3. Life is a school of probability... Walter Bagehot (English Economist) I don't believe in providence and fate, as a technologist I am used to reckoning with the formulae of probability… Max Frisch (German Architect and Novelist)

4. Most of the really important decisions in life will involve incomplete information. In one life time we cannot experience everything. Nor should we even want to! This is one reason why experience by way of sampling is so important. Statistics can help you have the experiences and yet maintain some control over mistakes. Remember, it is what you make out of experience (sample data) that is of real value…

5. AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1 § R§ R§ R§ Review of Statistical Terms –P–P–P–Population, from a statistical point of view, is considered as a set of measurements or counts, existing or conceptual –S–S–S–Sample is a subset of measurements from the population. Random Samples are considered for this section

6. AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1 § Review of Statistical Terms –P–P–P–Parameter is a numerical descriptive measure of a population. In statistical practice the value of a parameter is not know, it is not possible to examine the entire population –S–S–S–Statistic is a numerical descriptive measure of a sample, not depending of any unknown parameter. An statistic is used to estimate an unknown parameter

7. § C§ C§ C§ Common Statistics and Parameters MeasureStatistic AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1 Parameter Mean Variance Standard Deviation Proportion

8. § W§ W§ W§ Why Sample? At times, we’d like to know something about the population, but because our time, resources, and efforts are limited, we can take just a sample to learn about the population Ex: Take a sample of voters to learn about probable election results (before the final count). AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

9. § Why Sample? AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1 Inference is to draw conclusions for a entire population from the information of a sample So we must use measurements from a sample instead. In such cases, we will use a statistic (, s, or ) to make inferences about corresponding population parameters ( , , or p )

10. § T§ T§ T§ Types of Inference Estimation. In this case, we estimate or approximate the value of a population parameter Testing: In this case, we formulate a decision about a population parameter AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1 Regression: In this case, we make predictions or forecasts about the value of a statistical variable

11. § A§ A§ A§ Are Inferences Reliable? To evaluate the reliability of our inference, we need to know about the probability distribution of the statistic we are using AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1 Typically, we are interested in the sampling distributions for sample means and sample proportions

12. § S§ S§ S§ Sampling Distributions A Sampling Distribution is a probability distribution of a sample statistic based on all possible simple random samples of the same size from the same population AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

13. § S§ S§ S§ Sampling Distributions Example In a rural community with a children’s fishing pond, are posted rules stating that all fish under 6 inches must be returned to the pond, and the limit of five fish per day may be kept. 100 random samples of five trout are taken and recorded the lengths of the five trout. What is the average (mean) length of a trout taken from the pond ( tttt eeee xxxx tttt bbbb oooo oooo kkkk p p p p pppp.... 3333 6666 2222 t t t t aaaa bbbb llll eeee 7 7 7 7 ---- 1111) AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

14. Practice Textbook Section 8.1 Problems: pp. 365 CCCC hhhh eeee cccc kkkk iiii nnnn gggg f f f f oooo rrrr U U U U nnnn dddd eeee rrrr ssss tttt aaaa nnnn dddd iiii nnnn gggg HHHH MMMM S S S S TTTT AAAA TTTT S S S S pppp aaaa cccc eeee ( ( ( ( 7777.... 1111 )))) AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

15. Information: the negative reciprocal value of probability… Claude Shannon (American Mathematical Engineer) From principles is derived probability, but truth or certainty is obtained only from facts. Tom Stoppard (English Playwriths)

16. There is an old saying: All roads lead to Rome. In Statistics we can recast this saying: All probability distributions average out to the Normal distribution, (as the sample size increases)

17. § T§ T§ T§ The Central Limit Theorem (Normal) AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 For a Normal Probability Distribution, let x be a random variable with a normal distribution whose mean is , and whose standard deviation is . Let be the sample mean corresponding to random samples of size n taken form the x distribution. Then the following is true: – The distribution is a normal distribution – The mean of the distribution is  – The standard deviation of the distribution is:

18. § Central Limit Theorem. Converting to z n is the sample size  is the mean of the x distribution  is the standard deviation of the x distribution AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 We can convert the distribution to the standard normal z distribution using the following formulas

19. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § S§ S§ S§ Sample Size Considerations For the Central Limit Theorem (CLT) to be applicable: –I–I–I–If the x distribution is symmetric or reasonably symmetric, n ≥ 30 should suffice –I–I–I–If the x distribution is highly skewed or unusual, even larger sample sizes will be required –I–I–I–If possible, make a graph to visualize how the sampling distribution is behaving

20. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § C§ C§ C§ Central Limit Theorem. (Any Distribution) If x posses any distribution with mean  and standard deviation , then the sample mean based on a random sample of size n will have a normal distribution that approaches the distribution of a normal random variable with mean  and standard deviation, as n increases without limit

21. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 §F§F§F§Finding Probabilities Using Central Limit Theorem Given a probability distribution of x values with sample size n, mean , and standard deviation  : –I–I–I–If the x distribution is normal, then the –E–E–E–Even if the x distribution is not normal, if the sample of the size is n  33330, then by CLT, the distribution is normal distribution is approximately normal

22. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § Finding Probabilities Using Central Limit Theorem Given a probability distribution of x values with mean , standard deviation , and sample of size n –C–C–C–Convertto z using the formula:

23. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § Finding Probabilities Using Central Limit Theorem Given a probability distribution of x values with mean , standard deviation , and sample of size n –U–U–U–Use the standard normal distribution to find the corresponding probability for the events regarding

24. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § C§ C§ C§ Central Limit Theorem. Example The heights of 18-year old men are approximately normally distributed, with a mean  ==== 68 inches and a standard deviation  = 3 inches a. What is the probability that a randomly selected man is taller than 72 inches? Get the z score: Find Probability: P(z>72) = 1 – P(z<72) =.0918

25. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § Central Limit Theorem. Example The heights of 18-year old men are approximately normally distributed, with a mean  = 68 in and a standard deviation  = 3 in b. What’s the probability that the average height of 2 randomly selected men is greater than 72 in? Using CLT with n = 2:

26. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2 § Central Limit Theorem. Example The heights of 18-year old men are approximately normally distributed, with a mean  = 68 in and a standard deviation  = 3 in c. What is the probability that the average height of 16 randomly selected men is greater than 72 in? Using CLT with n = 16:

27. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2

28. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2

29. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2

30. AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2

31. Practice Textbook Section 7.2 Problems: pp. 373 – 379 CCCC hhhh eeee cccc kkkk iiii nnnn gggg f f f f oooo rrrr U U U U nnnn dddd eeee rrrr ssss tttt aaaa nnnn dddd iiii nnnn gggg HHHH MMMM S S S S TTTT AAAA TTTT S S S S pppp aaaa cccc eeee ( ( ( ( 7777.... 2222 )))) AP Statistics The Central Limit Theorem Braser–Braser Chapter 7.2

32. The scientific imagination always restrains itself within the limits of probability... Thomas Huxley (English Biologist) A property in the 100-year floodplain has a 96 percent chance of being flooded in the next hundred years without global warming. The fact that several years go by without a flood does not change that probability... Earl Blumenauer (Oregon Representative)

33. Many issues in life come down to success or failure. In most cases, we will not be successful all the time, so proportions of successes are very important. What is the probability sampling distributions for proportions?…

34. The annual crime rate in the Capital Hill of Denver is 1 11 victims per 1 000 residents. ( 111 out of 1 000 residents have been victim of a least one crime). These crimes range from minor crimes (stolen hubcaps or purse snatching) to major crimes (murder). The Arms is an apartment building on this neighborhood that has 50 year-round residents. Consider each of the n = 50 residents as a binomial trial. The random variable r, (1 = r = 50), represents the number of victims of a least one crime next year. (a)What is the population probability p that a resident a resident in the Capital Hill neighborhood will be / will not be a victim of a crime? (b)What is the probability that between 1 0% and 2 0% of the Arms residents will be victims of a crime next year? Hint: Use the binomial distribution. Use the normal approach to the binomial. Compare answers

35. § S§ S§ S§ Sampling Distribution for the Proportion Given:n = number of binomial trials (constant) If np > 5 and nq > 5, then the random variable AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 r = number of successes p = probability of success on each trial q = 1 – p = probability of failure on each trial can be approximated by a normal random variable x with

36. § C§ C§ C§ Continuity Corrections Since is discrete, but x is continuous, we have to make a continuity correction; for a small n, the correction is meaningful AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 How to make corrections to 1.I f r/n is the right end point of a intervals add 0.5/n to get the corresponding right end point of the x interval interval, we

37. § Continuity Corrections Since is discrete, but x is continuous, we have to make a continuity correction; for a small n, the correction is meaningful AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 How to make corrections to 2.I f r/n is the left end point of a intervals subtract 0.5/n to get the corresponding left end point of the x interval interval, we

38. § P§ P§ P§ Proportion Sampling Distribution. Example Suppose the annual crime rate in Denver is p = 0.111 If 50 people live in an apartment complex, what is the probability that between 10% and 20% of the residents will be victims of crimes next year? AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 Checking conditions (np >5, nq > 5): np = (50)(.111) = 5.55 n nq = (50)(.889) = 44.45 can be approximated with a normal distribution n = 50, p = 0.111, q = 1 – p = 1 – 0.111 = 0.899

39. § Proportion Sampling Distribution. Example Suppose the annual crime rate in Denver is p = 0.111 If 50 people live in an apartment complex, what is the probability that between 10% and 20% of the residents will be victims of crimes next year? AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 n = 50, p = 0.111, q = 0.899

40. § Proportion Sampling Distribution. Example Suppose the annual crime rate in Denver is p = 0.111 If 50 people live in an apartment complex, what is the probability that between 10% and 20% of the residents will be victims of crimes next year? AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 n = 50, p = 0.111, q = 0.899 Continuity Correction (0.5/n): 0.5/50 = 0.01

41. § Proportion Sampling Distribution. Example Suppose the annual crime rate in Denver is p = 0.111 If 50 people live in an apartment complex, what is the probability that between 10% and 20% of the residents will be victims of crimes next year? AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 n = 50, p = 0.111, q = 0.899 Using z-scores:  = 0.111,  = 0.044

42. § Proportion Sampling Distribution. Example Suppose the annual crime rate in Denver is p = 0.111 If 50 people live in an apartment complex, what is the probability that between 10% and 20% of the residents will be victims of crimes next year? AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 n = 50, p = 0.111, q = 0.899 Thus, there is about a 67% chance that between 10% and 20% of the residents will be victims of a crime next year.

43. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

44. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

45. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

46. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

47. § C§ C§ C§ Control Charts for Proportions Used to examine an attribute or quality of an observation (rather than a measurement). AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 –T–T–T–Then use the normal approximation of the sample proportion to determine the control limits –S–S–S–Select a fixed sample size, n, at fixed time intervals, and determine the sample proportions at each interval How to use it:

48. § H§ H§ H§ How to Make a P-Chart 1. Estimate AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 3. Control limits are located at: 2. Take the center line of control chart as:, the overall proportion of successes and

49. § P -Chart. Out of Control Signals Signal 1. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 At least two out of three consecutive points are beyond the control limits Run of nine consecutive points on one side of the center line Any point beyond control limit Signal 2. Signal 3.

50. § P-Chart. Out of Control Signals AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 If no out-of-control signals occur, we say that the process is in control, while keeping a watchful eye on what occurs next In some P-Charts the value of In this case, the control limits may drop below 0 or rise above 1. If this happens, follow the convention of rounding negative control limits to 0 and control limits above 1 to 1 may be near 0 or 1

51. § C§ C§ C§ Control Charts for Proportions. Example (pp. 384) (a) Estimate the overall proportion of successes AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

52. § Control Charts for Proportions. Example ( pp. 84) (b) Calculate AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 and

53. § Control Charts for Proportions. Example ( pp. 84) (c) Estimate AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 and Both are greater than 5, this means the normal distribution should be reasonable good

54. § Control Charts for Proportions. Example ( pp. 84) (d) Estimate the control limits of the P-Chart AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

55. § Control Charts for Proportions. Example ( pp. 84) (d) Estimate the control limits of the P-Chart AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

56. § Control Charts for Proportions. Example ( pp. 84) AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

57. § C§ C§ C§ Control Charts for Proportions. Example (pp. 84) AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 The Proportion of A’s given in class is in statistical control, with exception of the one unusually good class two semesters ago Out of control signals Signal 1. Semester 12 above 3s level (Very good class!) Signal 2. Not present Signal 3. Not present

58. Practice Textbook Section 7.3 Problems: pp. 387 – 389 CCCC hhhh eeee cccc kkkk iiii nnnn gggg f f f f oooo rrrr U U U U nnnn dddd eeee rrrr ssss tttt aaaa nnnn dddd iiii nnnn gggg HHHH MMMM S S S S TTTT AAAA TTTT S S S S pppp aaaa cccc eeee ( ( ( ( 7777.... 3333 )))) AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3

59. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 pp 389

60. AP Statistics Sampling Distributions for Proportions Braser–Braser Chapter 7.3 pp 389

61. Custom Shows

62. § Sampling Distributions Example AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

63. § Sampling Distributions Example AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

64. § Sampling Distributions Example AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

65. § Sampling Distributions Example AP Statistics Sampling Distributions? Braser–Braser Chapter 7.1

66. AP Statistics Normal Approximation to Binomial Distribution Braser–Braser Chapter 7.4 § N§ N§ N§ Normal Approximation to Binomial Error The error of the normal approximation to the binomial distribution decreases and becomes negligible as the number of trials n increases However, if the number of trials is not big, the error in this approximation can not be ignored…

67. AP Statistics Normal Approximation to Binomial Distribution § Normal Approximation to Binomial Error Binomial Probability Normal Approach n = 50 p = 0.111 q = 0.889  = 6.555  = 2.221 5 10 + Continuity Correction

68. AP Statistics Normal Approximation to Binomial Distribution § Normal Approximation to Binomial Error Binomial Probability Normal Approach 0.1 0.2 + Continuity Correction

69. AP Statistics Normal Approximation to Binomial Distribution Braser–Braser Chapter 7.4 § N§ N§ N§ Normal Approximation to Binomial Continuity Correction for Step 1. If p is a left-point of an interval, subtract 0.5/n to obtain the corresponding random variable x: Step 2. If p is a right-point of an interval, add 0.5/n to obtain the corresponding random variable x: ^ ^