1. Chapter 3 Basic Concepts in Statistics and Probability

2. 3.7 Choice of Statistical Distributions The distributions are simply models of reality (not reality themselves) Always check the assumptions

3. 3.8 Statistical Inference Methods of Statistical Inference used to estimate the parameters of the statistical distributions: –Central Limit Theorem –Point Estimation Maximum Likelihood Estimation –Confidence Intervals –Tolerance Intervals –Hypothesis Tests Probability Plots Likelihood Ratio Tests –Bonferroni Intervals

4. 4 3.8.1 Central Limit Theorem

5. Central Limit Theorem 5

6. Rule of Thumb For most populations, if the sample size is greater than 30, the Central Limit Theorem approximation is good. Normal approximation to the Binomial: If X ~ Bin(n,p) and if np > 5, and n(1– p) > 5, then X ~ N(np, np(1-p)) approximately. Normal Approximation to the Poisson: If X ~ Poisson(λ), where λ > 10, then X ~ N(λ, λ 2 ). 6

7. The statistical distributions have 1 or more parameters (usually represented by Greek letters) The value of these parameters are generally unknown and must be estimated by sample statistics. Point estimator: the form of the estimation Point estimates: the numerical value of an estimator 7 3.8.2 Point Estimation ParametersSample Statistics  22 S2S2 p

8. 3.8.2.1 Maximum Likelihood Estimation 8

9. An interval estimator will contain the unknown parameter with (approximately) a specified probability. The desired degree of confidence determines the width of the interval. For a fixed sample size, an increase in the width will increase the degree of confidence. Increasing the sample size will decrease the width of a confidence interval. Narrower confidence intervals are more meaningful (lower uncertainty) However, a low level of uncertainty is contingent upon the requisite assumptions being met. 9 3.8.3 Confidence Intervals

10. 10 Confidence Intervals

11. Constructing a CI To see how to construct a confidence interval, let  represent the unknown population mean and let  2 be the unknown population variance. Let X 1,…,X 100 be the 100 amperages of the sample batteries. The observed value of the sample mean is 185.5. Since is the mean of a large sample, and the Central Limit Theorem specifies that it comes from a normal distribution with mean  and whose standard deviation is. 11

12. Computing a 95% Confidence Interval The 95% confidence interval (CI) is. So, a 95% CI for the mean is 185.5  1.96 (5/√100) or 185.5  0.98 or (184.52, 186.48). We can use the sample standard deviation as an estimate for the population standard deviation, since the sample size is large. We can say that we are 95% confident, or confident at the 95% level, that the population mean amperage lies between 184.52 and 186.48. Warning: The methods described here require that the data be a random sample from a population. When used for other samples, the results may not be meaningful. 12

13. Illustration of Capturing True Mean Here is a normal curve, which represents the distribution of. The middle 95% of the curve, extending a distance of 1.96 on either side of the population mean , is indicated. The following illustrates what happens if lies within the middle 95% of the distribution: 13

14. Illustration of Not Capturing True Mean If the sample mean lies outside the middle 95% of the curve: Only 5% of all the samples that could have been drawn fall into this category. For those more unusual samples the 95% confidence interval fails to cover the true population mean . 14

15. Question?  Does this 95% confidence interval actually cover the population mean  ? It depends on whether this particular sample happened to be one whose mean came from the middle 95% of the distribution or whether it was a sample whose mean was unusually large or small, in the outer 5% of the population. There is no way to know for sure into which category this particular sample falls. In the long run, if we repeated these confidence intervals over and over, then 95% of the samples will have means in the middle 95% of the population. Then 95% of the confidence intervals will cover the population mean. 15

16. Pieces of CI Recall that the CI was 185.5  0.98. 185.5 was the sample mean which is a point estimate for the population mean. We call the plus-or-minus number 0.98 the margin of error The margin of error is the product of 1.96 and = 0.5. We refer to which is the standard deviation of, as the standard error. In general, the standard error is the standard deviation of the point estimator. The number 1.96 is called the critical value for the confidence interval. The reason that 1.96 is the critical value for a 95% CI is that 95% of the area under the normal curve is within – 1.96 and 1.96 standard errors of the population mean. 16

17. Other CI Levels Suppose we are interested in 68% confidence intervals, then we know that the middle 68% of the normal distribution is in an interval that extends 1.0 on either side of the population mean. It follows that an interval of the same length around specifically, will cover the population mean for 68% of the samples that could possibly be drawn. For our example, a 68% CI for the diameter of pistons is 185.5  1.0(0.5), or (185.0, 186.0). 17

18. 100(1 -  )% CI Let X 1,…,X n be a large (n > 30) random sample from a population with mean  and standard deviation , so that is approximately normal. Then a level 100(1 -  )% confidence interval for  is where. When the value of  is unknown, it can be replaced with the sample standard deviation s. 18

19. 100(1 -  )% CI 19

20. Specific Intervals for  is a 68% interval for . is a 90% interval for . is a 95% interval for . is a 99% interval for . is a 99.7% interval for . 20

21. More About CI’s The confidence level of an interval measures the reliability of the method used to compute the interval. A level 100(1 -  )% confidence interval is one computed by a method that in the long run will succeed in covering the population mean a proportion 1 -  of all the times that it is used. In practice, there is a decision about what level of confidence to use. This decision involves a trade-off, because intervals with greater confidence are less precise. 21

22. Probability vs. Confidence In computing CI, such as the one of amperage of batteries: (184.52, 186.48), it is tempting to say that the probability that  lies in this interval is 95%. The term probability refers to random events, which can come out differently when experiments are repeated. 184.52 and 186.48 are fixed, not random. The population mean is also fixed. The mean diameter is either in the interval or not. There is no randomness involved. So, we say that we have 95% confidence that the population mean is in this interval. 22

23. Probability vs. Confidence 23 a)68% CI b)95% CI c)99.7% CI

24. Determining Sample Size In Example 5.4, a 95% CI was given by 12.68  1.89. This interval specifies the mean to within  1.89. Now assume that the interval is too wide to be useful. Assume that it is desirable to produce a 95% confidence interval that specifies the mean to within  0.5. To do this, the sample size must be increased. The width of a CI is specified by If the desired with is  w then Solving this equation for n yields 24

25. One-Sided Confidence Intervals We are not always interested in CI’s with both an upper and lower bound. For example, we may want a confidence interval on battery life. We are only interested in a lower bound on the battery life. With the same conditions as with the two-sided CI, the level 100(1-  )% lower confidence bound for  is and the level 100(1-  )% upper confidence bound for  is 25

26. Small Sample CIs for a Population Mean The methods that we have discussed for a population mean previously require that the sample size be large. When the sample size is small, there are no general methods for finding CI’s. If the population is approximately normal, a probability distribution called the Student’s t distribution can be used to compute confidence intervals for a population mean. 26

27. More on CI’s What can we do if is the mean of a small sample? If the sample size is small, s may not be close to , and may not be approximately normal. If we know nothing about the population from which the small sample was drawn, there are no easy methods for computing CI’s. However, if the population is approximately normal, it will be approximately normal even when the sample size is small. It turns out that we can use the quantity, but since s may not be close to , this quantity has a Student’s t distribution. 27

28. Student’s t Distribution Let X 1,…,X n be a small (n < 30) random sample from a normal population with mean . Then the quantity has a Student’s t distribution with n -1 degrees of freedom (denoted by t n-1 ). When n is large, the distribution of the above quantity is very close to normal, so the normal curve can be used, rather than the Student’s t. 28

29. More on Student’s t The probability density of the Student’s t distribution is different for different degrees of freedom. The t curves are more spread out than the normal. Table C, called a t Distribution, provides probabilities associated with the Student’s t distribution. 29

30. Student’s t CI Let X 1,…,X n be a small random sample from a normal population with mean . Then a level 100(1 -  )% CI for  is To be able to use the Student’s t distribution for calculation and confidence intervals, you must have a sample that comes from a population that is approximately normal. Samples such as these rarely contain outliers. So if a sample contains outliers, this CI should not be used. 30

31. Other CI’s Let X 1,…,X n be a small random sample from a normal population with mean . Then a level 100(1 -  )% upper confidence bound for  is Then a level 100(1 -  )% lower confidence bound for  is Occasionally a small sample may be taken from a normal population whose standard deviation  is known. In these cases, we do not use the Student’s t curve, because we are not approximating  with s. The CI to use here is the one using the z table, which we discussed in the first section. 31

32. 3.8.4 Tolerance Intervals A confidence interval for a parameter is an interval that is likely to contain the true value of the parameter. Prediction and tolerance intervals are concerned with the population itself and with values that may be sampled from it in the future. These intervals are only useful when the shape of the population is known, here we assume the population is known to be normal. 32

33. Prediction Interval A prediction interval is an interval that is likely to contain the value of an item that will be sampled from the population at a future time. We “predict” that a value that is yet to be sampled from the population will fall within the predication interval. 33

34. 100(1 – α)% Prediction Interval 34

35. Comparing CI and PI The formula for the PI is similar to the formula for the CI of a mean of normal population. The prediction interval has a small adjustment to the standard error with the additional + 1 under the square root. This reflects the random variation in the value of the sampled item that is to be predicted. Prediction intervals are sensitive to the assumption that the population is normal. If the shape of the population differs much from the normal curve, the prediction interval may be misleading. Large samples do not help, if the population is not normal then the prediction interval is invalid. 35

36. Tolerance Intervals A tolerance interval is an interval that is likely to contain a specified proportion of the population. First assume that we have a normal population whose mean μ and standard deviation σ are known. To find an interval that contains 90% of the population, we have μ ± 1.645σ. In general, the interval μ ± z γ/2 σ will contain 100(1 – γ)% of the population. In practice, we do not know μ or σ. Instead we use the sample mean and sample standard deviation. 36

37. Consequences Since we are estimating the mean and standard deviation from the sample, –We must make the interval wider than it would be if μ and σ were known. –We cannot be 100% confident that the interval actually contains the required proportion of the population. 37

38. Construction of Tolerance Interval We must specify the proportion 100(1 – γ)% of the population that we wish the interval to contain. We must also specify the confidence 100(1 – α)% that the interval actually contains the specified proportion. It is then possible to find a number k n,α,γ such that the interval will contain at least 100(1 – γ)% of the population with confidence 100(1 – α)%. Values of k n,α,γ are presented in various statistical books (for example Table A.4 in Principles of Statistics for Engineers and Scientists) 38

39. Tolerance Interval Summary Let X 1,…,X n be a random sample from a normal population. A tolerance interval for containing at least 100(1 – γ)% of the population with confidence 100(1 – α)% is Of all the tolerance intervals that are computed by this method, 100(1 – α)% will actually contain at least 100(1 – γ)% of the population. 39

40. 1.Formulate a hypothesis that is to be tested 2.Use the data to test the hypothesis 3.Determine whether or not the hypothesis should be rejected The hypothesis that is being tested is called the null hypothesis (H 0 ), which is tested against an alternative hypothesis (H 1 ). Hypothesis tests are used implicitly when control charts are employed. 40 3.8.4 Hypothesis Tests

41. 41 Hypothesis Tests

42. In general, hypotheses that are tested are hardly ever true. A hypothesis that is being tested is more likely to be rejected for a very large sample size than for a very small sample size. 42 Hypothesis Tests

43. P-Value The P-value measures the plausibility of H 0. The smaller the P-value, the stronger the evidence is against H 0. If the P-value is sufficiently small, we may be willing to abandon our assumption that H 0 is true and believe H 1 instead. This is referred to as rejecting the null hypothesis. 43

44. Steps in Performing a Hypothesis Test 1.Define H 0 and H 1. 2.Assume H 0 to be true. 3.Compute a test statistic. A test statistic is a statistic that is used to assess the strength of the evidence against H 0. A test that uses the z-score as a test statistic is called a z-test. 4.Compute the P-value of the test statistic. The P-value is the probability, assuming H 0 to be true, that the test statistic would have a value whose disagreement with H 0 is as great as or greater than what was actually observed. The P-value is also called the observed significa nce level. 44

45. One and Two-Tailed Tests When H 0 specifies a single value for , both tails contribute to the P-value, and the test is said to be a two-sided or two-tailed test. When H 0 specifies only that  is greater than or equal to, or less than or equal to a value, only one tail contributes to the P- value, and the test is called a one-sided or one-tailed test. 45

46. Drawing Conclusions from the Results of Hypothesis Tests There are two conclusions that we draw when we are finished with a hypothesis test, –We reject H 0. In other words, we concluded that H 0 is false. –We do not reject H 0. In other words, H 0 is plausible. One can never conclude that H 0 is true. We can just conclude that H 0 might be true. We need to know what level of disagreement, measured with the P-value, is great enough to render the null hypothesis implausible. 46

47. More on the P-value The smaller the P-value, the more certain we can be that H 0 is false. The larger the P-value, the more plausible H 0 becomes but we can never be certain that H 0 is true. A rule of thumb suggests to reject H 0 whenever P  0.05. While this rule is convenient, it has no scientific basis. 47

48. Comments Some people report only that a test significant at a certain level, without giving the P-value. Such as, the result is “statistically significant at the 5% level.” This is poor practice. First, it provides no way to tell whether the P-value was just barely less than 0.05, or whether it was a lot less. Second, reporting that a result was statistically significant at the 5% level implies that there is a big difference between a P-value just under 0.05 and one just above 0.05, when in fact there is little difference. Third, a report like this does not allow readers to decide for themselves whether the P-value is small enough to reject the null hypothesis. Reporting the P-value gives more information about the strength of the evidence against the null hypothesis and allows each reader to decide for himself or herself whether to reject the null hypothesis. 48

49. Comments on P Let  be any value between 0 and 1. Then, if P  ,  The result of the test is said to be significantly significant at the 100  % level.  The null hypothesis is rejected at the 100  % level.  When reporting the result of the hypothesis test, report the P-value, rather than just comparing it to 5% or 1%. 49

50. Significance When a result has a small P-value, we say that it is “statistically significant.” In common usage, the word significant means “important.” It is therefore tempting to think that statistically significant results must always be important. Sometimes statistically significant results do not have any scientific or practical importance. 50

51. Hypothesis Tests and CI’s Both confidence intervals and hypothesis tests are concerned with determining plausible values for a quantity such as a population mean . In a hypothesis test for a population mean , we specify a particular value of  (the null hypothesis) and determine if that value is plausible. A confidence interval for a population mean  can be thought of as a collection of all values for  that meet a certain criterion of plausibility, specified by the confidence level 100(1-  )%. The values contained within a two-sided level 100(1-  )% confidence intervals are precisely those values for which the P-value of a two-tailed hypothesis test will be greater than . 51

52. Small Sample Test for a Population Mean When we had a large sample we used the sample standard deviation s to approximate the population deviation . When the sample size is small, s may not be close to , which invalidates this large-sample method. However, when the population is approximately normal, the Student’s t distribution can be used. The only time that we don’t use the Student’s t distribution for this situation is when the population standard deviation  is known. Then we are no longer approximating  and we should use the z- test. 52

53. Hypothesis Test Let X 1,…, X n be a sample from a normal population with mean  and standard deviation , where  is unknown. To test a null hypothesis of the form H 0 :    0, H 0 :  ≥  0, or H 0 :  =  0. Compute the test statistic 53

54. P-value Compute the P-value. The P-value is an area under the Student’s t curve with n – 1 degrees of freedom, which depends on the alternate hypothesis as follows. If the alternative hypothesis is H 1 :  >  0, then the P- value is the area to the right of t. If the alternative hypothesis is H 1 :  <  0, then the P- value is the area to the left of t. If the alternative hypothesis is H 1 :    0, then the P- value is the sum of the areas in the tails cut off by t and -t. 54

55. Fixed-Level Testing A hypothesis test measures the plausibility of the null hypothesis by producing a P-value. The smaller the P-value, the less plausible the null. We have pointed out that there is no scientifically valid dividing line between plausibility and implausibility, so it is impossible to specify a “correct” P-value below which we should reject H 0. If a decision is going to be made on the basis of a hypothesis test, there is no choice but to pick a cut-off point for the P-value. When this is done, the test is referred to as a fixed-level test. 55

56. Conducting the Test To conduct a fixed-level test: Choose a number , where 0 <  < 1. This is called the significance level, or the level, of the test. Compute the P-value in the usual way. If P  , reject H 0. If P > , do not reject H 0. 56

57. Comments In a fixed-level test, a critical point is a value of the test statistic that produces a P-value exactly equal to . A critical point is a dividing line for the test statistic just as the significance level is a dividing line for the P- value. If the test statistic is on one side of the critical point, the P-value will be less than , and H 0 will be rejected. If the test statistic is on the other side of the critical point, the P-value will be more than , and H 0 will not be rejected. The region on the side of the critical point that leads to rejection is called the rejection region. The critical point itself is also in the rejection region. 57

58. Errors When conducting a fixed-level test at significance level , there are two types of errors that can be made. These are  Type I error: Reject H 0 when it is true.  Type II error: Fail to reject H 0 when it is false. The probability of Type I error is never greater than . 58

59. Power of Tests A hypothesis test results in Type II error if H 0 is not rejected when it is false. The power of the test is the probability of rejecting H 0 when it is false. Therefore, Power = 1 – P(Type II error). To be useful, a test must have reasonable small probabilities of both type I and type II errors. 59

60. More on Power The type I error is kept small by choosing a small value of  as the significance level. If the power is large, then the probability of type II error is small as well, and the test is a useful one. The purpose of a power calculation is to determine whether or not a hypothesis test, when performed, is likely to reject H 0 in the event that H 0 is false. 60

61. Computing the Power This involves two steps: 1.Compute the rejection region. 2.Compute the probability that the test statistic falls in the rejection region if the alternate hypothesis is true. This is power. When power is not large enough, it can be increased by increasing the sample size. 61

62. Example Find the power of the 5% level test of H 0 :   80 versus H 1 :  > 80 for the mean yield of the new process under the alternative  = 82, assuming n = 50 and  = 5. 62

63. 3.8.5.1 Probability Plots Scientists and engineers often work with data that can be thought of as a random sample from some population. In many cases, it is important to determine the probability distribution that approximately describes the population. More often than not, the only way to determine an appropriate distribution is to examine the sample to find a sample distribution that fits. 63

64. Finding a Distribution Probability plots are a good way to determine an appropriate distribution. Here is the idea: Suppose we have a random sample X 1,…,X n. We first arrange the data in ascending order. Then assign evenly spaced values between 0 and 1 to each X i. There are several acceptable ways to this; the simplest is to assign the value (i – 0.5)/n to X i. The distribution that we are comparing the X’s to should have a mean and variance that match the sample mean and variance. We want to plot (X i, F(X i )), if this plot resembles the cdf of the distribution that we are interested in, then we conclude that that is the distribution the data came from. 64

65. Probability Plot: Example 65 iXiXi (i-.5)/nQiQi 13.010.12.4369 23.350.33.9512 34.790.55.0000 45.960.76.0488 57.890.97.5631

66. Probability Plot: Example 66

67. Software Many software packages take the (i – 0.5)/n assigned to each X i, and calculate the quantile (Q i ) corresponding to that number from the distribution of interest. Then it plots each (X i, Q i ). If this plot is a reasonably straight line then you may conclude that the sample came from the distribution that we used to find quantiles. 67

68. Normal Probability Plots The sample plotted on the left comes from a population that is not close to normal.

69. Normal Probability Plots The sample plotted on the left comes from a population that is not close to normal. The sample plotted on the right comes from a population that is close to normal.

70. 3.8.5.2 Likelihood Ratio Tests The general idea is to form the ratio of the likelihood function using the hypothesized value and the likelihood function using an alternative value. 70

71. 3.8.6 Bonferroni Intervals 71

72. Bonferroni Method in Hypothesis Tests Sometimes a situation occurs in which it is necessary to perform many hypothesis tests. The basic rule governing this situation is that as more tests are performed, the confidence that we can place in our results decreases. The Bonferroni method provides a way to adjust P- values upward when several hypothesis tests are performed. If a P-value remains small after the adjustment, the null hypothesis may be rejected. To make the Bonferroni adjustment, simply multiply the P-value by the number of test performed. 72

73. Example Four different coating formulations are tested to see if they reduce the wear on cam gears to a value below 100  m. The null hypothesis H 0 :   100  m is tested for each formulation and the results are Formulation A: P = 0.37 Formulation B: P = 0.41 Formulation C: P = 0.005 Formulation D: P = 0.21 The operator suspects that formulation C may be effective, but he knows that the P-value of 0.005 is unreliable, because several tests have been performed. Use the Bonferroni adjustment to produce a reliable P-value. 73

74. 3.9 Enumerative Studies v.s. Analytic Studies An enumerative study is conducted for the purpose of determining the “current state of affairs” relative to a fixed frame (population). Example of enumerative study: random sampling of typing errors made by clerical workers. Analytic study focuses on determining the cause(s) of the errors that were made, with an eye toward reducing the number. Making inferential and descriptive statements regarding a fixed frame <> Determining how to improve future performance.