INTRODUCTION TO STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL phần 5 potx

CHAPTER 3 DISTRIBUTIONS 83 FIGURE 3.6 Preparing to estimate difference in population means. FIGURE 3.7 Entering data and sample sizes in the BoxSampler worksheet. 3.7.2. Are Two Variables Correlated? Yet another example of the bootstrap’s application lies in the measurement of the correlation or degree of agreement between two variables. The Pearson correlation of two variables X and Y is defined as the ratio of the covariance between X and Y and the product of the standard deviations of X and Y. The covariance of X and Y is given by the formula . Recall that if X and Y are independent, the E(XY) = (EX)(EY), so that the expected value of the covariance and hence the correlation of X and Y is zero. If X and Y increase more or less together as do, for example, the height and weight of individuals, their covariance and their correlation will be positive so that we say that height and weight are positively correlated. I had a boss, more than once, who believed that the more abuse and criti- cism he heaped on an individual the more work he could get out of them. Not. Abuse and productivity are negatively correlated; heap on the abuse and work output declines. The reason we divide by the product of the standard deviations in assessing the degree of agreement between two variables is that it renders the correlation coefficient free of the units of measurement. If X =-Y, so that the two variables are totally dependent, the correla- tion coefficient, usually represented in symbols by the Greek letter r (rho) will be -1. In all cases, -1 £ r £ 1. Is systolic blood pressure an increasing function of age? To find out, I entered the data from 15 subjects in an Excel worksheet as shown in Fig. 3.8. Each row of the worksheet corresponds to a single subject. As described in Section 1.4.2, Resampling Stats was used to select a single bootstrap sample of subjects. That is, each row in the bootstrap sample corresponded to one of the rows of observations in the original sample. Making use of the data from the bootstrap samples, I entered the formula for the correlation of Systolic Blood Pressure and Age in a conve- nient empty cell of the worksheet as shown in Fig. 3.9 and then used the RS button to generate 100 values of the correlation coefficient. Exercise 3.25. Using the LSAT data from Exercise 1.16 and the boot- strap, obtain an interval estimate for the correlation between the LSAT score and the student’s subsequent GPA. Exercise 3.26. Trying to decide whether to take a trip to Paris or Tokyo, a student kept track of how many euros and yen his dollars would buy. Month by month he found that the values of both currencies were rising. XX n kk k n - () - () - () = Â YY 1 1 84 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® CHAPTER 3 DISTRIBUTIONS 85 FIGURE 3.8 Preparing to generate a bootstrap sample of subjects. FIGURE 3.9 Calculating the correlation between systolic blood pressure and age. Does this mean that improvements in the European economy are reflected by improvements in the Japanese economy? 3.7.3. Using Confidence Intervals to Test Hypotheses Suppose we have derived a 90% confidence interval for some parameter, for example, a confidence interval for the difference in means between two populations, one of which was treated and one that was not. We can use this interval to test the hypothesis that the difference in means is 4 units, by accepting this hypothesis if 4 is included in the confidence interval and rejecting it otherwise. If our alternative hypothesis is nondirectional and two-sided, q A π q B , the test will have a Type I error of 100% - 90% = 10%. Clearly, hypothesis tests and confidence intervals are intimately related. Suppose we test a series of hypotheses concerning a parameter q. For example, in the vitamin E experiment, we could test the hypothesis that vitamin E has no effect, q = 0, or that vitamin E increases life span by 25 generations, q = 25, or that it increases it by 50 generations, q = 50. In each case, whenever we accept the hypothesis, the corresponding value of the parameter should be included in the confidence interval. In this example, we are really performing a series of one-sided tests. Our hypotheses are that q = 0 against the one-sided alternative that q > 0, that q £ 25 against the alternative that q > 25 and so forth. Our corresponding confidence interval will be one-sided also; we will conclude q < q U if we accept the hypothesis q = q 0 for all values of q 0 < q U and reject it for all values of q 0 ≥ q U . One-sided tests lead to one-sided confidence intervals and two-sided tests to two-sided confidence intervals. Exercise 3.27. What is the relationship between the significance level of a test and the confidence level of the corresponding interval estimate? Exercise 3.28. In each of the following instances would you use a one- sided or a two-sided test? i. Determine whether men or women do better on math tests. ii. Test the hypothesis that women can do as well as men on math tests. iii. In Commonwealth v. Rizzo et al., 466 F. Supp 1219 (E.D. Pa 1979), help the judge decide whether certain races were discriminated against by the Philadelphia Fire Department by means of an unfair test. iv. Test whether increasing a dose of a drug will increase the number of cures. Exercise 3.29. Use the data of Exercise 3.18 to derive an 80% upper con- fidence bound for the effect of vitamin E to the nearest 5 cell generations. 86 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® 3.8. SUMMARY AND REVIEW In this chapter, we considered the form of four common distributions, two discrete—the binomial and the Poisson—and two continuous—the normal and the exponential. We provided the R functions necessary to generate random samples from the various distributions and to display plots side by side on the same graph. We noted that, as sample size increases, the observed or empirical distri- bution of values more closely resembles the theoretical. The distributions of sample statistics such as the sample mean and sample variance are differ- ent from the distribution of individual values. In particular, under very general conditions with moderate-size samples, the distribution of the sample mean will take on the form of a normal distribution. We consid- ered two nonparametric methods—the bootstrap and the permutation test—for estimating the values of distribution parameters and for testing hypotheses about them. We found that because of the variation from sample to sample, we run the risk of making one of two types of error when testing a hypothesis, each with quite different consequences. Normally when testing hypotheses, we set a bound called the significance level on the probability of making a Type I error and devise our tests accordingly. Finally, we noted the relationship between our interval estimates and our hypothesis tests. Exercise 3.30. Make a list of all the italicized terms in this chapter. Provide a definition for each one, along with an example. Exercise 3.31. A farmer was scattering seeds in a field so they would be at least a foot apart 90% of the time. On the average, how many seeds should he sow per square foot? The answer to Exercise 3.0 is yes, of course; an observation or even a sample of observations from one population may be larger than observa- tions from another population even if the vast majority of observations are quite the reverse. This variation from observation to observation is why before a drug is approved for marketing its effects must be demonstrated in a large number of individuals and not just in one or two. CHAPTER 3 DISTRIBUTIONS 87 [...]... 2880, 56 70, 11620, 8660, 6010, 11620, 8600, 12860, 21420, 55 10, 12270, 650 0, 1 650 0, 4930, 10 650 , 16310, 157 30, 4610, 86260, 652 20, 3820, 34040, 91270, 51 450 , 16010, 6010, 156 40, 49170, 62200, 62640, 58 80, 2700, 4900, 55 820, 9960, 28130, 34 350 , 4120, 61340, 24220, 3 153 0, 3890, 49410, 2820, 58 850 , 4100, 3020, 52 80, 3160, 64710, 250 70 4.2.4 Two-Sample t-Test For the same reasons that Student’s t was an excellent... worms and died, only to wake to discover that reincarnation is real and that to expiate your sins in the previous life you’ve been reborn as a consulting statistician I’m sure that’s what must have happened in my case Introduction to Statistics Through Resampling Methods & Microsoft Office Excel ®, by Phillip I Good Copyright © 20 05 John Wiley & Sons, Inc 106 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT. .. steps: FIGURE 4.3 Preparing to shuffle the data 96 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL 1 Use the cursor to outline the two columns that we wish to shuffle, that is, to rearrange again in two columns, one with 15 observations and one with 14 2 Press the S on the Resampling Stats in Excel menu 3 Note the location of the top left cell where you wish to position the reshuffled... need to consider the assumptions 98 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL under which the test is valid, the effect of violations of these assumptions, and the Type I and Type II errors associated with each test 4.3.1 p Values and Significance Levels In the preceding sections we have referred several times to p values and significance levels We have used both in helping us to. .. are (139, 137), (140, 138 .5) , (141, 140), (142 .5, 141), (143 .5, 142) Both arm spans and heights are in increasing order Is this just coincidence? Or is there a causal relationship between them or between them and a third hidden variable? What is the probability that an event like this could happen by chance alone? 102 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL n The test statistic... GPA? 104 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL 4.4 SUMMARY AND REVIEW In this chapter, we derived permutation, parametric, and bootstrap tests of hypothesis for a single sample, for comparing two samples, and for bivariate correlation We showed how to improve the accuracy and precision of bootstrap confidence intervals We explored the relationships and distinctions among p values,... Section 3.6.1 to the hockey data is next to impossible Imagine how many Ê 14 + 15 years it would take us to look at all Á ˜ possible rearrangements! Ë 15 ¯ What we can do today—something not possible with the primitive calculators that were all that was available in the 1930s when permutation methods were first introduced—is to look at a large random sample of rearrangements We prepare to reshuffle the... both arm span and height along with the value of S, but this won’t be necessary We can get exactly the same result if we fix the order of one of the variables, the height, for example, and look at the 5! = 120 ways in which we could rearrange the arm span readings: (140, 137) (139, 138 .5) (141, 140) (142 .5, 141) (143 .5, 142) (141, 137) (140, 138 .5) (139, 140) (142 .5, 141) (143 .5, 142) and so forth Obviously,... about the shape of the distribution from which the observations are taken 1 2 Any bootstrap but the parametric bootstrap We need to modify our testing procedure if we suspect this to be the case; see Chapter 8 100 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL When the variances of the populations from which the observations are drawn are not the same, the significance level of the... us to make a decision whether to accept or reject a hypothesis and, in consequence, to take a course of action that might result in gains or losses To see the distinction between the two concepts, please go through the following steps: 1 Use BoxSampler to generate a sample of size 10 from a Normal Distribution with mean 0 .5 and variance 1 2 Use this sample and the t-test to test the hypothesis that the . distributed. 96 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® Hospital Billing Data 4181, 2880, 56 70, 11620, 8660, 6010, 11620, 8600, 12860, 21420, 55 10, 12270, 650 0, 1 650 0, 4930, 10 650 ,. 3.18 to derive an 80% upper con- fidence bound for the effect of vitamin E to the nearest 5 cell generations. 86 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® 3.8. SUMMARY AND. 157 30, 4610, 86260, 652 20, 3820, 34040, 91270, 51 450 , 16010, 6010, 156 40, 49170, 62200, 62640, 58 80, 2700, 4900, 55 820, 9960, 28130, 34 350 , 4120, 61340, 24220, 3 153 0, 3890, 49410, 2820, 58 850 ,