Find more at www.downloadslide.com 392 Applied Statistics in Business and Economics Medicine Half of a group of 18,882 healthy men with no sign of prostate cancer were given an experimental drug called finasteride, while half were given a placebo, based on a random selection process Participants underwent annual exams and blood tests Over the next years, 571 men in the placebo group developed prostate cancer, compared with only 435 in the finasteride group Is the difference in cancer rates significant? Education In a certain college class, 20 randomly chosen students were given a tutorial, while 20 others used a self-study computer simulation On the same 20-point quiz, the tutorial students’ mean score was 16.7 with a standard deviation of 2.5, compared with a mean of 14.5 and a standard deviation of 3.2 for the simulation students Did the tutorial students better, or is it just due to chance? Is there any significant difference in the degree of variation in the two groups? Mini Case 10.1 Early Intervention Saves Lives Statistics is helping U.S hospitals prove the value of innovative organizational changes to deal with medical crisis situations At the Pittsburgh Medical Center, “SWAT teams” were shown to reduce patient mortality by cutting red tape for critically ill patients They formed a Rapid Response Team (RRT) consisting of a critical care nurse, intensive care therapist, and a respiratory therapist, empowered to make decisions without waiting until the patient’s doctor could be paged Statistics were collected on cardiac arrests for two months before and after the RRT concept was implemented The sample data revealed more than a 50 percent reduction in total cardiac deaths and a decline in average ICU days after cardiac arrest from 163 days to only 33 days after RRT These improvements were both statistically significant and of practical importance because of the medical benefits and the large cost savings in hospital care Statistics played a similar role at the University of California San Francisco Medical Center in demonstrating the value of a new method of expediting treatment of heart attack emergency patients (See The Wall Street Journal, December 1, 2004, p D1; and “How Statistics Can Save Failing Hearts,” The New York Times, March 7, 2007, p C1.) Basis of Two-Sample Tests Two-sample tests are especially useful because they possess a built-in point of comparison You can think of many situations where two groups are to be compared (e.g., before and after, old and new, experimental and control) Sometimes we don’t really care about the actual value of the population parameter, but only whether the parameter is the same for both populations Usually, the null hypothesis is that both samples were drawn from populations with the same parameter value, but we can also test for a given degree of difference The logic of two-sample tests is based on the fact that two samples drawn from the same population may yield different estimates of a parameter due to chance For example, exhaust emission tests could yield different results for two vehicles of the same type Only if the two sample statistics differ by more than the amount attributable to chance can we conclude that the samples came from populations with different parameter values, as illustrated in Figure 10.1 Test Procedure The testing procedure is like that of one-sample tests We state our hypotheses, set up a decision rule, insert the sample statistics, and make a decision Because the true parameters are unknown, we rely on statistical theory to help us reach a defensible conclusion about our hypotheses Our decision could be wrong—we could commit a Type I or Type II error—but at least we can specify our acceptable level of risk of making an error Larger samples are always doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 393 Find more at www.downloadslide.com Chapter 10 Two-Sample Hypothesis Tests 393 FIGURE 10.1 Same Population or Different? ␪1 ϭ ␪2 ␪ˆ1 ␪1 ␪ˆ2 ␪ˆ1 Samples came from the same population Any differences are due to sampling variation ␪2 ␪ˆ2 Samples came from populations with different parameter values desirable because they permit us to reduce the chance of making either a Type I error or Type II error (i.e., increase the power of the test) Comparing two population means is a common business problem Is there a difference between the average customer purchase at Starbucks on Saturday and Sunday mornings? Is there a difference between the average satisfaction scores from a taste test for two versions of a new menu item at Noodles & Company? Is there a difference between the average age of full-time and part-time seasonal employees at a Vail Resorts ski mountain? The process of comparing two means starts by stating null and alternative hypotheses, just as we did in Chapter If a company is simply interested in knowing if a difference exists between two populations, they would want to test the null hypothesis H0 : μ1 − μ2 = But there might be situations in which the business would like to know if the difference is equal to some value other than zero, using the null hypothesis H0 : μ1 − μ2 = D0 For example, we might ask if the difference between the average number of years worked at a Vail Resorts ski mountain for full-time and part-time seasonal employees is greater than two years In this situation we would formulate the null hypothesis as: H0 : μ1 − μ2 = where D0 = years Format of Hypotheses 10.2 COMPARING TWO MEANS: INDEPENDENT SAMPLES LO1 Recognize and perform a test for two means with known σ1 and σ2 In this section we will focus on the more common situation of simply comparing two population means The possible pairs of null and alternative hypotheses are Left-Tailed Test H0 : μ1 − μ2 ≥ H1 : μ1 − μ2 < Two-Tailed Test H0 : μ1 − μ2 = H1 : μ1 − μ2 = Right-Tailed Test H0 : μ1 − μ2 ≤ H1 : μ1 − μ2 > Test Statistic The sample statistic used to test the parameter μ1 − μ2 is X − X where both X and X are calculated from independent random samples taken from normal populations The test statistic will follow the same general format as the z- and t-scores we calculated in Chapter The test statistic is the difference between the sample statistic and the parameter divided by the standard error of the sample statistic As always, the formula for the test statistic is determined by the sampling distribution of the sample statistic and whether or not we know the population variances LO2 Recognize and perform a test for two means with unknown σ1 and σ2 doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 394 Find more at www.downloadslide.com 394 Applied Statistics in Business and Economics Case 1: Known Variances For the case where we know the values of the population variances, σ12 and σ22 , the test statistic is a z-score We would use the standard normal distribution to find p-values or zcrit values LO4 Explain the assumptions underlying the twosample test of means Case 1: Known Variances z calc = (10.1) ( x¯1 − x¯2 ) − (μ1 − μ2 ) σ12 σ2 + n1 n2 Case 2: Unknown Variances but Assumed Equal For the case where we don’t know the values of the population variances but we have reason to believe they are equal, we would use the Student’s t distribution We would need to rely on sample estimates s12 and s22 for the population variances, σ12 and σ22 By assuming that the population variances are equal, we are allowed to pool the sample variances by taking a weighted average of s12 and s22 to calculate an estimate of the common population variance Weights are assigned to s12 and s22 based on their respective degrees of freedom (n − 1) and (n − 1) Because we are pooling the sample variances, the common variance estimate is called the pooled variance and is denoted s p2 Case is often called the pooled t test Case 2: Unknown Variances Assumed Equal tcalc = ( x¯1 − x¯2 ) − (μ1 − μ2 ) s p2 (10.2) n1 s p2 = + where s p2 n2 (n − 1)s12 + (n − 1)s22 and d f = n + n − n1 + n2 − Case 3: Unknown Variances but Assumed Unequal If the unknown variances σ12 and σ22 are assumed unequal, we not pool the variances This is a more conservative assumption than Case because we are not assuming equal variances Under these conditions the distribution of the random variable X − X is no longer certain, a difficulty known at the Behrens-Fisher problem One solution to this problem is the Welch-Satterthwaite test which replaces σ12 and σ22 with s12 and s22 in the known variance z formula, but then uses a Student’s t test with Welch’s adjusted degrees of freedom Case 3: Unknown Variances Assumed Unequal (10.3) tcalc = ( x¯1 − x¯2 ) − (μ1 − μ2 ) s12 s2 + n1 n2 with d f = s12 s2 + n1 n2 2 s12 s22 n1 n2 + n1 − n2 − Finding Welch’s degrees of freedom requires a tedious calculation, but this is easily handled by Excel, MegaStat, or MINITAB When doing these calculations with a calculator, a conservative quick rule for degrees of freedom is to use d f = min(n − 1, n − 1) If the sample sizes are equal, the value of tcalc will be the same as in Case 2, although the degrees of freedom may differ The formulas for Case and Case will usually yield the same decision about the hypotheses unless the sample sizes and variances differ greatly doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 395 Find more at www.downloadslide.com Chapter 10 Two-Sample Hypothesis Tests 395 Table 10.1 summarizes the formulas for the test statistic in each of the three cases described above We have simplified the formulas based on the assumption that we will usually be testing for equal population means Therefore we have left off the expression μ1 − μ2 because we are assuming it is equal to All of these test statistics presume independent random samples from normal populations, although in practice they are robust to non-normality as long as the samples are not too small and the populations are not too skewed TABLE 10.1 Case Case Case Known Variances Unknown Variances, Assumed Equal Unknown Variances, Assumed Unequal zcalc = x¯1 − x¯2 σ12 n1 + σ22 tcalc = n2 s p2 n1 s p2 = For critical value, use standard normal distribution ( x¯1 − x¯2 ) + where s p2 n2 tcalc = Test Statistic for Zero Difference of Means ( x¯1 − x¯2 ) s12 s2 + n1 n2 (n1 − 1)s12 + (n2 − 1)s22 n1 + n2 − For critical value, use Student’s t with d.f = n1 + n2 − For critical value, use Student’s t with Welch’s adjusted degrees of freedom or min(n1 − 1, n2 − 1) The formulas in Table 10.1 require some calculations, but most of the time you will be using a computer As long as you have raw data (i.e., the original samples of n and n observations) Excel’s Data Analysis menu handles all three cases, as shown in Figure 10.2 Both MegaStat and MINITAB also perform these tests and will so for summarized data as well (i.e., when you have x¯1 , x¯2 , s1 , s2 instead of the n and n data columns) FIGURE 10.2 Excel’s Data Analysis Menu The price of prescription drugs is an ongoing national issue in the United States Zocor is a common prescription cholesterol-reducing drug prescribed for people who are at risk for heart disease Table 10.2 shows Zocor prices from 15 randomly selected pharmacies in two states At α = 05, is there a difference in the mean for all pharmacies in Colorado and Texas? From the dot plots shown in Figure 10.3, it seems unlikely that there is a significant difference, but we will a test of means to see whether our intuition is correct Step 1: State the Hypotheses To check for a significant difference without regard for its direction, we choose a two-tailed test The hypotheses to be tested are H0 : μ1 − μ2 = H1 : μ1 − μ2 = Step 2: Specify the Decision Rule We will assume equal variances For the pooled-variance t test, degrees of freedom are d f = n + n – = 16 + 13 − = 27 From Appendix D we get the two-tail critical value t = ±2.052 The decision rule is illustrated in Figure 10.4 EXAMPLE Drug Prices in Two States doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 396 Find more at www.downloadslide.com 396 Applied Statistics in Business and Economics TABLE 10.2 Zocor Prices (30-Day Supply) in Two States Zocor Colorado Pharmacies City Texas Pharmacies Price ($) Alamosa Avon Broomfield Buena Vista Colorado Springs Colorado Springs Denver Denver Eaton Fort Collins Gunnison Pueblo Pueblo Pueblo Sterling Walsenburg City 125.05 137.56 142.50 145.95 117.49 142.75 121.99 117.49 141.64 128.69 130.29 142.39 121.99 141.30 153.43 133.39 Price ($) Austin Austin Austin Austin Austin Dallas Dallas Dallas Dallas Houston Houston Houston Houston x¯1 = $133.994 s1 = $11.015 n1 = 16 pharmacies 145.32 131.19 151.65 141.55 125.99 126.29 139.19 156.00 137.56 154.10 126.41 114.00 144.99 x¯2 = $138.018 s2 = $12.663 n2 = 13 pharmacies Source: Public Research Interest Group (www.pirg.org) Surveyed pharmacies were chosen from the telephone directory in 2004 Data used with permission FIGURE 10.3 Zocor Prices from Sampled Pharmacies in Two States TX CO 115 FIGURE 10.4 125 135 155 Two-Tailed Decision Rule for Student’s t with α = 05 and d.f = 27 Reject H0 Do not reject H0 ␣/2 ϭ 025 Ϫ2.052 Step 3: Calculate the Test Statistic The sample statistics are x¯1 = 133.994 s1 = 11.015 n = 16 145 x¯2 = 138.018 s2 = 12.663 n = 13 Reject H0 ␣/2 ϭ 025 ϩ2.052 doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 397 Find more at www.downloadslide.com Chapter 10 Two-Sample Hypothesis Tests 397 Because we are assuming equal variances, we use the formulas for Case The pooled variance s p2 is s p2 = (n − 1)s12 + (n − 1)s22 (16 − 1)(11.015) + (13 − 1)(12.663) = = 138.6737 n1 + n2 − 16 + 13 − Using s p2 the test statistic is tcalc = x¯1 − x¯2 s p2 n1 + s p2 133.994 − 138.018 = 138.6737 138.6737 + 16 13 n2 −4.024 = −0.915 4.39708 = √ The pooled standard deviation is s p = 138.6737 = 11.776 Notice that sp always lies between s1 and s2 (if not, you have an arithmetic error) This is because s p2 is a weighted average of s12 and s22 Step 4: Make the Decision The test statistic tcalc = −0.915 does not fall in the rejection region so we cannot reject the hypothesis of equal means Excel’s menu and output are shown in Figure 10.5 Both onetailed and two-tailed tests are shown FIGURE 10.5 Excel’s Data Analysis with Unknown but Equal Variances The p-value can be calculated using Excel’s two-tail function =TDIST(.915,27,2) which gives p = 3681 This large p-value says that a result this extreme would happen by chance about 37 percent of the time if μ1 = μ2 The difference in sample means seems to be well within the realm of chance The sample variances in this example are similar, so the assumption of equal variances is reasonable But if we instead use the formulas for Case (assuming unequal variances) the test statistic is tcalc = x¯1 − x¯2 s12 s2 + n1 n2 133.994 − 138.018 = (11.015) (12.663) + 16 13 = −4.024 = −0.902 4.4629 The formula for degrees of freedom for the Welch-Satterthwaite test is d f = s2 s12 + n1 n2 2 s12 s22 n1 n2 + n1 − n2 − = (11.015) (12.663) + 16 13 (11.015) 16 16 − + (12.663) 13 13 − = 24 The degrees of freedom are rounded to the next lower integer, to be conservative LO7 Use Excel to find p-values for two-sample tests using z or t doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 398 Find more at www.downloadslide.com 398 Applied Statistics in Business and Economics For the unequal-variance t test with d f = 24, Appendix D gives the two-tail critical value t.025 = ±2.064 The decision rule is illustrated in Figure 10.6 FIGURE 10.6 Two-Tail Decision Rule for Student’s t with α = 05 and d.f = 24 Reject H0 Do not reject H0 ␣/2 ϭ 025 Reject H0 ␣/2 ϭ 025 Ϫ2.064 ϩ2.064 The calculations are best done by computer Excel’s menu and output are shown in Figure 10.7 Both one-tailed and two-tailed tests are shown FIGURE 10.7 Excel’s Data Analysis with Unknown and Unequal Variances For the Zocor data, either assumption leads to the same conclusion: Assumption Test Statistic d.f Critical Value Decision Case (equal variances) Case (unequal variances) tcalc = −0.915 tcalc = −0.902 27 24 t.025 = ±2.052 t.025 = ±2.064 Don’t reject Don’t reject Which Assumption Is Best? If the sample sizes are equal, the Case and Case test statistics will be identical, although the degrees of freedom may differ If the variances are similar, the two tests usually agree If you have no information about the population variances, then the best choice is Case The fewer assumptions you make about your populations, the less likely you are to make a mistake in your conclusions Case (known population variances) is not explored further here because it is so uncommon in business Must Sample Sizes Be Equal? Unequal sample sizes are common, and the formulas still apply However, there are advantages to equal sample sizes We avoid unbalanced sample sizes when possible But many times, we have to take the samples as they come doa73699_ch10_390-437.qxd 11/23/09 1:39 PM Page 399 Find more at www.downloadslide.com Chapter 10 Two-Sample Hypothesis Tests Large Samples For unknown variances, if both samples are large (n ≥ 30 and n ≥ 30) and you have reason to think the population isn’t badly skewed (look at the histograms or dot plots of the samples), it is common to use formula 10.4 with Appendix C Although it usually gives results very close to the “proper” t tests, this approach is not conservative (i.e., it may increase Type I risk) x¯1 − x¯2 z calc = (large samples, symmetric populations) (10.4) s12 s22 + n1 n2 Caution: Three Issues Bear in mind three questions when you are comparing two sample means: • Are the populations skewed? Are there outliers? • Are the sample sizes large (n ≥ 30)? • Is the difference important as well as significant? Skewness or outliers can usually be seen in a histogram or dot plot of each sample The t tests (Case and Case 3) are probably OK in the face of moderate skewness, especially if the samples are large (e.g., sample sizes of at least 30) Outliers are more serious and might require consultation with a statistician In such cases, you might ask yourself whether a test of means is appropriate With small samples or skewed data, the mean may not be a very reliable indicator of central tendency, and your test may lack power In such situations, it may be better merely to describe the samples, comment on similarities or differences in the data, and skip the formal t-tests Regarding importance, note that a small difference in means or proportions could be significant if the sample size is large, because the standard error gets smaller as the sample size gets larger So, we must separately ask if the difference is important The answer depends on the data magnitude and the consequences to the decision maker How large must a price differential be to make it worthwhile for a consumer to drive from A to B to save 10 percent on a loaf of bread? A DVD player? A new car? Research suggests, for example, that some cancer victims will travel far and pay much for treatments that offer only small improvement in their chances of survival, because life is so precious But few consumers compare prices or drive far to save money on a gallon of milk or other items that are unimportant in their overall budget Mini Case 10.2 Length of Statistics Articles Are articles in leading statistics journals getting longer? It appears so, based on a comparison of the June 2000 and June 1990 issues of the Journal of the American Statistical Association (JASA), shown in Table 10.3 TABLE 10.3 Article Length in JASA June 1990 JASA June 2000 JASA x¯1 = 7.1333 pages s1 = 1.9250 pages n1 = 30 articles x¯2 = 11.8333 pages s2 = 2.5166 pages n2 = 12 articles Source: Journal of the American Statistical Association 85, no 410, and 95, no 450 We will a left-tailed test at α = 01 The hypotheses are H0 : μ1 − μ2 ≥ H1 : μ1 − μ2 < 399 doa73699_index.qxd 11/20/09 5:56 PM Page 826 Find more at www.downloadslide.com 826 Index Redundancy, 186–188 applications of, 187–188 Space Shuttle case study, 186–187 Refrigerators, prices of, 567–568 Regional binary predictors, 566 Regional voting patterns, and predictors, 566–567, 574–575 Regions Financial, 167 Regression, 488–533, 545 best subsets, 576 bivariate, 494–496, 545 cockpit noise, applied to, 514–515 criteria for assessment, 549 and data, 529–530 estimated, 495 exam scores, applied to, 506–508 fitted, 548 fuel consumption, applied to, 498 and intercepts, 496–497 multiple, 545–549 and outliers, 528 prediction using, 496 problems with, 528–531 and residual tests, 518–524 retail sales, applied to, 508–509 on scatter plots, 497 simple regression defined, 495 and slopes, 496–497 standard error of the, 504, 558 stepwise, 576 variation explained by the, 511 Regression equation, 497 Regression lines, 497 Regression modeling, 549, 577–578 Rejection region, 350 Relative frequencies, 64 Relative frequency approach (see Empirical approach) Relative index, 628 Replacement, sampling with and without, 35–36 Replication, 464 two-factor ANOVA with, 464–470 two-factor ANOVA without, 456–464 Reports: appearance of, 804 presenting, 805–806 writing, 803–805 Research Industry Coalition, Inc., 47 Residual plots, heteroscedastic, 576–577 Residual tests, 518–524, 575 Residuals: body fat, applied to, 527–528 defined, 497 histogram of, 518, 519 standardized, 518, 519 in two-predictor regression, 551 unusual, 524–525, 579 Response error, 42, 43 Response variable, 439, 495, 547 Restaurant quality, Mann-Whitney test applied to, 692–694 Retail sales: interval estimate, applied to, 516–517 regression example for, 508–509 Return Exchange, 366–367 Reuters/Zogby poll, 324–325 Rex Stores, 690 RFID (radio frequency identification) tags, 366 Right-sided tests, 348–350 Right-skewed (positively skewed), 121 Right-skewed histograms, 71 Right-tail chi-square test, 645–647 Right-tailed test, 348, 350, 353–356, 370, 371, 689 Risk, 744 Robust design, 745 Rockwell, 167 Rockwell Collins, 39 Roderick, J A., 169 Roell, Stephen A., 39 Rohm & Haas Co., 60, 691 Romig, Harry G., 719 Roosevelt, Franklin D., 140 Roosevelt, Theodore, 140 Rose Bowl, 101 Rotated graphs, 98 Rounding, 25–26 Row/column data arrays, 37 Rubin, Donald B., 169 Rudestam, Kjell Erik, 21 Rule of Three, 325 Runs test, one-sample, 686–688 Russell 3000 Index, 36 Ryan, Thomas M., 39 Ryan, Thomas P., 591 S S (see Sample space; Seasonal) s (sample standard deviation), 131 S charts, 741 s2 (sample variance), 131 Saab, 63 Safety applications, two-sample tests for, 391 Sahei, Hardeo, 21 Sales data: deseasonalization of, 622–625 exponential smoothing of, 618–619 seasonal binaries in, 625–626 Sally Beauty Co., Inc., 690 Salmon, wild vs farm-bred, 53 Sample accuracy, 325 Sample correlation coefficient, 149–150, 490 Sample covariance, 150, 151 Sample mean(s), 299, 300, 303–304 Sample proportions, 299 Sample size, 375 determination for mean, 327–328 determination for proportions, 329–331 and difference of two means, 403 effect of, 376–377 and effectively infinite populations, 36 and standard error, 304–305 Sample space (S), 173–174 Sample standard deviation (s), 131, 299 Sample statistics, and test statistics, 393 Sample variance (s2), and dispersion, 131 Samples: census vs., 32–33 defined, 31–32 effect of larger, 371 errors in reading, 15 good, 314–315 large, 15, 362–363 non-normality, 373–374 nonrandom, 15 preliminary, 327, 330 doa73699_index.qxd 11/20/09 5:56 PM Page 827 Find more at www.downloadslide.com Index prior, 330 size of, 42 small, 15, 373–374, 688 Sampling, 32–33 acceptance, 744 cluster, 40–41 computer methods for, 36–37 convenience, 41 focus groups, 42 judgment, 41 quota, 41 random vs nonrandom, 35 randomizing in Excel, 37 with replacement, 35 without replacement, 35, 238–242 scanner accuracy case example, 42 simple random, 35 single vs double, 744 size needed for, 42 stratified, 39 systematic, 37–38 using data arrays, 37 Sampling distributions, of an estimator, 297 Sampling error, 42–43, 60, 297–299 Sampling frames, 33 Sampling variation, 295–296 Sara Lee, 60, 691, 705, 711 Sarbanes-Oxley Act, 337 SARS, 53 SAT scores, 160, 385 Saturn, 63 Scales: arithmetic, 79 log, 79–80 logarithmic, 79 Scallops, 54 Scanner accuracy, sampling example about, 42 Scatter plots, 86–89, 489 in Excel, 88–89 MBA applicants, applied to, 491–492 and regressions in Excel, 497 regressions on, 497 Schaeffer, Richard L., 57 Scheffé, H., 434 Schenker, Nathan, 434 Schindler, Pamela S., 57 Schlumberger, 144, 145 Schmidt, Eric E., 39 Scholarships, for athletes, 209 Scientific inquiry, 343–344 Scion (automobile brand), 63 Scott, H Lee, Jr., 39 SE (standard error of the regression), 558 Sears, Roebuck, 42 Seasonal (S), 598–599 Seasonal binaries, 625, 627 Seasonal forecasts, using binary predictors, 625 Seasonality, 622–626 deseasonalization of data, 622–626 exponential smoothing with, 620 Seglin, Jeffrey L., 21 Selection bias, 42, 43 Sensitivity, to α, 360 Sensitivity of test, 346 Sentencing, of convicts, 568 “Serial exchangers,” 366 Shape, 59, 71, 113, 226, 227 Shenton, L R., 682 Sherringham, Philip R., 39 Sherwin-Williams Co., 690 Shewhart, Walter A., 719, 721 Shift variable, 561 Shingo, Shigeo, 721 Shivery, Charles W., 39 Shoemaker, Lewis F., 434 Sigma (σ): confidence intervals for, 333 confidence intervals for a mean with known σ, 307–310 confidence intervals for a mean with unknown σ, 310–317 estimating, 140, 327 σ2 (see Population variance) Significance: F test for, 551–552 level of, 345, 346, 375 practical importance vs., 16, 351, 357, 361 statistical, 392, 575 testing for, 561 Simple events, 174 Simple line charts, 77 Simple random sampling, 35 Simple regression (see Regression) Singh, Ravindra, 57 Single exponential smoothing, in MINITAB, 618–620 Single sampling, 744 Six Sigma, 723, 745–746 Size effect, 530 Skewed left, 121 Skewed population, 303–304 Skewed right, 121 Skewness, 154–155 and central tendency, 121 of histogram, 71 Skewness coefficient, 154–155 Sky (aircraft manufacturer), 167, 536 Slopes, 499–500 confidence intervals for, 504–505 in exponential trend calculations, 604 in linear trend calculations, 601 and regressions, 496–497 Small samples, 15, 688 Smartphone devices, 364 Smith, Gerald M., 756 Smith, Neil, 39 Smith International, 144, 145 Smithfield Foods, 711 Smoking: and gender (case study), 193–194, 202–203 vaccine for, 433 Smoothing constant (␣), 617 Smoothing models, 599 Socata (aircraft manufacturer), 167, 536 Software packages: binary dependent variables in, 576 contingency tables in, 651 for visual description, 61 Solutions, search for, 718 Som, R K., 57 Sony, 540 Sources of variation, 439, 465–466 SouthTrust Corp., 167 Southwest Airlines, 107, 342 Sovereign Bancorp, 39 827 doa73699_index.qxd 11/20/09 5:56 PM Page 828 Find more at www.downloadslide.com 828 Index Space launch, patterns in, 599 Space Shuttle, 14, 186–187 Spam e-mail, 52 SPC (statistical process control), 723 Spearman’s rank correlation test, 702–704 Spearman’s rho, 702 Special cause variation, 717, 723 Special law of addition, 181 Specification limits, upper and lower, 736 Spelling, 804 Spirit Airlines, 107 Sports drinks, potassium content of, 358 Sports drinks, sodium content of, 386 Spurious correlation, 530–531 SQC (see Statistical quality control) Squares, sum of, 490 SSE (unexplained variation errors, random error), 439, 502, 511–512 SSR (variation explained by the regression), 502, 511–512 SST (total variation around the mean), 502, 511–512 Stacked bar charts, 83, 84 Stacked data, 448n Stacked dot plots, 62–63 Standard & Poor’s 500 Index, 60 Standard deviation, 130–133 calculating, 132 characteristics of, 133 and discrete distributions, 219–220 in grouped data, 153–154 true, 375 two-sum formula for calculating, 132 Standard error: as a measure of fit, 612–614 of a regression, 504 and sample size, 304–305 true, 525 Standard error of the mean, 300, 304–305, 728 Standard error of the proportion, 321, 366 Standard error of the regression (SE), 558 Standard normal distributions, 262–274 Standard normal distributions, cumulative, 764–765 Standardized data, 137–141 Chebyshev’s Theorem, 137 defining variables with, 139 Empirical Rule, 137, 138 estimating sigma using, 140 and outliers, 138–140 unusual observations in, 138 Standardized residuals, 518, 519 in Excel, 524, 525 in MegaStat, 525 in MINITAB, 525 Standardized variables, 139, 268 Staples Inc., 60 Starwood Hotels, 60 State and Metropolitan Area Data Book, 45 State and Metropolitan Area Data Book, 44 State Farm, 186 State St Corp., 167 Statistical Abstract of the United States, 44, 45 Statistical estimation, 296 Statistical generalization, 16 Statistical hypothesis tests, 348 Statistical process control (SPC), 723, 746 Statistical quality control (SQC), 722, 723 Statisticians: and quality control, 717 traits of, 10 Statistics (data), 3, 33, 113 leverage, 526–527 t statistic, 491 Statistics (discipline), challenges in, 10 descriptive, inferential, pitfalls of, 14–16 Statistics Canada (Web site), 45 Stefanski, Leonard A., 177n Stephens, Michael A., 682 Stepwise regression, 576 Stochastic process, 208 Stock prices, MINITAB sample of, 156–157 Stocks, 596 Stratified sampling, 39 Stryker Corporation, 60 Student work, 652–653 Student’s t, for unknown population variance, 359–361 Student’s t distribution, 310–314, 316–317 Sturges’ Rule, 65, 667 Subaru, 63, 364 Subgroup size, 724 Subjective approach (probability), 178–179 Sum of squares, 490 Sum of squares, partitioned, 444 Sum of squares error, 502 Sums of random variables: and discrete distributions, 245 gasoline application of, 245 project scheduling application of, 245–246 SunTrust Banks, 167 Supply-chain management, 744–745 Surveys, 45–50 coding, 48 on colleges, role of, 50 of customer satisfaction, 325 and data file formats, 49 data screening, 48 Likert scales, 50 poor quality methods, 15 questionnaire design for, 47 response rates for, 46 sources of error in, 42–43 types of, 46 wording of, 47–48 Sutter Home, 536, 703 Suzuki, 63 Swordfish, 54 Symantec Corp., 60 Symmetric data, 120 Symmetric histograms, 71 Symmetric triangular distributions, 284 Syms Corporation, 690 Synovus Financial Corp., 167 Systematic error, 298 Systematic population, 302–303 Systematic sampling, 37–38 T T (see Trend) t distribution (see Student’s t distribution) t statistic, 491 t test, paired, 404, 408–409 doa73699_index.qxd 11/20/09 5:56 PM Page 829 Find more at www.downloadslide.com Index Tables, 92–94 contingency, 192–193 presenting, 13 in technical reports, 804, 805 3-way, 651 Tabulated data, for Poisson goodness-of-fit-tests, 661–662 Taft, William, 140 Taguchi, Genichi, 719, 721, 745 Taguchi method, 723 Talbots, Inc., 690 Target (store), 42, 689 Target population, 33 Tax returns, estimated time for preparing, 53 Taylor, Zachary, 140 TD Ameritrade Holding, 39 Technical literacy, studying statistics and, Technical reports, 804–805 Telefund calling, 243 Telemarketers, predictive dialing by, 385 Telephone surveys, 45, 46 Television ratings, 34 Test preparation companies, 385 Test statistic, calculating, 350, 357 Tests for one variance, 381–382 Thode, Henry C., Jr., 693 Thompson, Steven K., 57 Three P’s, 806 3-D bar charts, 82–83 3-D graphs, 98 3-D pie charts, 95, 96 3-way tables, 651 Tiffany & Co., 690 Tiger (aircraft manufacturer), 167, 536 Time, 100 Time series: for growth rates, 127 in software packages, 633 Time-series analysis, 594–631 Time-series data, 30–31, 595–596 bar charts for, 84 in line charts, 80 Time-series decomposition, 597 Time-series graphs, 595–596, 598–599 Time-series variable, 595 TINV (Excel function), 316 TMA (trailing moving average), 614–616 TMR Inc., 50 Tootsie Roll, 270, 335 Total cost, 245 Total quality control, 721 Total quality management (TQM), 721, 744 Total variation around the mean (see SST) Totals, problem of, 530 Touchstar, 385 Toyota, 6, 63, 90, 329, 401, 536, 585 TQM (see Total quality management) Trade, arithmetic scale example in, 80 Traffic fatalities, uniform GOF test of, 656–657 Trailing moving average (TMA), 614–616 Transformations, linear, 244 Transformations of random variables, 244–246 Transocean, 144, 145 Transplants, graph of trends for, 599 Transportation, U.S Department of, 342 TransUnion, 186 Treatment, 439, 456 Tree diagrams, 196, 197 Trend (T), 597–599, 620 Trend fitting: criteria for, 608–609 in Excel, 607, 609 and U.S trade deficit, 611–612 Trend forecasting, 599–609 Trend line, 88, 89 Trend models, 600 Trend pattern (control charts), 734–736 Trendless data, 614 Trendline, in Excel, 497n Triangular distributions, 282–285 characteristics of, 282–284 symmetric, 284 uses of, 285 Tribune Company, 60 Tri-Cities Tobacco Coalition, 372 Trimmed mean, and central tendency, 116, 127–128 Troppo Malo, 290 True mean, 375 True standard deviation, 375 True standard error, 525 Truman, Harry, 140 Tufte, Edward R., 109 Tukey, John Wilder, 450 Tukey t tests, 451–452 Tukey’s studentized range tests, 450–451 decision rule for, 450–451 of pairs and means, 470 Turbine data, stepwise regression of, 576 Two events: intersection of, 180 union of, 179–180 Two means, comparing, 393–401 hypotheses for, 393 and independent samples, 393–401 issues of, 399 with known variances, 394 and paired samples, 404–409 with unknown variances, 394 Two means, difference of, 401–403 2k models, 476 Two-factor ANOVA with replication, 464–470 hypothesis for, 464–465 sources of variation in, 465–466 and turbine engine thrust example, 472–473 using MegaStat for, 468 Two-factor ANOVA without replication, 456–464 calculation of nonreplicated, 458–460 limitations of, 461–462 using MegaStat, 460–461 Two-factor model, 457 Two-predictor regression, 551 Two-sample hypothesis tests (see Hypothesis tests, two-sample) Two-scale line charts, 77 Two-sided tests, 348–350 Two-sum formula, for standard deviations, 132 Two-tailed p-values, and software packages, 554 Two-tailed tests, 348, 350, 356–357 in bivariate regression, 512 for means with known population variance, 356–357 for power curves and OC curves, 380 p-value method in, 369, 414 in Wilcoxan signed-rank test, 689 for a zero difference, 406–407 829 doa73699_index.qxd 11/20/09 5:56 PM Page 830 Find more at www.downloadslide.com 830 Index 2004 election, 324–325 Type I error(s) (α), 344, 346, 392–393, 734 choosing, 347 effect of varying, 369–370 in one-sample hypothesis tests, 344, 345–353 probability of, 345, 346 sensitivity to, 360 and Type II error, 352–353 Type II error(s) (β), 344, 347, 392–393, 734 consequences of, 347 in one-sample hypothesis tests, 344 and power of curve, 374–379 and Type I error, 352–353 U UCL (upper control limit), 725 Unbiased estimator, 298 Unconditional probability (prior probability), 198 Unconscious bias, 16 Unexplained variation errors (see SSE) Uniform continuous distributions, 257–259 Uniform discrete distributions, 221–224 copier codes application of, 224 gas pumping application of, 223 lotteries application of, 223 Uniform distributions, 302–303, 656 Uniform goodness-of-fit tests, 654–659 for grouped data, 656–657 for raw data, 657–659 Uniform model, 259–260 Uniform population: all possible samples from, 305–306 assuming, 327 Uniform random integers, 223 Unimodal classes, 71 Uninsured patients, 225 Union, of two events, 179–180 Union Pacific, 39, 60 Union Planters Corp., 167 Unit rectangular distribution, 259 United Airlines, 107 customer service, 342 dot plot example using, 148–149 United Auto Group, 680 United Nations Department of Economic and Social Affairs, 45 U.S Bureau of the Census, 45 U.S Centers for Disease Control and Prevention, 17, 54 U.S Congress, 32 U.S Customs Service, 338 U.S Energy Information Administration, 34 U.S Federal Statistics (Web site), 45 U.S Fisheries and Wildlife Service, 54 U.S Food and Drug Administration (FDA), 18, 45, 53, 54 U.S News & World Report, 100 U.S presidents, standardized data example using, 140–141 U.S trade deficit, and trend fitting, 611–612 Units of measure, 328, 331 Univariate data sets, 24, 59 Unknown population variances, 359–363 hypothesis tests of mean with, 359–363 p-value method for, 361 Student’s t method for, 359–361 using MegaStat to find, 362 Unstacked data, 448n Unusual data, 138, 146–147 Unusual leverage, 580 Unusual observations, 579–580 Unusual residuals, 524–525, 579 Upper control limit (UCL), 725 Upper specification limit (USL), 736–739 US Airways, 107 US Airways Flight 1549, 609 USA Today, 100 USL (upper specification limit), 736–739 Utts, Jessica, 16, 21 V Vail Resorts, Inc.: correlation matrix for, 152–153 employee age, 433 employee pay, 272, 351, 364 employee rehiring, 427 employee seniority, 401 guest age, 310 ISO certification, 747 Likert scales for, 28, 30 market research surveys, 85 regression modeling, 550, 557 satisfaction surveys, 338, 401, 550, 557, 575 use of statistics by, 4–5 Vallee, Roy, 39 Van Belle, Gerald, 339 Van Buren, Martin, 140 Vanguard, Varco International, 144, 145 Vardeman, Stephen B., 21 Variable(s): binary, 25 continuous, 255 continuous random, 256–257 defined, 23 dependent, 495 discrete, 255 dummy, 561 independent, 495, 644 indicator, 561 random, 208, 215–216, 261–262 response, 439, 495, 547 shift, 561 standardized, 139 time-series, 595 Variable control charts, 724 Variable data, 724 Variable selection, 548 Variable transform, 531 Variance(s): analysis of, 511–512 Bay Street Inn example of, 219–220 comparing two, 417–424 comparing two means with known variances, 394 comparing two means with unknown variances, 394 comparison of, in one-tailed test, 421–422 comparison of, in two-tailed test, 419–421 of continuous random variable, 256–257 critical values for two, 418 decomposition of, 511 and discrete distributions, 219–220 F test for two, 417–418 folded F test for two, 421 homogeneity of, 452–454 hypothesis tests for one, 381–382 doa73699_index.qxd 11/20/09 5:56 PM Page 831 Find more at www.downloadslide.com Index hypothesis tests for two, 417 nonconstant, 519 population, 130–131, 331–332 sample, 131 tests for one, 381–382 two, applied to collision damage, 418–419 Variance inflation, 571 Variance inflation factor (VIF), 572–573 Variation around the mean, total (SST), 502, 511–512 Variation explained by the regression (SSR), 502, 511–512 Variations: among observations, 31 coefficient of, 133 common cause, 717, 723 reduced, 716–717 special cause, 717, 723 unexplained error (SSE), 439, 502, 511–512 zero, 716–717 Vellemen, Paul F., 57 Venn diagram, 174 Verbal anchors, for interval data, 28 Verify-1 (software), 366 Verizon, 372–373 Video preference, viewer age and, 194 VIF (see Variance inflation factor) Visual data, 59–100 bar charts, 82–84 dot plots, 61–64 frequency distributions, 67–69 histograms, 66–71 line charts, 77–80 measuring, 60 pie charts, 95–96 pivot tables, 92–94 scatter plots, 86–89 sorting, 60–61 stacked dot plots, 62–63 tables, 92–94 Visual displays, 61, 489, 490 Visual Statistics (software program): Anderson-Darling test in, 672 equal expected frequencies in, 667, 668 Poisson GOF test in, 664 Volkswagen, 63, 536, 585 Volunteers, effect on sampling, 34 Volvo, 63, 90, 536, 585 Voting patterns, regional, 566–567 W The W Edwards Deming Institute, 720 Wachovia Corp., 167 Wainscott, James L., 39 Wald, Abraham, 686 Wald-Wolfowitz test (one-sample runs test), 686–688 The Wall Street Journal, 5, 6, 100, 119, 148, 179 The Wall Street Journal Index, 45 Wallis, W Allen, 695 Walmart, 6, 39, 401, 563, 564 Wang, Y., 434 Warehousing, 246 Washington, George, 140 Web surveys, 46 Weight Watchers, 712 Weighted index, 629 Welch’s adjusted degrees of freedom, 394 Welch’s formula, 401–402 Welch-Satterthwaite test, 394 Well-conditioned data, 529–530 Wells Fargo, 167 Wendy’s International Incorporated, 60, 691 Wheelwright, Steven C., 641 Whirlpool Corporation, 60, 691 Whitaker, D., 20 Whitney, D R., 692 Wichern, Dean W., 641 Wilcoxan, Frank, 689 Wilcoxan rank-sum test (Mann-Whitney test), 692–694 Wilcoxan signed-rank test, 689–691 Wilkinson, Leland, 57 William of Occam, 608 William Wrigley, Jr., 705 Williamson, Bruce A., 39 Williams-Sonoma, 680 Willoughby, Floyd G., 695 Wilson, J Holton, 641 Wilson, Woodrow, 140 Wireless routers, encryption for, 428 Wirthlin Worldwide, 53 Wm Wrigley, Jr (company), 711 Wolfowitz, Jacob, 686 World Bank, 45 World Demographics (Web site), 45 World Health Organization, 45 Written communication skills, 803–805 executive summaries, 805 technical reports, 804–805 Wynn, Stephen A., 39 Wynn Resorts, 39 X x charts, 724, 725, 733 Y Y: confidence intervals for, 558–559 interval estimate for, 515–516 Yahoo, 108 Young, James R., 39 Yum! Brands Inc., 60, 691 Z z: in sample size, 328 Student’s t distribution vs., 316 Zale, 680, 690 Zelazny, Gene, 109 Zero, meaningful, 29 Zero correlation, zero slope and, 506 Zero difference: testing for, 409 two-tailed test for a, 406–407 Zero slope: in bivariate regression, 512 test for, 506 Zero variation, 716–717 Zions Bancorp, 167 Zocor, 395–398 Zone charts, 742 z-scores, 139 z-values, 367 831 doa73699_index.qxd 11/20/09 5:56 PM Page 832 Find more at www.downloadslide.com doa73699_index.qxd 11/20/09 5:56 PM Page 833 Find more at www.downloadslide.com doa73699_index.qxd 11/20/09 5:56 PM Page 834 Find more at www.downloadslide.com doa73699_index.qxd 11/20/09 5:56 PM Page 835 Find more at www.downloadslide.com doa73699_index.qxd 11/20/09 5:56 PM Page 836 Find more at www.downloadslide.com doa73699_index.qxd 11/20/09 5:56 PM Page 837 Find more at www.downloadslide.com doa73699_Endpaper_spread 11/14/09 8:53 PM Page Find more at www.downloadslide.com STANDARD NORMAL AREAS This table shows the normal area between and z Example: P(0 < z < 1.96) = 4750 z z 00 01 02 03 04 05 06 07 08 09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 0000 0398 0793 1179 1554 1915 2257 2580 2881 3159 3413 3643 3849 4032 4192 4332 4452 4554 4641 4713 4772 4821 4861 4893 4918 4938 4953 4965 4974 4981 49865 49903 49931 49952 49966 49977 49984 49989 0040 0438 0832 1217 1591 1950 2291 2611 2910 3186 3438 3665 3869 4049 4207 4345 4463 4564 4649 4719 4778 4826 4864 4896 4920 4940 4955 4966 4975 4982 49869 49906 49934 49953 49968 49978 49985 49990 0080 0478 0871 1255 1628 1985 2324 2642 2939 3212 3461 3686 3888 4066 4222 4357 4474 4573 4656 4726 4783 4830 4868 4898 4922 4941 4956 4967 4976 4982 49874 49910 49936 49955 49969 49978 49985 49990 0120 0517 0910 1293 1664 2019 2357 2673 2967 3238 3485 3708 3907 4082 4236 4370 4484 4582 4664 4732 4788 4834 4871 4901 4925 4943 4957 4968 4977 4983 49878 49913 49938 49957 49970 49979 49986 49990 0160 0557 0948 1331 1700 2054 2389 2704 2995 3264 3508 3729 3925 4099 4251 4382 4495 4591 4671 4738 4793 4838 4875 4904 4927 4945 4959 4969 4977 4984 49882 49916 49940 49958 49971 49980 49986 49991 0199 0596 0987 1368 1736 2088 2422 2734 3023 3289 3531 3749 3944 4115 4265 4394 4505 4599 4678 4744 4798 4842 4878 4906 4929 4946 4960 4970 4978 4984 49886 49918 49942 49960 49972 49981 49987 49991 0239 0636 1026 1406 1772 2123 2454 2764 3051 3315 3554 3770 3962 4131 4279 4406 4515 4608 4686 4750 4803 4846 4881 4909 4931 4948 4961 4971 4979 4985 49889 49921 49944 49961 49973 49981 49987 49992 0279 0675 1064 1443 1808 2157 2486 2794 3078 3340 3577 3790 3980 4147 4292 4418 4525 4616 4693 4756 4808 4850 4884 4911 4932 4949 4962 4972 4979 4985 49893 49924 49946 49962 49974 49982 49988 49992 0319 0714 1103 1480 1844 2190 2517 2823 3106 3365 3599 3810 3997 4162 4306 4429 4535 4625 4699 4761 4812 4854 4887 4913 4934 4951 4963 4973 4980 4986 49896 49926 49948 49964 49975 49983 49988 49992 0359 0753 1141 1517 1879 2224 2549 2852 3133 3389 3621 3830 4015 4177 4319 4441 4545 4633 4706 4767 4817 4857 4890 4916 4936 4952 4964 4974 4981 4986 49900 49929 49950 49965 49976 49983 49989 49992 doa73699_Endpaper_spread 11/14/09 8:53 PM Page Find more at www.downloadslide.com CUMULATIVE STANDARD NORMAL DISTRIBUTION This table shows the normal area less than z Example: P(z < −1.96) = 0250 This table shows the normal area less than z Example: P(z < 1.96) = 9750 z z z 00 01 02 03 04 05 06 07 08 09 z 00 01 02 03 04 05 06 07 08 09 −3.7 −3.6 −3.5 −3.4 −3.3 −3.2 −3.1 −3.0 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2.0 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1.0 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 −0.0 00011 00016 00023 00034 00048 00069 00097 00135 0019 0026 0035 0047 0062 0082 0107 0139 0179 0228 0287 0359 0446 0548 0668 0808 0968 1151 1357 1587 1841 2119 2420 2743 3085 3446 3821 4207 4602 5000 00010 00015 00022 00032 00047 00066 00094 00131 0018 0025 0034 0045 0060 0080 0104 0136 0174 0222 0281 0351 0436 0537 0655 0793 0951 1131 1335 1562 1814 2090 2389 2709 3050 3409 3783 4168 4562 4960 00010 00015 00022 00031 00045 00064 00090 00126 0018 0024 0033 0044 0059 0078 0102 0132 0170 0217 0274 0344 0427 0526 0643 0778 0934 1112 1314 1539 1788 2061 2358 2676 3015 3372 3745 4129 4522 4920 00010 00014 00021 00030 00043 00062 00087 00122 0017 0023 0032 0043 0057 0075 0099 0129 0166 0212 0268 0336 0418 0516 0630 0764 0918 1093 1292 1515 1762 2033 2327 2643 2981 3336 3707 4090 4483 4880 00009 00014 00020 00029 00042 00060 00084 00118 0016 0023 0031 0041 0055 0073 0096 0125 0162 0207 0262 0329 0409 0505 0618 0749 0901 1075 1271 1492 1736 2005 2296 2611 2946 3300 3669 4052 4443 4841 00009 00013 00019 00028 00040 00058 00082 00114 0016 0022 0030 0040 0054 0071 0094 0122 0158 0202 0256 0322 0401 0495 0606 0735 0885 1056 1251 1469 1711 1977 2266 2578 2912 3264 3632 4013 4404 4801 00008 00013 00019 00027 00039 00056 00079 00111 0015 0021 0029 0039 0052 0069 0091 0119 0154 0197 0250 0314 0392 0485 0594 0721 0869 1038 1230 1446 1685 1949 2236 2546 2877 3228 3594 3974 4364 4761 00008 00012 00018 00026 00038 00054 00076 00107 0015 0021 0028 0038 0051 0068 0089 0116 0150 0192 0244 0307 0384 0475 0582 0708 0853 1020 1210 1423 1660 1922 2206 2514 2843 3192 3557 3936 4325 4721 00008 00012 00017 00025 00036 00052 00074 00104 0014 0020 0027 0037 0049 0066 0087 0113 0146 0188 0239 0301 0375 0465 0571 0694 0838 1003 1190 1401 1635 1894 2177 2483 2810 3156 3520 3897 4286 4681 00008 00011 00017 00024 00035 00050 00071 00100 0014 0019 0026 0036 0048 0064 0084 0110 0143 0183 0233 0294 0367 0455 0559 0681 0823 0985 1170 1379 1611 1867 2148 2451 2776 3121 3483 3859 4247 4641 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 5000 5398 5793 6179 6554 6915 7257 7580 7881 8159 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 9821 9861 9893 9918 9938 9953 9965 9974 9981 99865 99903 99931 99952 99966 99977 99984 99989 5040 5438 5832 6217 6591 6950 7291 7611 7910 8186 8438 8665 8869 9049 9207 9345 9463 9564 9649 9719 9778 9826 9864 9896 9920 9940 9955 9966 9975 9982 99869 99906 99934 99953 99968 99978 99985 99990 5080 5478 5871 6255 6628 6985 7324 7642 7939 8212 8461 8686 8888 9066 9222 9357 9474 9573 9656 9726 9783 9830 9868 9898 9922 9941 9956 9967 9976 9982 99874 99910 99936 99955 99969 99978 99985 99990 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 8485 8708 8907 9082 9236 9370 9484 9582 9664 9732 9788 9834 9871 9901 9925 9943 9957 9968 9977 9983 99878 99913 99938 99957 99970 99979 99986 99990 5160 5557 5948 6331 6700 7054 7389 7704 7995 8264 8508 8729 8925 9099 9251 9382 9495 9591 9671 9738 9793 9838 9875 9904 9927 9945 9959 9969 9977 9984 99882 99916 99940 99958 99971 99980 99986 99991 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 8531 8749 8944 9115 9265 9394 9505 9599 9678 9744 9798 9842 9878 9906 9929 9946 9960 9970 9978 9984 99886 99918 99942 99960 99972 99981 99987 99991 5239 5636 6026 6406 6772 7123 7454 7764 8051 8315 8554 8770 8962 9131 9279 9406 9515 9608 9686 9750 9803 9846 9881 9909 9931 9948 9961 9971 9979 9985 99889 99921 99944 99961 99973 99981 99987 99992 5279 5675 6064 6443 6808 7157 7486 7794 8078 8340 8577 8790 8980 9147 9292 9418 9525 9616 9693 9756 9808 9850 9884 9911 9932 9949 9962 9972 9979 9985 99893 99924 99946 99962 99974 99982 99988 99992 5319 5714 6103 6480 6844 7190 7517 7823 8106 8365 8599 8810 8997 9162 9306 9429 9535 9625 9699 9761 9812 9854 9887 9913 9934 9951 9963 9973 9980 9986 99896 99926 99948 99964 99975 99983 99988 99992 5359 5753 6141 6517 6879 7224 7549 7852 8133 8389 8621 8830 9015 9177 9319 9441 9545 9633 9706 9767 9817 9857 9890 9916 9936 9952 9964 9974 9981 9986 99900 99929 99950 99965 99976 99983 99989 99992 doa73699_Endpaper_1.qxd 11/20/09 6:38 PM Page Find more at www.downloadslide.com STUDENT’S t CRITICAL VALUES This table shows the t-value that defines the area for the stated degrees of freedom (d.f.) .80 d.f Confidence Level 90 95 98 99 80 20 Significance Level for Two-Tailed Test 10 05 02 01 10 Significance Level for One-Tailed Test 05 025 01 005 Confidence Level 90 95 t 98 99 20 Significance Level for Two-Tailed Test 10 05 02 01 d.f .10 Significance Level for One-Tailed Test 05 025 01 005 3.078 1.886 1.638 1.533 1.476 6.314 2.920 2.353 2.132 2.015 12.706 4.303 3.182 2.776 2.571 31.821 6.965 4.541 3.747 3.365 63.656 9.925 5.841 4.604 4.032 36 37 38 39 40 1.306 1.305 1.304 1.304 1.303 1.688 1.687 1.686 1.685 1.684 2.028 2.026 2.024 2.023 2.021 2.434 2.431 2.429 2.426 2.423 2.719 2.715 2.712 2.708 2.704 10 1.440 1.415 1.397 1.383 1.372 1.943 1.895 1.860 1.833 1.812 2.447 2.365 2.306 2.262 2.228 3.143 2.998 2.896 2.821 2.764 3.707 3.499 3.355 3.250 3.169 41 42 43 44 45 1.303 1.302 1.302 1.301 1.301 1.683 1.682 1.681 1.680 1.679 2.020 2.018 2.017 2.015 2.014 2.421 2.418 2.416 2.414 2.412 2.701 2.698 2.695 2.692 2.690 11 12 13 14 15 1.363 1.356 1.350 1.345 1.341 1.796 1.782 1.771 1.761 1.753 2.201 2.179 2.160 2.145 2.131 2.718 2.681 2.650 2.624 2.602 3.106 3.055 3.012 2.977 2.947 46 47 48 49 50 1.300 1.300 1.299 1.299 1.299 1.679 1.678 1.677 1.677 1.676 2.013 2.012 2.011 2.010 2.009 2.410 2.408 2.407 2.405 2.403 2.687 2.685 2.682 2.680 2.678 16 17 18 19 20 1.337 1.333 1.330 1.328 1.325 1.746 1.740 1.734 1.729 1.725 2.120 2.110 2.101 2.093 2.086 2.583 2.567 2.552 2.539 2.528 2.921 2.898 2.878 2.861 2.845 55 60 65 70 75 1.297 1.296 1.295 1.294 1.293 1.673 1.671 1.669 1.667 1.665 2.004 2.000 1.997 1.994 1.992 2.396 2.390 2.385 2.381 2.377 2.668 2.660 2.654 2.648 2.643 21 22 23 24 25 1.323 1.321 1.319 1.318 1.316 1.721 1.717 1.714 1.711 1.708 2.080 2.074 2.069 2.064 2.060 2.518 2.508 2.500 2.492 2.485 2.831 2.819 2.807 2.797 2.787 80 85 90 95 100 1.292 1.292 1.291 1.291 1.290 1.664 1.663 1.662 1.661 1.660 1.990 1.988 1.987 1.985 1.984 2.374 2.371 2.368 2.366 2.364 2.639 2.635 2.632 2.629 2.626 26 27 28 29 30 1.315 1.314 1.313 1.311 1.310 1.706 1.703 1.701 1.699 1.697 2.056 2.052 2.048 2.045 2.042 2.479 2.473 2.467 2.462 2.457 2.779 2.771 2.763 2.756 2.750 110 120 130 140 150 1.289 1.289 1.288 1.288 1.287 1.659 1.658 1.657 1.656 1.655 1.982 1.980 1.978 1.977 1.976 2.361 2.358 2.355 2.353 2.351 2.621 2.617 2.614 2.611 2.609 31 32 33 34 35 1.309 1.309 1.308 1.307 1.306 1.696 1.694 1.692 1.691 1.690 2.040 2.037 2.035 2.032 2.030 2.453 2.449 2.445 2.441 2.438 2.744 2.738 2.733 2.728 2.724 ∞ 1.282 1.645 1.960 2.326 2.576 Note: As n increases, critical values of Student’s t approach the z-values in the last line of this table A common rule of thumb is to use z when n > 30, but that is not conservative ... against a left-tailed, two-tailed, or right-tailed alternative: Left-Tailed Test H0: σ 12 ≥ 22 H1: σ 12 < 22 Two-Tailed Test H0: σ 12 = 22 H1: σ 12 = 22 Right-Tailed Test H0: σ 12 ≤ 22 H1: σ 12. .. proportion in place of π: n1 p1 = (23 09) (20 48 /23 09) = 20 48 n2 p2 = (23 86) (20 14 /23 86) = 20 14 n1 (1 − p1) = (23 09)(1 − 20 48 /23 09) = 26 1 n2(1 − p2) = (23 86)(1 − 20 14 /23 86) = 3 72 The normality requirement... Test 2 H0: 12 = 2 2 H1: 12 = 2 Right-Tailed Test 2 H0: 12 ≤ 2 2 H1: 12 > 2 The F Test In a left-tailed or right-tailed test, we actually test only at the equality, with the understanding