1. 1 Ch6. Sampling distribution Dr. Deshi Ye yedeshi@zju.edu.cn

2. 2/38 Outline  Population and sample  The sampling distribution of the mean ( known)  The sampling distribution of the mean ( unknown)  The sampling distribution of the variance

3. 3/38 Statistics  Descriptive statistics  Inferential statistics  Remarks: many thanks to Paul Resnick for some slides

4. 4/38 Inferential Statistics  1.Involves: Estimation Hypothesis Testing  2.Purpose Make Inferences about Population Characteristics Population?

5. 5/38 Inference Process Population Sample Sample statistic (X) Estimates & tests

6. 6/38 Key terms  Population All items of interest  Sample Portion of population  Parameter Summary Measure about Population  Statistic Summary Measure about sample

7. 7/38 6.1 Population and Sample  Population: refer to a population in term of its probability distribution or frequency distribution. Population f(x) means a population described by a frequency distribution, a probability distribution f(x)  Population might be infinite or it is impossible to observe all its values even finite, it may be impractical or uneconomical to observe it.

8. 8/38 Sample  Sample: a part of population.  Random samples (Why we need?): such results can be useful only if the sample is in some way “representative”.  Negative example: performance of a tire if it is tested only on a smooth roads; family incomes based on the data of home owner only.

9. 9/38 Sampling  Representative sample Same characteristics as the population  Random sample Every subset of the population has an equal chance of being selected

10. 10/38 Random sample  Random sample: A set of observations constitutes a random sample of size n from a finite population of size N, if its value are chosen so that each subset of n of the N elements of the population has the same probability of being selected.

11. 11/38 Discussion  Ways assuring the selection of a sample is at least approximately random  Both finite population and infinite population

12. 12/38 6.2 The sampling distribution of the Mean ( known)  Random sample of n observations, and its mean has been computed.  Another random sample of n observation, and also its mean has been computed.  Probably no two of them are alike.

13. 13/38  Suppose There’s a Population...  Population Size, N = 4  Random Variable, x, Is # Errors in Work  Values of x: 1, 2, 3, 4  All values equally likely  Estimate based on a sample of two © 1984-1994 T/Maker Co.

14. 14/38 Checking list  What is the experiment corresponding to random variable X?  What is the experiment corresponding to the random variable ?  What is “the sampling distribution of the mean”?

15. 15/38 Population Characteristics Population Distribution Summary Measures

16. 16/38 All Possible Samples of Size n = 2 16 Samples Sample with replacement

17. 17/38 All Possible Samples of Size n = 2 16 Samples 16 Sample Means Sample with replacement

18. 18/38 Sampling Distribution of All Sample Means 16 Sample Means Sampling Distribution

19. 19/38 Comparison Population Sampling Distribution

20. 20/38 EX  Suppose that 50 random samples of size n=10 are to be taken from a population having the discrete uniform distribution sampling is with replacement, so to speak, so that we sampling from an infinite population.

21. 21/38 Sample means  We get 50 samples whose means are 4.4 3.2 5.0 3.5 4.1 4.4 3.6 6.5 5.3 4.4 3.1 5.3 3.8 4.3 3.3 5.0 4.9 4.8 3.1 5.3 3.0 3.0 4.6 5.8 4.6 4.0 3.7 5.2 3.7 3.8 5.3 5.5 4.8 6.4 4.9 6.5 3.5 4.5 4.9 5.3 3.6 2.7 4.0 5.0 2.6 4.2 4.4 5.6 4.7 4.3

22. 22/38 Theorem  If a random sample of size n is taken from a population having the mean and the variance, then is a random variable whose distribution has the mean For samples from infinite populations the variance of this distribution is For samples from a finite population without replacement of size N the variance is

23. 23/38 Central limit theorem  If is the mean of a sample of size n taken from a population having the mean and the finite variance, then is a random variable whose distribution function approaches that of the standard normal distribution as

24. 24/38 Central Limit Theorem As sample size gets large enough (n  30)... sampling distribution becomes almost normal.

25. 25/38 EX  If a 1-gallon can of paint covers on the average 513.3 square feet with a standard variation of 31.5 square feet.  Question: what is the probability that the sample mean area covered by a sample of 40 of these 1-gallon cans will be anywhere from 510 to 520 square feet?

26. 26/38 Solution  We shall have to find the normal curve area between and Check from the cumulative standard normal distribution Table Hence, the probability is

27. 27/38 Another example  You’re an operations analyst for AT&T. Long-distance telephone calls are normally distributed with  = 8 min. &  = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes?

28. 28/38 Solution Sampling Distribution.3830.3830.1915.1915 Standardized Normal Distribution

29. 29/38  If n is large, it doesn’t matter whether is known or not, as it is reasonable in that case to substitute for it the sample standard deviation s. Question: how about n is a small value? We need to make the assumption that the sample comes from a normal population. 6.2 The sampling distribution of the Mean ( unknown)

30. 30/38 Assumption: population having normal distribution  If is the mean of a random sample of size n taken from a normal population having the mean and the variance, and, then is a random variable having the t distribution with the parameter

31. 31/38 t-distribution

32. 32/38 EX.  A manufacturer of fuses claims that with a 20% overload, the fuses will blow in 12.4 minutes on the average. To test this claim, sample of 20 of the fuses was subjected to a 20% overload, and the times it took them to blow had a mean of 10.63 minutes and a standard deviation of 2.48 minutes. If it can be assumed that the data constitute a random sample from a normal population. Question: do they tend to support or refute the manufacturer’s claim?

33. 33/38 Solution  First, we calculate Rule to reject the claim: t value is larger than 2.86 or less than -2.86 where And

34. 34/38 6.4 The Sampling distribution of the variance  Theorem 6.4. If is the variance of a random sample of size n taken from a normal population having the variance then is a random variable having the chi- square distribution with the parameter

35. 35/38 Chi-square distribution

36. 36/38 F distribution  Theorem. If and are the variances of independent random samples of size and, respectively, taken from two normal populations having the same variance, then is a random variable having the F distribution with the parameter

37. 37/38 F distribution

38. 38/38 Thanks!