Sampling Distributions

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BPS - 5th Ed.. Chapter 11. 2. Sampling Terminology. Parameterfixed, unknown number that describes the populationStatisticknown value calculated from a samplea statistic is often used to estimate a parameterVariabilitydifferent samples from the same population may yield different values of the

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Sampling Distributions

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1. BPS - 5th Ed. Chapter 11 1 Chapter 11 Sampling Distributions

2. BPS - 5th Ed. Chapter 11 2 Sampling Terminology Parameter fixed, unknown number that describes the population Statistic known value calculated from a sample a statistic is often used to estimate a parameter Variability different samples from the same population may yield different values of the sample statistic Sampling Distribution tells what values a statistic takes and how often it takes those values in repeated sampling

3. BPS - 5th Ed. Chapter 11 3 Parameter vs. Statistic From Seeing Through Statistics, 2nd Edition by Jessica M. Utts. From Seeing Through Statistics, 2nd Edition by Jessica M. Utts.

4. BPS - 5th Ed. Chapter 11 4 Parameter vs. Statistic From Seeing Through Statistics, 2nd Edition by Jessica M. Utts. From Seeing Through Statistics, 2nd Edition by Jessica M. Utts.

5. BPS - 5th Ed. Chapter 11 5 The Law of Large Numbers Consider sampling at random from a population with true mean µ. As the number of (independent) observations sampled increases, the mean of the sample gets closer and closer to the true mean of the population.

6. BPS - 5th Ed. Chapter 11 6 The Law of Large Numbers Gambling The “house” in a gambling operation is not gambling at all the games are defined so that the gambler has a negative expected gain per play (the true mean gain after all possible plays is negative) each play is independent of previous plays, so the law of large numbers guarantees that the average winnings of a large number of customers will be close the the (negative) true average

7. BPS - 5th Ed. Chapter 11 7 Sampling Distribution The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size (n) from the same population to describe a distribution we need to specify the shape, center, and spread we will discuss the distribution of the sample mean (x-bar) in this chapter

8. BPS - 5th Ed. Chapter 11 8 Case Study

9. BPS - 5th Ed. Chapter 11 9 Case Study

10. BPS - 5th Ed. Chapter 11 10 Where should 95% of all individual threshold values fall? mean plus or minus two standard deviations 25 ? 2(7) = 11 25 + 2(7) = 39 95% should fall between 11 & 39 What about the mean (average) of a sample of n adults? What values would be expected?

11. BPS - 5th Ed. Chapter 11 11 Sampling Distribution What about the mean (average) of a sample of n adults? What values would be expected?

12. BPS - 5th Ed. Chapter 11 12 Case Study

13. BPS - 5th Ed. Chapter 11 13 Case Study

14. BPS - 5th Ed. Chapter 11 14 Mean and Standard Deviation of Sample Means

15. BPS - 5th Ed. Chapter 11 15 Mean and Standard Deviation of Sample Means

16. BPS - 5th Ed. Chapter 11 16 Sampling Distribution of Sample Means

17. BPS - 5th Ed. Chapter 11 17 Case Study

18. BPS - 5th Ed. Chapter 11 18

19. BPS - 5th Ed. Chapter 11 19

20. BPS - 5th Ed. Chapter 11 20 Central Limit Theorem

21. BPS - 5th Ed. Chapter 11 21 Central Limit Theorem: Sample Size

22. BPS - 5th Ed. Chapter 11 22 Central Limit Theorem: Sample Size and Distribution of x-bar

23. BPS - 5th Ed. Chapter 11 23

24. BPS - 5th Ed. Chapter 11 24 Statistical Process Control Goal is to make a process stable over time and keep it stable unless there are planned changes All processes have variation Statistical description of stability over time: the pattern of variation remains stable (does not say that there is no variation)

25. BPS - 5th Ed. Chapter 11 25 Statistical Process Control A variable described by the same distribution over time is said to be in control To see if a process has been disturbed and to signal when the process is out of control, control charts are used to monitor the process distinguish natural variation in the process from additional variation that suggests a change most common application: industrial processes

26. BPS - 5th Ed. Chapter 11 26 Charts There is a true mean m that describes the center or aim of the process Monitor the process by plotting the means (x-bars) of small samples taken from the process at regular intervals over time Process-monitoring conditions: measure quantitative variable x that is Normal process has been operating in control for a long period know process mean m and standard deviation s that describe distribution of x when process is in control

27. BPS - 5th Ed. Chapter 11 27 Control Charts Plot the means (x-bars) of regular samples of size n against time Draw a horizontal center line at m

28. BPS - 5th Ed. Chapter 11 28 Case Study Need to control the tension in millivolts (mV) on the mesh of fine wires behind the surface of the screen. Proper tension is 275 mV (target mean m) When in control, the standard deviation of the tension readings is s=43 mV

29. BPS - 5th Ed. Chapter 11 29 Case Study Proper tension is 275 mV (target mean m). When in control, the standard deviation of the tension readings is s=43 mV.

30. BPS - 5th Ed. Chapter 11 30 Case Study

31. BPS - 5th Ed. Chapter 11 31 Case Study

32. BPS - 5th Ed. Chapter 11 32 Case Study

33. BPS - 5th Ed. Chapter 11 33 Natural Tolerances

34. Exercise 11.34: Airline passengers average 190 pounds (including carry on luggage) with a standard deviation of 35 pounds. Weights are not Normally distributed but they are not very non-Normal. A commuter plane carries 19 passengers. What is the approximate probability that the total weight of the passengers exceeds 400 pounds? BPS - 5th Ed. Chapter 11 34

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