1. BPS - 3rd Ed. Chapter 101 Sampling Distributions

2. BPS - 3rd Ed. Chapter 102 Parameters and Statistics u Parameter –Is a fixed number that describes the location or spread of a population –Its value is NOT known in statistical practice u Statistic –A calculated number from data in the sample –Its value IS known in statistical practice –Some statistics are used to estimate parameters u Sampling variability –Different samples or experiments from the same population yield different values of the statistic

3. BPS - 3rd Ed. Chapter 103 Parameters and statistics u The mean of a population is called µ  this is a parameter u The mean of a sample is called “x-bar”  this is a statistic u Illustration: –Average age of all SJSU students (µ) is 26.5 –A SRS of 10 San Jose State students yields a mean age (x-bar) of 22.3 –x-bar and µ are related but are not the same thing!

4. BPS - 3rd Ed. Chapter 104 The Law of Large Numbers

5. BPS - 3rd Ed. Chapter 105 Example 10.2 (p. 251) u Dimethyl sulfide (DMS) is sometimes present in wine causing off-odors u Different people have different thresholds for smelling DMS u Winemakers want to know the odor threshold that the human nose can detect. u Population values: –Mean threshold of all adults  is 25 µg/L wine –Standard deviation  of all adults is 7 µg/L –Distribution is Normal Does This Wine Smell Bad?

6. BPS - 3rd Ed. Chapter 106 Simulation: Example 10.2 (cont.) u Suppose you take a simple random sample (SRS) from the population and the first individual in sample has a value of 28 u The second individual has a value of 40 u The average of the first two individuals = (28+40)/2 = 34 u Continue sampling individuals at random and calculating means u Plot the means

7. BPS - 3rd Ed. Chapter 107 Simulation: law of large numbers Fig 10.1 The sample mean gets close to population mean  as we take more and more samples

8. BPS - 3rd Ed. Chapter 108 Sampling distribution of xbar Key questions: What would happen if we took many samples or did the experiment many times? How would the statistics from these repeated samples vary?

9. BPS - 3rd Ed. Chapter 109 Case Study Recall  = 25 µg / L,  = 7 µg / L and the distribution is Normal Suppose you take 1,000 repetitions of samples, each of n =10 from this population You calculate x-bar in each sample You plot the x-bars as a histogram You study the histogram (next slide) Does This Wine Smell Bad?

10. BPS - 3rd Ed. Chapter 1010 Simulation: 1000 sample means Example 10.4

11. BPS - 3rd Ed. Chapter 1011 Mean and Standard Deviation of “x-bar”

12. BPS - 3rd Ed. Chapter 1012 Mean and Standard Deviation of Sample Means  Since the mean of is , we say that is an unbiased estimator of  u Individual observations have standard deviation , but sample means from samples of size n have standard deviation. Averages are less variable than individual observations.

13. BPS - 3rd Ed. Chapter 1013 Case Study Does This Wine Smell Bad? (Population distribution)

14. BPS - 3rd Ed. Chapter 1014 Illustration Exercise 10.8 (p. 258) Suppose blood cholesterol in a population of men is Normal with  = 188 and  = 41. You select 100 men at random (a) What is mean and standard deviation of the 100 x-bars from these samples? –x-bar x-bar = 188 (same as µ) –s x-bar = 41 / sqrt(100) = 4.1 (one-tenth of  ) (b) What is probability a given x-bar is less than 180? Pr(x-bar < 180)  standardize  z = (180 – 188) / 4.1 = -1.95 = Pr(Z < –1.95)  TABLE A  =.0256

15. BPS - 3rd Ed. Chapter 1015 Central Limit Theorem No matter what the shape of the population, the sampling distribution of xbar, will follow a Normal distribution when the sample size is large.

16. BPS - 3rd Ed. Chapter 1016 Central Limit Theorem in Action Example 10.6 (p. 259) u Data = time to perform an activity –NOT Normal (Fig a) –µ = 1 hour –σ = 1 hour Fig (a) is for single observations Fig (b) is for x-bars based on n = 2 Fig (c) is for x-bars based on n = 10 Fig (d) is for x-bars based on n = 25 u Notice how distribution becomes increasingly Normal as n increases!

17. BPS - 3rd Ed. Chapter 1017 Example 10.7 (cont.) u Sample n = 70 activities u What is the distribution of x-bars? –x-bar x-bar = 1 (same as µ) –s x-bar = 1 / sqrt(70) = 0.12 (by formula) –Normal (central limit theorem) u What % of x-bars will be less than 0.83 hours? Pr(x-bar < 0.83)  standardize  z = ( 0.83 – 1) / 0.12 = -1.42 Pr(Z < - 1.42)  TABLE A  = 0.0778 Notice that if Pr(x-bar 0.83) = 1 – 0.0778 = 0.9222

18. BPS - 3rd Ed. Chapter 1018

19. BPS - 3rd Ed. Chapter 1019 Statistical process control* Skip pp. 262 – 269