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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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