1 Ch6. Sampling distribution Dr. Deshi Ye

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

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

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

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

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

7/ 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.

8/38 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.

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

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.

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

12/ 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.

13/38  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 © T/Maker Co.

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”?

15/38 Population Characteristics Population Distribution Summary Measures

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

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

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

19/38 Comparison Population Sampling Distribution

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.

21/38 Sample means  We get 50 samples whose means are

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

23/38 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

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

25/38 EX  If a 1-gallon can of paint covers on the average 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?

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

27/38 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?

28/38 Solution Sampling Distribution Standardized Normal Distribution

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)

30/38 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

31/38 t-distribution

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 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?

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

34/ 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

35/38 Chi-square distribution

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

37/38 F distribution

38/38 Thanks!