1. 1 l Using Statistics l Sample Statistics as Estimators of Population Parameters l Sampling Distributions l Estimators and Their Properties l Degrees of Freedom l Summary and Review of Terms Sampling and Sampling Distributions 5

2. 2 Take random samples from populations Distinguish between population parameters and sample statistics Apply the central limit theorem Derive sampling distributions of sample means and proportions Explain why sample statistics are good estimators of population parameters Judge one estimator as better than another based on desirable properties of estimators Apply the concept of degrees of freedom Identify special sampling methods Compute sampling distributions and related results using templates LEARNING OBJECTIVES 5 After studying this chapter you should be able to:

3. 3 Statistical Inference: Predict and forecast values of population parameters... Test hypotheses about values of population parameters... Make decisions... On basis of sample statistics derived from limited and incomplete sample information Make generalizations about the characteristics of a population... On the basis of observations of a sample, a part of a population Inference( 推論 ) 5-1 Statistics is a Science of Inference( 推論 )

4. 4 Democrats Republicans People who have phones and/or cars and/or are Digest readers. Biased Sample Population Democrats Republicans Unbiased Sample Population Unbiased, representative sample drawn at random from the entire population. Biased, unrepresentative sample drawn from people who have cars and/or telephones and/or read the Digest. The Literary Digest Poll (1936)

5. 5 estimator( 估計式 ) An estimator( 估計式 ) of a population parameter is a sample statistic used to estimate or predict the population parameter. estimate( 估計 ) An estimate( 估計 ) of a parameter is a particular numerical value of a sample statistic obtained through sampling. point estimate A point estimate is a single value used as an estimate of a population parameter. population parameter A population parameter is a numerical measure of a summary characteristic of a population. 5-2 Sample Statistics as Estimators of Population Parameters(p.176) sample statisticA sample statistic is a numerical measure of a summary characteristic of a sample.

6. 6 The sample mean,, is the most common estimator of the population mean,  The sample variance, s 2, is the most common estimator of the population variance,  2. The sample standard deviation, s, is the most common estimator of the population standard deviation, . The sample proportion,, is the most common estimator of the population proportion, p. The sample mean,, is the most common estimator of the population mean,  The sample variance, s 2, is the most common estimator of the population variance,  2. The sample standard deviation, s, is the most common estimator of the population standard deviation, . The sample proportion,, is the most common estimator of the population proportion, p. Estimators( 估計式, 樣本統計量 )

7. Estimators 7

8. 8 population proportion The population proportion is equal to the number of elements in the population belonging to the category of interest, divided by the total number of elements in the population: sample proportion The sample proportion is the number of elements in the sample belonging to the category of interest, divided by the sample size: Population and Sample Proportions  p x n  p X N 

9. 9 XXXX XXXXXX XXXX XXXXXX XXXXXX XXXXXX XXXX Population mean (  ) Sample points Frequency distribution of the population Sample mean ( ) A Population Distribution, a Sample from a Population, and the Population and Sample Means X

10. sampling: Stratified sampling ( 分層抽樣 ) : in stratified sampling, the population is partitioned into two or more subpopulation called strata( 層 ), and from each stratum a desired sample size is selected at random.( 層層有顯著 差異， ex. 性別、變異數 ) Cluster sampling: Cluster sampling ( 分群抽樣 ) : in cluster sampling, a random sample of the strata is selected and then samples from these selected strata are obtained. Single-stage Cluster sampling ( 一階段分群抽樣 ) : in single-stage cluster sampling, a cluster is chosen at random and every item or person in that cluster is sampled. Other Sampling Methods 10

11. Two-stage Cluster sampling ( 二階段分群抽樣 ) : in two-stage cluster sampling, a cluster is chosen at random and random samples of the items or persons are taken from that cluster. Multistage Cluster sampling: Multistage Cluster sampling ( 多階段分群抽樣 ) : in multistage cluster sampling, a random sample of the strata is selected, then samples of items or persons from these strata are taken and then samples from these items and persons are selected. Systematic sampling Systematic sampling ( 系統抽樣 ) Other Sampling Methods 11

12. 12 亂數表之使用 (p.178 or p.702) Example, 10 samples from 001~600 numbered data 習題 5-5

13. 13 sampling distribution The sampling distribution of a statistic is the probability distribution of all possible values the statistic may assume, when computed from random samples of the same size, drawn from a specified population. sampling distribution of X The sampling distribution of X is the probability distribution of all possible values the random variable may assume when a sample of size n is taken from a specified population. 5-3 Sampling Distributions ( 抽樣分配 ) (p.182)

14. 14 Ex. Uniform population of integers from 1 to 8: XP(X)XP(X)(X-  x )(X-  x ) 2 P(X)(X-  x ) 2 10.1250.125-3.512.251.53125 20.1250.250-2.5 6.250.78125 30.1250.375-1.5 2.250.28125 40.1250.500-0.5 0.250.03125 50.1250.625 0.5 0.250.03125 60.1250.750 1.5 2.250.28125 70.1250.875 2.5 6.250.78125 80.1251.000 3.512.251.53125 1.0004.500 5.25000 87654321 0.2 0.1 0.0 X P ( X ) Uniform Distribution (1,8) E(X) =  = 4.5 V(X) =  2 = 5.25 SD(X) =  = 2.2913 Sampling Distributions (Continued)

15. 15 There are 8*8 = 64 different but equally-likely samples of size 2 that can be drawn (with replacement) from a uniform population of the integers from 1 to 8: Each of these samples has a sample mean. For example, the mean of the sample (1,4) is 2.5, and the mean of the sample (8,4) is 6. Sampling Distributions (Continued)

16. 16 Sampling Distribution of the Mean sampling distribution of the the sample mean The probability distribution of the sample mean is called the sampling distribution of the the sample mean.(p.184) X P(X) XP(X)X-  X (X-  X ) 2 P(X)(X-  X ) 2 1.00.0156250.015625-3.512.250.191406 1.50.0312500.046875-3.0 9.000.281250 2.00.0468750.093750-2.5 6.250.292969 2.50.0625000.156250-2.0 4.000.250000 3.00.0781250.234375-1.5 2.250.175781 3.50.0937500.328125-1.0 1.000.093750 4.00.1093750.437500-0.5 0.250.027344 4.50.1250000.562500 0.0 0.000.000000 5.00.1093750.546875 0.5 0.250.027344 5.50.0937500.515625 1.0 1.000.093750 6.00.0781250.468750 1.5 2.250.175781 6.50.0625000.406250 2.0 4.000.250000 7.00.0468750.328125 2.5 6.250.292969 7.50.0312500.234375 3.0 9.000.281250 8.00.0156250.125000 3.512.250.191406 1.0000004.5000002.625000 EX VX SDX X X X () () ().       4.5 2.625 16202 2 Sampling Distributions (Continued)

17. 17 Comparing the population distribution and the sampling distribution of the mean: The sampling distribution is more bell-shaped and symmetric. Both have the same center. The sampling distribution of the mean is more compact, with a smaller variance.(p.184) 87654321 0.2 0.1 0.0 X P ( X ) Uniform Distribution (1,8) X Properties of the Sampling Distribution of the Sample Mean

18. 18 expected value of the sample mean The expected value of the sample mean is equal to the population mean: variance of the sample mean The variance of the sample mean is equal to the population variance divided by the sample size: standard deviation of the sample mean, known as the standard error of the mean The standard deviation of the sample mean, known as the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size: Relationships between Population Parameters and the Sampling Distribution of the Sample Mean

19. 19 normal population normal sampling distribution When sampling from a normal population with mean  and standard deviation , the sample mean, X, has a normal sampling distribution: This means that, as the sample size increases, the sampling distribution of the sample mean remains centered on the population mean, but becomes more compactly distributed around that population mean Normal population 0.4 0.3 0.2 0.1 0.0 f ( X ) Sampling Distribution of the Sample Mean  Sampling Distribution: n =2 Sampling Distribution: n =16 Sampling Distribution: n =4 Sampling from a Normal Population Normal population

20. 20 When sampling from a population with mean  and finite standard deviation , the sampling distribution of the sample mean will tend to a normal distribution with mean  and standard deviation as the sample size becomes large (n >30). For “large enough” n: When sampling from a population with mean  and finite standard deviation , the sampling distribution of the sample mean will tend to a normal distribution with mean  and standard deviation as the sample size becomes large (n >30). For “large enough” n: P ( X ) X 0.25 0.20 0.15 0.10 0.05 0.00 n = 5 P ( X ) 0.2 0.1 0.0 X n = 20 f ( X ) X - 0.4 0.3 0.2 0.1 0.0  Large n The Central Limit Theorem(p.186)

21. 21 NormalUniformSkewed Population n = 2 n = 30 X  X  X  X  General Any The Central Limit Theorem Applies to Sampling Distributions from Any Population(p.187)

22. 22 Mercury makes a 2.4 liter V-6 engine, the Laser XRi, used in speedboats. The company’s engineers believe the engine delivers an average power of 220 horsepower and that the standard deviation of power delivered is 15 HP. A potential buyer intends to sample 100 engines (each engine is to be run a single time). What is the probability that the sample mean will be less than 217HP? The Central Limit Theorem (Example 5-1, p.188)

23. Mercury makes a 2.4 liter V-6 engine, the Laser XRi, used in speedboats. The company’s engineers believe the engine delivers an average power of 220 horsepower and that the standard deviation of power delivered is 15 HP. A potential buyer intends to sample 100 engines (each engine is to be run a single time). What is the probability that the sample mean will be less than 217HP? The Central Limit Theorem (Example 5-1) – Using the Template 23

24. 24 Example 5-2 (p.189)

25. 25 unknown If the population standard deviation, , is unknown, replace  with the sample standard deviation, s. If the population is normal, the resulting statistic: t distribution with (n - 1) degrees of freedom has a t distribution with (n - 1) degrees of freedom. unknown If the population standard deviation, , is unknown, replace  with the sample standard deviation, s. If the population is normal, the resulting statistic: t distribution with (n - 1) degrees of freedom has a t distribution with (n - 1) degrees of freedom. The t is a family of bell-shaped and symmetric distributions, one for each number of degree of freedom. The expected value of t is 0. The variance of t is greater than 1, but approaches 1 as the number of degrees of freedom increases. The t is flatter and has fatter tails than does the standard normal. The t distribution approaches a standard normal as the number of degrees of freedom increases. The t is a family of bell-shaped and symmetric distributions, one for each number of degree of freedom. The expected value of t is 0. The variance of t is greater than 1, but approaches 1 as the number of degrees of freedom increases. The t is flatter and has fatter tails than does the standard normal. The t distribution approaches a standard normal as the number of degrees of freedom increases. Standard normal t, df=20 t, df=10  Student’s t Distribution (p.191)

26. 26 sample proportion The sample proportion is the percentage of successes in n binomial trials. It is the number of successes, X, divided by the number of trials, n. normal distribution As the sample size, n, increases, the sampling distribution of approaches a normal distribution with mean p and standard deviation How to prove it? normal distribution As the sample size, n, increases, the sampling distribution of approaches a normal distribution with mean p and standard deviation How to prove it? Sample proportion: 1514131211109876543210 0.2 0.1 0.0 P ( X ) n=15, p = 0.3 X 14 15 13 15 12 15 11 15 10 15 9 15 8 15 7 15 6 15 5 15 4 15 3 15 2 15 1 15 0 15 ^p^p 210 0.5 0.4 0.3 0.2 0.1 0.0 X P ( X ) n=2, p = 0.3 109876543210 0.3 0.2 0.1 0.0 P ( X ) n=10,p=0.3 X The Sampling Distribution of the Sample Proportion,

27. 27 公式證明

28. 28 In recent years, convertible sports coupes have become very popular in Japan. Toyota is currently shipping Celicas to Los Angeles, where a customizer does a roof lift and ships them back to Japan. Suppose that 25% of all Japanese in a given income and lifestyle category are interested in buying Celica convertibles. A random sample of 100 Japanese consumers in the category of interest is to be selected. What is the probability that at least 20% of those in the sample will express an interest in a Celica convertible? n p npEp pp n Vp pp n SDp        100 025 100025 1 75 100 0001875 1 0 004330127. ()(.)(  ) () (.)(.).(  ) ()..(  ) PpP pp pp n p pp n PzPz Pz ( .)  (). ().. (.)(.).. (.).                                      020 1 1 25 75 100 05 0433 11508749 Sample Proportion (Example 5-3)(p.193)

29. In recent years, convertible sports coupes have become very popular in Japan. Toyota is currently shipping Celicas to Los Angeles, where a customizer does a roof lift and ships them back to Japan. Suppose that 25% of all Japanese in a given income and lifestyle category are interested in buying Celica convertibles. A random sample of 100 Japanese consumers in the category of interest is to be selected. What is the probability that at least 20% of those in the sample will express an interest in a Celica convertible? Sample Proportion (Example 5-3) – Template Solution 29

30. 30 Exercise (p.193, 15min) Exercise 5-12 (Ans:1.Binomial, 2.No, 不適用 ) Exercise 5-13 (Ans: 0.075 ) Exercise 5-14 (Ans: 0.9923 ) Exercise 5-15 (Ans: 0.2308 ) Exercise 5-16 (Ans: 0.0497 )

31. 31 estimator An estimator of a population parameter is a sample statistic used to estimate the parameter. The most commonly-used estimator of the: Population Parameter Sample Statistic Mean (  ) is theMean (X) Variance (  2 )is the Variance (s 2 ) Standard Deviation (  )is the Standard Deviation (s) Proportion (p)is the Proportion ( ) estimator An estimator of a population parameter is a sample statistic used to estimate the parameter. The most commonly-used estimator of the: Population Parameter Sample Statistic Mean (  ) is theMean (X) Variance (  2 )is the Variance (s 2 ) Standard Deviation (  )is the Standard Deviation (s) Proportion (p)is the Proportion ( ) Desirable properties of estimators include: Unbiasedness ( 不偏性 ) Efficiency ( 有效性 ) Consistency ( 一致性 ) Sufficiency ( 充分性 ) Desirable properties of estimators include: Unbiasedness ( 不偏性 ) Efficiency ( 有效性 ) Consistency ( 一致性 ) Sufficiency ( 充分性 ) 5-4 Estimators and Their Properties (p.177)

32. 32 unbiased An estimator is said to be unbiased if its expected value is equal to the population parameter it estimates. For example, E(X)=  so the sample mean is an unbiased estimator of the population mean. Unbiasedness is an average or long-run property. The mean of any single sample will probably not equal the population mean, but the average of the means of repeated independent samples from a population will equal the population mean. systematic deviation bias Any systematic deviation of the estimator from the population parameter of interest is called a bias. unbiased An estimator is said to be unbiased if its expected value is equal to the population parameter it estimates. For example, E(X)=  so the sample mean is an unbiased estimator of the population mean. Unbiasedness is an average or long-run property. The mean of any single sample will probably not equal the population mean, but the average of the means of repeated independent samples from a population will equal the population mean. systematic deviation bias Any systematic deviation of the estimator from the population parameter of interest is called a bias. Unbiasedness (p.194)

33. 33 unbiased An unbiased estimator is on target on average. biased A biased estimator is off target on average. { Bias Unbiased and Biased Estimators

34. 34 efficient An estimator is efficient if it has a relatively small variance (and standard deviation). efficient An efficient estimator is, on average, closer to the parameter being estimated.. inefficient An inefficient estimator is, on average, farther from the parameter being estimated. Efficiency (p.195)

35. 35 consistent An estimator is said to be consistent if its probability of being close to the parameter it estimates increases as the sample size increases. sufficient An estimator is said to be sufficient if it contains all the information in the data about the parameter it estimates. n = 100 n = 10 Consistency Consistency and Sufficiency(p.196)

36. 36 unbiased estimators efficient sufficient For a normal population, both the sample mean and sample median are unbiased estimators of the population mean, but the sample mean is both more efficient (because it has a smaller variance), and sufficient. Every observation in the sample is used in the calculation of the sample mean, but only the middle value is used to find the sample median. best In general, the sample mean is the best estimator of the population mean. The sample mean is the most efficient unbiased estimator of the population mean. It is also a consistent estimator. Properties of the Sample Mean(p.196)

37. 37 sample variance unbiased estimator The sample variance (the sum of the squared deviations from the sample mean divided by (n-1) is an unbiased estimator of the population variance. In contrast, the average squared deviation from the sample mean is a biased (though consistent) estimator of the population variance. EsE xx n E xx n () () () () 2 2 2 2 2 1                       Properties of the Sample Variance(p.197)

38. 38 公式證明 :

39. 39 公式證明 (Continued)

40. 40 Exercise, p.197, 5min 5-19 (Ans. 1300) 5-21 (Ans. n ＞ 100 )

41. 41 Consider a sample of size n=4 containing the following data points: x 1 =10x 2 =12x 3 =16x 4 =? and for which the sample mean is: Given the values of three data points and the sample mean, the value of the fourth data point can be determined: 5-5 Degrees of Freedom x 4 = 14

42. 42 If only two data points and the sample mean are known: x 1 =10x 2 =12x 3 =?x 4 =? The values of the remaining two data points cannot be uniquely determined: Degrees of Freedom (Continued)

43. 43 The number of degrees of freedom is equal to the total number of measurements (these are not always raw data points), less the total number of restrictions on the measurements. A restriction is a quantity computed from the measurements. The sample mean is a restriction on the sample measurements, so after calculating the sample mean there are only (n-1) degrees of freedom remaining with which to calculate the sample variance. The sample variance is based on only (n-1) free data points: The number of degrees of freedom is equal to the total number of measurements (these are not always raw data points), less the total number of restrictions on the measurements. A restriction is a quantity computed from the measurements. The sample mean is a restriction on the sample measurements, so after calculating the sample mean there are only (n-1) degrees of freedom remaining with which to calculate the sample variance. The sample variance is based on only (n-1) free data points: s xx n 2 2 1     () () Degrees of Freedom (Continued)

44. 44 A sample of size 10 is given below. We are to choose three different numbers from which the deviations are to be taken. The first number is to be used for the first five sample points; the second number is to be used for the next three sample points; and the third number is to be used for the last two sample points. Example 5-4 (p.201) Sample #12345678910 Sample Point 93976072968359668853 i.What three numbers should we choose in order to minimize the SSD (sum of squared deviations from the mean).? Note:

45. 45 Solution: Solution: Choose the means of the corresponding sample points. These are: 83.6, 69.33, and 70.5. ii.Calculate the SSD with chosen numbers. Solution: Solution: SSD = 2030.367. See table on next slide for calculations. iii.What is the df for the calculated SSD? Solution: Solution: df = 10 – 3 = 7. iv.Calculate an unbiased estimate of the population variance. Solution: Solution: An unbiased estimate of the population variance is SSD/df = 2030.367/7 = 290.05. Solution: Solution: Choose the means of the corresponding sample points. These are: 83.6, 69.33, and 70.5. ii.Calculate the SSD with chosen numbers. Solution: Solution: SSD = 2030.367. See table on next slide for calculations. iii.What is the df for the calculated SSD? Solution: Solution: df = 10 – 3 = 7. iv.Calculate an unbiased estimate of the population variance. Solution: Solution: An unbiased estimate of the population variance is SSD/df = 2030.367/7 = 290.05. Example 5-4 (continued)

46. 46 Example 5-4 (continued) Sample #Sample PointMeanDeviationsDeviation Squared 19383.69.488.36 29783.613.4179.56 36083.6-23.6556.96 47283.6-11.6134.56 59683.612.4153.76 68369.3313.6667186.7778 75969.33-10.3333106.7778 86669.33-3.333311.1111 98870.517.5306.25 105370.5-17.5306.25 SSD2030.367 SSD/df290.0524

47. Sampling Distribution of a Sample Mean 5-6 Using the Computer Using the Template 47

48. Sampling Distribution of a Sample Mean (continued) Using the Template 48

49. Sampling Distribution of a Sample Proportion Using the Template 49

50. Using the Template Sampling Distribution of a Sample Proportion (continued) 50

51. Constructing a sampling distribution of the mean from a uniform population (n = 10) using EXCEL (use RANDBETWEEN(0, 1) command to generate values to graph): Histogram of Sample Means 0 50 100 150 200 250 123456789101112131415 Sample Means (Class Midpoints) Frequency CLASS MIDPOINTFREQUENCY 0.150 0.20 0.253 0.326 0.3564 0.4113 0.45183 0.5213 0.55178 0.6128 0.6565 0.720 0.753 0.83 0.850 999 Using Excel to Generate Random Data 51

52. 52 Exercise, p.206, 15min Exercise5-27 (Ans. 1065, 2500 ) Exercise5-28 (Ans. 53, 0.5 ) Exercise5-29 (Ans. 0.2, 0.04216) Exercise5-30 (Ans. 0.923) Exercise5-31 (Ans. 0.9544) Exercise5-34 (Ans. 1.000) Exercise5-35 (Ans. 0.9503) Exercise5-36 (Ans. 不是最小的（ n ＝ 1 對常態性已經夠了 ) ) Exercise5-40 (Ans. P(Z< － 5) ＝ 0.0000003 不太可能 )