1. 1 Nonparametric Methods （非参数统计） Chapter 15

2. Nonparametric Methods 15.1The Sign Test: A Hypothesis Test about the Median （符号检验）The Sign Test: A Hypothesis Test about the Median 15.2The Wilcoxon Rank Sum TestThe Wilcoxon Rank Sum Test （ Wilcoxon 符号和检验）

3. 1.1 Nonparametric Tests ( 非参数检验 ) A. One-Sample Mean Test Many tests are concern with testing some parameter under a certain distribution. Test under a normal population if is known, the Z-test is recommended, where is the sample mean and n is the sample size.

4. 1.1 Nonparametric Tests B. Two-Sample Mean Tests Test under two respective normal populations and. If a t-test is suggested.

5. In most cases the variances are unknown. Comparing Means of Two Populations

6. 1.1 Nonparametric Tests If the data are not normal distributed, the distribution of the t-statistic is unknown and depends the distribution of the populations. There are a huge amount of underlying distributions. Can we have some tests that are distribution free? The nonparametric test is one of such kinds of tests.

7. A local pizza restaurant located close to a college campus advertises that their delivery time to a college dormitory is less than for a local branch of a national pizza chain. In order to determine whether this advertisement is valid, you and some friends have decided to order 10 pizzas from the local pizza restaurant and 10 pizzas from the national chain, all at different times. The delivery times in minutes PIZZATIME are shown. Example 1.1 Delivery times

8. Testing for the difference in the mean delivery times LocalChain 16.818.122.019.5 11.714.115.217.0 15.621.818.719.5 16.713.915.616.5 17.520.8 24.0 Example 1.1 Delivery times

9. We can use t-test for this comparison if the delivery times are normal distributed. Since the distribution of delivery times is not normal distributed, we might have difficulty to use the t-test. Example 1.1 Delivery times

10. We can consider the following way to compare these two restaurants Local16.811.715.616.715.718.114.121.813.920.8 Chain22.015.218.715.620.819.517.019.516.524.0 result+++-+++-++

11. If two restaurants have the same level of the delivery time, there is a half chance for “+” and another half for “-”. The number of “+”, denoted by T, follows the binomial distribution with p=0.5. The number of “-” also follows the binomial distribution with p=0.5. T=8 in this example. 1.2 Sign Test ( 符号检验 )

12. Review: Binomial Distribution A. Bernoulli trials A trial with only two outcomes (yes or no, success or fail, boy or girl, win or loss, 1 or 0) and related probabilities p and 1-p, is called a Bernoulli trial. B. Several Bernoulli trials Let X be the number of success in n independently identical Bernoulli trials. Random variable is said to follow a binomial distribution B(n;p).

13. Review: Binomial Distribution C. Binomial probability distribution ( 二项概率分布 ) The probability of X=k is given by

14. 1.2 Sign Test: Example 1.1 One tailed test SPSS result:

15. 1.2 Sign Test: Example 1.1 Two tailed test SPSS result:

16. An Italian restaurant, close to a college campus, contemplated a new recipe for the sauce used on its pizza. A random sample of eight students was chosen, and each was asked to rate on a scale from 1 to 10 the tastes of the original sauce and the propose new one. The scores of the tests comparison are: Example 1.2: Product Preference

17. We can’t use the t-test for this data as the score is not normal distributed. The statistic T, the number of “+”, follows B (7;0.5) as the score of case “G” is zero. This sample gives T=2. Example 1.2: Product Preference

18. 1.2 Sign Test: Example 1.2 One tailed test SPSS result: There is no overall tendency to prefer one product to the other A majority prefer the new product (or fewer than 50% prefer the old product)

19. 1.2 Sign Test: Example 1.2 Two tailed test SPSS result: Also, note that

20. Review: Binomial Distribution C. Properties of the binomial distribution The expectation of B(n;p) is The variance of B(n;p) is The standard deviation of B(n;p) is

21.  D. Normal Approximation (Section 6.4 of the book ） Review: Binomial Distribution where is the distribution function of

22. Example 1.3 Customer Sales (Example 6.8, p. 213) A saleswoman makes initial telephone contact with potential customers in an effort to assess whether a follow-up visit to their homes is likely to be worthwhile. Her experience suggests that 40% of the initial contacts lead to follow-up visit. If she contacts 100 people by telephone, what is the probability that between 45 and 50 home visits will result?

23. Solution to Example 1.3: Customer Sales Solution Let X be the number of follow-up visits. Then X has a binomial distribution with n=100 and p=0.40. Approximating the required probability gives This probability is shown as an area under the standard normal curve below.

24. Solution to Example 1.3: Customer Sales Number of Successes

25. The continuity correction Since the binomial distribution is discrete and the normal distribution is continuous, it is common practice to use continuity correction in the approximation: Return to Example 1.3

26. 1.2 Sign test: normal approximation The approximation test-statistic where corrected for continuity defined as follows: a. For a two-tail test c. For an lower tail test b. For an upper tail test

27. Example 1.4 Ice Cream Solution: Use the normal approximation equations:

28. Example 1.4 Ice Cream The SPSS output:

29. 1.3 Sign test for single population median Example 1.5 Solution: The dean of the School of Business Administration at a particular university would like information about the starting incomes of recent college graduates. A random sample of 23 recent graduates indicated the following starting salaries: 29250 29900 28070 31400 31100 29000 33000 50000 28500 31000 34800 42100 33200 36000 65800 34000 29900 32000 31500 29900 32890 36000 35000 Do the data indicate that the median starting income differs from $35000?

30. Since the distribution of incomes is often skewed, the sign test is recommended. There is a half chance that the income is greater than $35,000 if the hypothesis is true. Let T be the number of the income > $35,000. N=23-1=22 as one data=$35,000. T=17 Solution to Example 1.5

31. SPSS output to Example 1.5

32. 1.4 Wilcoxon Rank Sum Test Two population identical test Take a sample of size from the first population, and a sample of size from the second population, We Want to test

33. 1.4 Wilcoxon Rank Sum Test The sign test does not use all the information from the data set. The sign test for the delivery time in Example 1.1 ignores the time length. The Wilcoxon rank sum test provides a method to incorporate information about the magnitude of the differences between two populations.

34. 1.4 Wilcoxon Rank Sum Test Two samples are pooled and sorted them in ascending order. Let T denote the sum of the ranks of the observations from the first population.

35. Wilcoxon Rank Sum Test: Example 1.1 Sort the Local data 11.7, 13.9, 14.1, 15.6, 16.7, 16.8, 17.5, 18.1, 20,8, 21.8 Sort the Chain data 15.2, 15.6, 16.5, 17.0, 18.7, 19.5, 19.5, 20,8, 22.0, 24.0 Sort the mixed data Rank12345678910 Local11.713.914.115.616.716.8 Chain15.215.616.517.0 Rank11121314151617181920 Local17.518.120.821.8 Chain18.719.5 20.822.024.0

36. Wilcoxon Rank Sum Test: Example 1.1 Sum of the rank Test-statistic Normal approximation

37. SPSS output to Example 1.1

38. Example 1.6

39. Solution: