1. Copyright 2011 by W. H. Freeman and Company. All rights reserved.1 Introductory Statistics: A Problem-Solving Approach by Stephen Kokoska Chapter 8: Confidence Intervals Based on a Single Sample

2. Introduction A single value of a statistic computed from a sample conveys little information about confidence and reliability. Alternative method: use a single value to construct an interval in which we are fairly certain the true value lies. Copyright 2011 by W. H. Freeman and Company. All rights reserved.2

3. 3 Point Estimation Point estimate of a population parameter: a single number computed from a sample which serves as a best guess for the parameter.

4. Copyright 2011 by W. H. Freeman and Company. All rights reserved.4 Estimator An estimator (statistic) is a rule used to produce a point estimate of a population parameter.

5. Copyright 2011 by W. H. Freeman and Company. All rights reserved.5 Estimator and Estimate Estimator: a statistic of interest, a random variable. An estimator has a distribution, a mean, a variance, and a standard deviation. Estimate: a specific value of an estimator.

6. Unbiased Estimator Copyright 2011 by W. H. Freeman and Company. All rights reserved.6

7. Unbiased and Biased Estimators Copyright 2011 by W. H. Freeman and Company. All rights reserved.7

8. Copyright 2011, W.H.Freeman and Company, all rights reserved8 Some Unbiased Estimators The sample mean is an unbiased statistic for estimating the population mean. The sample proportion is an unbiased statistic for estimating the population proportion. The sample variance is an unbiased statistic for estimating the population variance.

9. Copyright 2011 by W. H. Freeman and Company. All rights reserved.9 Minimum-Variance Unbiased Estimator If several statistics from which to choose, select the one with the smallest possible variance. If one of these statistics has the smallest possible variance, it is called the minimum-variance unbiased estimator (MVUE).

10. Remarks If the underlying population is normal, the sample mean is the MVUE for estimating the population mean. So, if the population is normal, the sample mean is a really good statistic to use for estimating the populations mean. Copyright 2011 by W. H. Freeman and Company. All rights reserved.10

11. Copyright 2011 by W. H. Freeman and Company. All rights reserved.11 Confidence Interval (CI) A confidence interval (CI) for a population parameter is an interval of values constructed so that, with a specified degree of confidence, the value of the population parameter lies in this interval.

12. Copyright 2011 by W. H. Freeman and Company. All rights reserved.12 Confidence Coefficient The confidence coefficient is the probability the CI encloses the population parameter in repeated samplings.

13. Copyright 2011 by W. H. Freeman and Company. All rights reserved.13 Confidence Level The confidence level is the confidence coefficient expressed as a percentage.

14. Copyright 2011 by W. H. Freeman and Company. All rights reserved.14 Critical Value z α/2 is a critical value. It is a value on the measurement axis in a standard normal distribution such that P(Z ≥ z α/2 ) = α/2.

15. Copyright 2011 by W. H. Freeman and Company. All rights reserved.15 Remarks 1.The subscript on z could be any variable, or letter. 2.Z /2 : a z value such that there is /2 of the area (probability) to the right of z /2.  z /2 : the negative critical value.

16. Remarks (cont.) 3.Critical values are always defined in terms of right-tail probability. Copyright 2011 by W. H. Freeman and Company. All rights reserved.16

17. Remarks (cont.) 4. z α/2 critical values are easy to find by using the complement rule and working backward. P(Z ≥ z /2 ) = /2 P(Z ≤ z /2 ) = 1  /2 Work backward in Table 3 to find z /2. Copyright 2011 by W. H. Freeman and Company. All rights reserved.17

18. Critical Values Copyright 2011 by W. H. Freeman and Company. All rights reserved.18

19. Example: Critical Values for 95% Confidence Level Copyright 2011 by W. H. Freeman and Company. All rights reserved.19

20. Constructing a General 100(1  )% Confidence Interval for a Population Mean 1.Find a symmetric interval about 0 such that the probability Z lies in this interval is 1 . P(  z α/2 < Z < z α/2 ) = 1 . Copyright 2011 by W. H. Freeman and Company. All rights reserved.20

21. Constructing a General 100(1  )% Confidence Interval for a Population Mean 2.Substitute for Z. Copyright 2011 by W. H. Freeman and Company. All rights reserved.21

22. Constructing a General 100(1  )% Confidence Interval for a Population Mean 3.Manipulate this equation to obtain the probability statement. Copyright 2011 by W. H. Freeman and Company. All rights reserved.22

23. Critical Values Copyright 2011 by W. H. Freeman and Company. All rights reserved.23

24. Illustration of a (1  ) 100% CI Copyright 2011 by W. H. Freeman and Company. All rights reserved.24

25. Illustration of 95% CI Copyright 2011 by W. H. Freeman and Company. All rights reserved.25

26. Copyright 2011 by W. H. Freeman and Company. All rights reserved.26 How to Find a 100(1  α )% CI for µ When σ is Known Given a random sample of size n from a population with mean µ, if 1.the underlying population distribution is normal and/or n is large, and 2.the population standard deviation σ is known, then a 100(1  α )% confidence interval for µ has as endpoints the values

27. Copyright 2011 by W. H. Freeman and Company. All rights reserved.27 Remarks 1.This CI for µ can only be used if σ is known. 2.If n large and σ unknown, some statisticians substitute s for σ. This produces an approximate confidence interval. Next section: an exact confidence interval for when is unknown. 3.As confidence coefficient increases (n, constant), CI is wider.

28. Copyright 2011 by W. H. Freeman and Company. All rights reserved.28 Concepts to Remember 1.The population parameter, µ, is fixed. The confidence interval varies from sample to sample. Correct statement: We are 95% confident the interval captures the true mean µ. Incorrect statement: We are 95% confident µ lies in the interval.

29. Copyright 2011 by W. H. Freeman and Company. All rights reserved.29 Concepts to Remember (cont.) 2.Confidence coefficient: a probability, a long- run limiting relative frequency. In repeated samples, the proportion of confidence intervals that capture the true value of µ approaches the confidence coefficient, in this case, 0.95. Cannot be certain about any one specific confidence interval. The confidence is in the long-run process.

30. Copyright 2011 by W. H. Freeman and Company. All rights reserved.30

31. Example: Tire Weight The total weight of a filled tire can dramatically affect the performance and safety of an automobile. Some transportation officials argue that mechanics should check the tire weights of every vehicle as part of an annual inspection. Copyright 2011 by W. H. Freeman and Company. All rights reserved.31

32. Example (cont.) Suppose the weight of a 185/60/14 filled tire is normally distributed with standard deviation 1.25 pounds. In a random sample of 15 filled tires, the sample mean weight was 18.75 pounds. Find a 95% confidence interval for the true mean weight of 185/60/14 tires. Copyright 2011 by W. H. Freeman and Company. All rights reserved.32

33. Example (cont.) Copyright 2011 by W. H. Freeman and Company. All rights reserved.33

34. Example (cont.) Copyright 2011 by W. H. Freeman and Company. All rights reserved.34

35. Example (cont.) (18.12, 19.38) is a 95% confidence interval for the true mean weight (in pounds) of 185/60/14 tires. Copyright 2011 by W. H. Freeman and Company. All rights reserved.35

36. Copyright 2011 by W. H. Freeman and Company. All rights reserved.36 Necessary Sample Size 1.Suppose n is large (unknown), is known, confidence level is 100(1  )%. 2.Desired width W. B = W/2: bound on the error of estimation. B is half the width of the confidence interval.

37. Necessary Sample Size (cont.) 3.Confidence interval endpoints: Copyright 2011 by W. H. Freeman and Company. All rights reserved.37

38. Necessary Sample Size (cont.) Copyright 2011 by W. H. Freeman and Company. All rights reserved.38

39. Necessary Sample Size (cont.) 4.Solve for n and the resulting formula is: Copyright 2011 by W. H. Freeman and Company. All rights reserved.39

40. Copyright 2011 by W. H. Freeman and Company. All rights reserved.40 A Confidence Interval for a Population Mean When σ Is Unknown Confidence interval for µ based on Z: valid only when σ is known which is unrealistic.

41. Theorem Copyright 2011 by W. H. Freeman and Company. All rights reserved.41

42. Comparison of Density Curves Copyright 2011 by W. H. Freeman and Company. All rights reserved.42

43. Properties of t Distribution Copyright 2011 by W. H. Freeman and Company. All rights reserved.43

44. Copyright 2011 by W. H. Freeman and Company. All rights reserved.44 Critical Value t α is a critical value related to a t distribution with df degrees of freedom. If T has a t distribution with df degrees of freedom then P(T ≥ t α ) = α.

45. Illustration of Critical Values Copyright 2011 by W. H. Freeman and Company. All rights reserved.45

46. Example: Visualization of t 0.01 = 2.5176. Copyright 2011 by W. H. Freeman and Company. All rights reserved.46

47. Copyright 2011 by W. H. Freeman and Company. All rights reserved.47 Remarks t α :a t value such that there is of the area to the right of t α  t α : the negative critical value Because the t distribution is symmetric, P(T ≤  t α ) = P(T ≥ t α ) = α.

48. How to Find a 100(1  α )% CI for µ When σ is Unknown Copyright 2011 by W. H. Freeman and Company. All rights reserved.48

49. Copyright 2011 by W. H. Freeman and Company. All rights reserved.49 Remarks 1.This CI can be used with any sample size n (≥ 2). This produces an exact CI for µ. 2.This CI for µ is valid only if the underlying population is normal.

50. A Large-Sample CI for a Population Proportion Let p = true population proportion of a success. Use the sample proportion to construct a CI for p. Copyright 2011 by W. H. Freeman and Company. All rights reserved.50

51. A Large-Sample CI for a Population Proportion Sample of n individuals. X = number of individuals with the characteristic or number of successes. Copyright 2011 by W. H. Freeman and Company. All rights reserved.51

52. Sample Proportion Copyright 2011 by W. H. Freeman and Company. All rights reserved. 52

53. Standardize the Sample Proportion Copyright 2011 by W. H. Freeman and Company. All rights reserved.53

54. A Large-Sample Confidence Interval for a Population Proportion Copyright 2011 by W. H. Freeman and Company. All rights reserved.54

55. Example: Americans with Hypertension High blood pressure, or hypertension, occurs when the force of blood against the artery walls is too strong. In a recent survey, 1100 adult Americans were randomly selected and examined for high blood pressure. The number of patients classified with hypertension is 319. Copyright 2011 by W. H. Freeman and Company. All rights reserved.55

56. Example (cont.) Find a 95% confidence interval for the true proportion of adult Americans with hypertension. Copyright 2011 by W. H. Freeman and Company. All rights reserved.56

57. Example (cont.) = 0.29 ± 0.0268 = (0.2632, 0.3168) (0.2632, 0.3168) is a 95% confidence interval for the true proportion, p, of adult Americans with hypertension. Copyright 2011 by W. H. Freeman and Company. All rights reserved.57

58. Copyright 2011 by W. H. Freeman and Company. All rights reserved.58 Necessary Sample Size 1.Suppose n is large (unknown), confidence level is 100(1  )%, bound on the error of estimation is B.

59. Necessary Sample Size (cont.) 2.The bound on the error of estimation is the step in each direction: Copyright 2011 by W. H. Freeman and Company. All rights reserved.59

60. Necessary Sample Size (cont.) 3.Solve for n and the resulting formula is: Copyright 2011 by W. H. Freeman and Company. All rights reserved.60

61. Necessary Sample Size (cont.) Problem: sample proportion is unknown. Don't know sample proportion until we have n. Copyright 2011 by W. H. Freeman and Company. All rights reserved.61

62. Necessary Sample Size (cont.) Solutions: 1.Use a reasonable estimate for sample proportion from previous experience. 2.If no prior information is available, use 0.5 as the estimate of the sample proportion. This produces a very conservative, large value of n. Copyright 2011 by W. H. Freeman and Company. All rights reserved.62

63. Copyright 2011 by W. H. Freeman and Company. All rights reserved.63 A Confidence Interval for a Population Variance 1.Seems reasonable to use S 2 as an estimator for σ 2. 2.A CI for σ 2 is based on a new standardization and a chi-square distribution. 3.Chi-square (  2 ) distribution: a)Positive probability only for non-negative values. b)Focus on the properties and a method for finding critical values.

64. Copyright 2011 by W. H. Freeman and Company. All rights reserved.64 Properties of a Chi-Square Distribution 1.A chi-square distribution is completely determined by one parameter, df, the number of degrees of freedom, a positive integer (df = 1, 2, 3, 4,…). There is a different chi-square distribution corresponding to each value of df.

65. Copyright 2011 by W. H. Freeman and Company. All rights reserved.65 Properties of a Chi-Square Distribution (cont.) 2.If X has a chi-square distribution with df degrees of freedom, denoted X ~  df 2, then µ X = df and σ X 2 = 2df. The mean of X is df, the number of degrees of freedom, and the variance is 2 twice the number of degrees of freedom.

66. Copyright 2011 by W. H. Freeman and Company. All rights reserved.66 Properties of a Chi-Square Distribution (cont.) 3.Suppose X ~  df 2. The density curve for X is positively skewed (not symmetric), and as x increases it gets closer and closer to the x axis but never touches it. As df increases, the density curve becomes flatter and actually looks more normal.

67. Examples of Density Curvers for Chi-Square Distributions Copyright 2011 by W. H. Freeman and Company. All rights reserved.67

68. Critical Values Copyright 2011 by W. H. Freeman and Company. All rights reserved.68

69. Illustration of Critical Values Copyright 2011 by W. H. Freeman and Company. All rights reserved.69

70. Illustration of Critical Values Copyright 2011 by W. H. Freeman and Company. All rights reserved.70

71. Example: Visualization of 2 0.05 = 18.3070 Copyright 2011 by W. H. Freeman and Company. All rights reserved.71

72. Example: Visualization of 2 0.99 = 1.2390. Copyright 2011 by W. H. Freeman and Company. All rights reserved.72

73. Copyright 2011 by W. H. Freeman and Company. All rights reserved.73 Theorem Let S 2 be the sample variance of a random sample of size n from a normal distribution with variance σ 2. The random variable has a chi-square distribution with n  1 degrees of freedom.

74. CI for a Population Variance Copyright 2011 by W. H. Freeman and Company. All rights reserved.74

75. Example: Kiln-Fired Dishes Earthenware dishes are made from clay and are fired, or exposed to heat, in a large kiln. Large fluctuations in the kiln temperature can cause cracks, bumps, or other flaws (and increase cost). Copyright 2011 by W. H. Freeman and Company. All rights reserved.75

76. Example (cont.) With the kiln set at 800  C, a random sample of 19 temperature measurements (in  C) was obtained. The sample variance was s 2 = 17.55. Copyright 2011 by W. H. Freeman and Company. All rights reserved.76

77. Example (cont.) a)Find a 95% confidence interval for the true population variance in temperature of the kiln when it is set to 800  C. Assume that the underlying distribution is normal. Copyright 2011 by W. H. Freeman and Company. All rights reserved.77

78. Example (cont.) s 2 = 17.55, n = 19, df = 19  1 =18 2 /2 = 2 0.025 = 31.5264 2 1- /2 = 2 0.975 = 8.2307 Copyright 2011 by W. H. Freeman and Company. All rights reserved.78

79. Example (cont.): Left Endpoint Copyright 2011 by W. H. Freeman and Company. All rights reserved.79

80. Example (cont.): Right Endpoint Copyright 2011 by W. H. Freeman and Company. All rights reserved.80

81. Example (cont.) (10.0202, 38.3807) is a 95% confidence interval for the true population variance in temperature when the kiln is set to 800  C. Copyright 2011 by W. H. Freeman and Company. All rights reserved.81

82. Example (cont.) b)Quality control engineers have determined that the maximum variance in temperature during firing should be 16  C. Using the confidence interval constructed in part a), is there any evidence to suggest that the true temperature variance is greater than 16  C? Justify your answer. Copyright 2011 by W. H. Freeman and Company. All rights reserved.82

83. Example (cont.) Claim: 2 = 16 Experiment: s 2 = 17.55 Likelihood: The likelihood is expressed as a 95% confidence interval, an interval of likely values for 2, (10.0202, 38.3807), from part a). Copyright 2011 by W. H. Freeman and Company. All rights reserved.83

84. Example (cont.) Conclusion: The required kiln temperature variance, 16  C, is included in this confidence interval. There is no evidence to suggest that 2 is different from 16  C. Copyright 2011 by W. H. Freeman and Company. All rights reserved.84

85. Example (cont.) Copyright 2011 by W. H. Freeman and Company. All rights reserved.85

86. Copyright 2011 by W. H. Freeman and Company. All rights reserved.86 Remarks 1.This CI for σ 2 is valid only if the underlying population is normal. 2.Take the square root of each endpoint to obtain a 100(1  α )% CI for σ.