1. Chapter Estimation 1 of 83 8 © 2012 Pearson Education, Inc. All rights reserved.

2. Chapter Outline © 2012 Pearson Education, Inc. All rights reserved. 2 of 83

3. Section 8.1 © 2012 Pearson Education, Inc. All rights reserved. 3 of 83

4. Section 8.1 Objectives Find a point estimate and a margin of error Construct and interpret confidence intervals for the population mean Determine the minimum sample size required when estimating μ © 2012 Pearson Education, Inc. All rights reserved. 4 of 83

5. Point Estimate for Population μ Point Estimate A single value estimate for a population parameter Most unbiased point estimate of the population mean μ is the sample mean Estimate Population Parameter… with Sample Statistic Mean: μ © 2012 Pearson Education, Inc. All rights reserved. 5 of 83

6. Exercise 1: Point Estimate for Population μ A social networking website allows its users to add friends, send messages, and update their personal profiles. The following represents a random sample of the number of friends for 40 users of the website. Find a point estimate of the population mean, µ. (Source: Facebook) 140 105 130 97 80 165 232 110 214 201 122 98 65 88 154 133 121 82 130 211 153 114 58 77 51 247 236 109 126 132 125 149 122 74 59 218 192 90 117 105 © 2012 Pearson Education, Inc. All rights reserved. 6 of 83

7. Solution: Point Estimate for Population μ The sample mean of the data is Your point estimate for the mean number of friends for all users of the website is 130.8 friends. © 2012 Pearson Education, Inc. All rights reserved. 7 of 83

8. 115120125130135140150145 Point estimate 115120125130135140150145 Point estimate Interval Estimate © 2012 Pearson Education, Inc. All rights reserved. 8 of 83 ( ) Interval estimate Right endpointLeft endpoint Interval is centered at point estimate Interval Width: 146.5 – 115.1 = 31.4 Interval estimate will predict value of parameter µ

9. Confidence Levels

10. Level of Confidence Level of confidence c The probability that the interval estimate contains the population parameter. z z = 0–zc–zc zczc Critical values ½(1 – c) c is the area under the standard normal curve between the critical values. The remaining area in the tails is 1 – c. c Use the Standard Normal Table to find the corresponding z-scores. © 2012 Pearson Education, Inc. All rights reserved. 10 of 83

11. Common Confidence Levels Appendix II, Table 5: Page A23

12. zczc Level of Confidence If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean μ. z z = 0zczc The corresponding z-scores are ±1.645. c = 0.90 ½(1 – c) = 0.05 –z c = –1.645 z c = 1.645 © 2012 Pearson Education, Inc. All rights reserved. 12 of 83

13. Sampling Error Sampling error The difference between the point estimate and the actual population parameter value. For μ:  the sampling error is the difference – μ  μ is generally unknown  varies from sample to sample © 2012 Pearson Education, Inc. All rights reserved. 13 of 83

14. Margin of Error When n ≥ 30, the sample standard deviation, s, can be used for σ © 2012 Pearson Education, Inc. All rights reserved. 14 of 83

15. Distribution of Sample Means © 2012 Pearson Education, Inc. All rights reserved. 15 of 83

16. Distribution of Sample Means © 2012 Pearson Education, Inc. All rights reserved. 16 of 83

17. Distribution of Sample Means © 2012 Pearson Education, Inc. All rights reserved. 17 of 83 STATEMENTREASON Shaded area in Fig. 8-3 Equivalent inequalities Shaded area in diagram

18. Exercise 2: Finding the Margin of Error Use the social networking website data and a 95% confidence level to find the margin of error for the mean number of friends for all users of the website. Assume that the population standard deviation is about 53.0. © 2012 Pearson Education, Inc. All rights reserved. 18 of 83

19. zczc Solution: Finding the Margin of Error Step 1: Find the critical values [Table 5 – Page A23] z zczc z = 0 0.95 0.025 –z c = –1.96 95% of the area under the standard normal curve falls within 1.96 standard deviations of the mean. (You can approximate the distribution of the sample means with a normal curve by the Central Limit Theorem, because n = 40 ≥ 30.) z c = 1.96 © 2012 Pearson Education, Inc. All rights reserved. 19 of 83

20. Solution: Finding the Margin of Error We are 95% confident that the margin of error for the population mean is about 16.4 friends. Our estimate might be too big, or too small. © 2012 Pearson Education, Inc. All rights reserved. 20 of 83 Step 2: Compute E

21. Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ The probability that the confidence interval contains μ is c. © 2012 Pearson Education, Inc. All rights reserved. 21 of 83

22. Constructing Confidence Intervals for μ © 2012 Pearson Education, Inc. All rights reserved. 22 of 83

23. Exercise 3: Construct a Confidence Interval Construct a 95% confidence interval for the mean number of friends for all users of the website. Solution: Recall and E ≈ 16.4 114.4 < μ < 147.2 Left Endpoint:Right Endpoint: © 2012 Pearson Education, Inc. All rights reserved. 23 of 83

24. Solution: Constructing a Confidence Interval 114.4 < μ < 147.2 With 95% confidence, you can say that the population mean number of friends is between 114.4 and 147.2. © 2012 Pearson Education, Inc. All rights reserved. 24 of 83

25. Exercise 4: Construct Confidence Interval: Case σ is Known A college admissions director wishes to estimate the mean age of all students currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, the population standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval of the population mean age. © 2012 Pearson Education, Inc. All rights reserved. 25 of 83

26. zczc Solution: Constructing a Confidence Interval σ Known First find the critical values z z = 0zczc c = 0.90 ½(1 – c) = 0.05 –z c = –1.645 z c = 1.645 © 2012 Pearson Education, Inc. All rights reserved. 26 of 83

27. Margin of error: Confidence interval: Solution: Constructing a Confidence Interval σ Known Left Endpoint:Right Endpoint: 22.3 < μ < 23.5 © 2012 Pearson Education, Inc. All rights reserved. 27 of 83

28. Solution: Constructing a Confidence Interval σ Known 22.3 < μ < 23.5 ( ) 22.922.323.5 With 90% confidence, we can say that the mean age of all the students is between 22.3 and 23.5 years. Point estimate © 2012 Pearson Education, Inc. All rights reserved. 28 of 83

29. Interpreting the Results μ is a fixed number. Once the interval is fixed, it either contains or does not contain μ. Incorrect: “There is a 90% probability that the actual mean μ is in the interval (22.3, 23.5).” Correct: “If a large number of samples is collected and a confidence interval is created for each sample, approximately 90% of these intervals will contain μ. Correct: “We are 90% confident that the interval (22.3,23.5) is one which contains the actual population parameter μ.” © 2012 Pearson Education, Inc. All rights reserved. 29 of 83

30. Interpreting the Results The horizontal segments represent 90% confidence intervals for different samples of the same size. In the long run, 9 of every 10 such intervals will contain μ. μ © 2012 Pearson Education, Inc. All rights reserved. 30 of 83

31. Interpreting the Results μ © 2012 Pearson Education, Inc. All rights reserved. 31 of 83

32. TI-84: Z-Interval Function The ZInterval (STAT / TESTS / 7: ZInterval) function computes a confidence interval for a population mean µ when the population standard deviation is known. © 2012 Pearson Education, Inc. All rights reserved. 32 of 83 Input Output Margin of Error

33. Exercise 5: Construct Confidence Interval via TI-84: Case σ is Known The ZInterval (STAT / TESTS / 7: ZInterval) function computes a confidence interval for a population mean µ when the population standard deviation is known. © 2012 Pearson Education, Inc. All rights reserved. 33 of 83 Inputs Output(22.3, 23.5) Margin of Error

34. Minimum Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean µ is given by Round up using CELING © 2012 Pearson Education, Inc. All rights reserved. 34 of 83 If σ is unknown, you can estimate it using s, provided you have a preliminary sample with at least 30 Result should be rounded up by using the CEILING function

35. Exercise 6: Minimum Sample Size You want to estimate the mean number of friends for all users of the website. How many users must be included in the sample if you want to be 95% confident that the sample mean is within seven friends of the population mean? Assume the population standard deviation is about 53.0. © 2012 Pearson Education, Inc. All rights reserved. 35 of 83

36. zczc Solution: Minimum Sample Size First find the critical values z c = 1.96 z z = 0zczc 0.95 0.025 –z c = –1.96 z c = 1.96 © 2012 Pearson Education, Inc. All rights reserved. 36 of 83

37. Solution: Minimum Sample Size z c = 1.96 σ =53.0 E = 7 When necessary, round up to obtain a whole number. Interpretive Statement: You should include at least 221 users in your sample in order to be 95% confident that the sample mean will be within 7 friends of the population mean. © 2012 Pearson Education, Inc. All rights reserved. 37 of 83

38. Exercise 7: Salmon Sample Size A sample of 50 salmon is caught and weighed. The sample standard deviation of the 50 weights is 2.15 lb. How large of a sample should be taken to be 97% confident that the sample mean is within 0.20 lb of the mean weight of the population? Summarize your results in a complete sentence relevant to this application. © 2012 Pearson Education, Inc. All rights reserved. 38 of 83

39. Solution: Salmon Sample Size © 2012 Pearson Education, Inc. All rights reserved. 39 of 83 Critical Values Apply Ceiling Function 545 salmon Interpretive StatementA sample size of 545 salmon will be large enough to ensure that we are 97% confident that the sample mean is within 0.20 lb. of the population mean.

40. Section 8.1 Summary Found a point estimate and a margin of error Constructed and interpreted confidence intervals for the population mean Determined the minimum sample size required when estimating μ © 2012 Pearson Education, Inc. All rights reserved. 40 of 83

41. Section 8.2 © 2012 Pearson Education, Inc. All rights reserved. 41 of 83

42. Section 8.2 Objectives Interpret the t-distribution and use a t-distribution table Construct confidence intervals when n < 30, the population is normally distributed, and σ is unknown © 2012 Pearson Education, Inc. All rights reserved. 42 of 83

43. The t-Distribution When the population standard deviation is unknown, the sample size is less than 30, and the random variable x is approximately normally distributed, it follows a t-distribution. Critical values of t are denoted by t c. Distribution was discovered in 1908 by Louis Gosset while he worked for the Guinness Brewing Company © 2012 Pearson Education, Inc. All rights reserved. 43 of 83

44. The t-Distribution © 2012 Pearson Education, Inc. All rights reserved. 44 of 83

45. Properties of the t-Distribution 1.The t-distribution is bell shaped and symmetric about the mean. 2.The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size.  d.f. = n – 1 Degrees of freedom © 2012 Pearson Education, Inc. All rights reserved. 45 of 83

46. Example: Two Degrees of Freedom © 2012 Pearson Education, Inc. All rights reserved. 46 of 83

47. Properties of the t-Distribution 3.The total area under a t-curve is 1 or 100%. 4.The mean, median, and mode of the t-distribution are equal to zero. 5.As the degrees of freedom increase, the t-distribution approaches the normal distribution. After 30 d.f., the t- distribution is very close to the standard normal z- distribution. t 0 Standard normal curve The tails in the t- distribution are “thicker” than those in the standard normal distribution. d.f. = 5 d.f. = 2 © 2012 Pearson Education, Inc. All rights reserved. 47 of 83

48. Exercise 1: Critical Values of t Appendix II, Table 6– p. A24 Find the critical value t c for a 95% confidence level when the sample size is 15. Table 5: t-Distribution t c = 2.145 Solution: d.f. = n – 1 = 15 – 1 = 14 © 2012 Pearson Education, Inc. All rights reserved. 48 of 83

49. Example: Critical Values of t Appendix II, Table 5(a) – p. A23 Larson/Farber 5th ed 49 Rounding Convention: t-scores should be rounded to three decimal places

50. Solution: Critical Values of t 95% of the area under the t-distribution curve with 14 degrees of freedom lies between t = ±2.145. t –t c = –2.145 t c = 2.145 c = 0.95 © 2012 Pearson Education, Inc. All rights reserved. 50 of 83

51. Confidence Intervals for the Population Mean: Case: σ is Unknown A c-confidence interval for the population mean μ The probability that the confidence interval contains μ is c. © 2012 Pearson Education, Inc. All rights reserved. 51 of 83

52. Confidence Intervals and t-Distributions 1.Identify the sample statistics n,, and s. 2.Identify the degrees of freedom, the level of confidence c, and the critical value t c. 3.Find the margin of error E. d.f. = n – 1 In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 52 of 83

53. Confidence Intervals and t-Distributions 4.Find the left and right endpoints and form the confidence interval. Left endpoint: Right endpoint: Interval: In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 53 of 83

54. Exercise 2: Construct a Confidence Interval You randomly select 16 coffee shops and measure the temperature of the coffee sold at each. The sample mean temperature is 162.0ºF with a sample standard deviation of 10.0ºF. Find the 95% confidence interval for the population mean temperature. Assume the temperatures are approximately normally distributed. Solution: Use the t-distribution (n < 30, σ is unknown, temperatures are approximately normally distributed). © 2012 Pearson Education, Inc. All rights reserved. 54 of 83

55. Solution: Construct a Confidence Interval n =16, x = 162.0 s = 10.0 c = 0.95 df = n – 1 = 16 – 1 = 15 Critical Value Table 5: t-Distribution t c = 2.131 © 2012 Pearson Education, Inc. All rights reserved. 55 of 83

56. Solution: Construct a Confidence Interval Margin of error: Confidence interval: Left Endpoint:Right Endpoint: 156.7 < μ < 167.3 © 2012 Pearson Education, Inc. All rights reserved. 56 of 83

57. Solution: Construct a Confidence Interval 156.7 < μ < 167.3 ( ) 162.0156.7167.3 With 95% confidence, you can say that the population mean temperature of coffee sold is between 156.7ºF and 167.3ºF. Point estimate © 2012 Pearson Education, Inc. All rights reserved. 57 of 83

58. No Normal or t-Distribution? Is n ≥ 30? Is the population normally, or approximately normally, distributed? Cannot use the normal distribution or the t-distribution. Yes Is σ known? No Use the normal distribution with If σ is unknown, use s instead. YesNo Use the normal distribution with Yes Use the t-distribution with and n – 1 degrees of freedom. © 2012 Pearson Education, Inc. All rights reserved. 58 of 83

59. Example: Normal or t-Distribution? You randomly select 25 newly constructed houses. The sample mean construction cost is $181,000 and the population standard deviation is $28,000. Assuming construction costs are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 95% confidence interval for the population mean construction cost? Solution: Case: σ is known (n < 30): Use the normal distribution (the population is normally distributed and the population standard deviation is known) © 2012 Pearson Education, Inc. All rights reserved. 59 of 83

60. Exercise 3: Construct Confidence Interval An archeologist discovered a new, but extinct, species of miniature horse. The only seven known samples show shoulder heights (in cm) of 45.3, 47.1, 44.2, 46.8, 46.5, 45.5, and 47.6. Find the 99% confidence interval for the mean height of the entire population of ancient horses and the margin of error E. Then summarize your results in a complete sentence relevant to this application. © 2012 Pearson Education, Inc. All rights reserved. 60 of 83

61. Exercise 3: Construct Confidence Interval 6 46.14 1.19 46.14 – 1.67 = 44.5 46.14 + 1.67 = 47.8 Confidence Interval © 2012 Pearson Education, Inc. All rights reserved. 61 of 83

62. TI-84: T-Interval Function The TInterval (STAT / TESTS / 8: TInterval) function computes a confidence interval for a population mean µ when the population standard deviation is unknown. © 2012 Pearson Education, Inc. All rights reserved. 62 of 83 Input (Stats) Input (Data)List:, Freq: Output Margin of Error

63. Exercise 4: Construct Confidence Interval using T-Interval Function (Data) © 2012 Pearson Education, Inc. All rights reserved. 63 of 83 InputsInput: DATA, List L1 Freq: 1 (Data is not grouped) Output(44.5,47.8) Margin of Error

64. Exercise 4: Construct Confidence Interval via TI-84 An archeologist discovered a new, but extinct, species of miniature horse. The only seven known samples show shoulder heights (in cm) of 45.3, 47.1, 44.2, 46.8, 46.5, 45.5, and 47.6. Find the 99% confidence interval for the mean height of the entire population of ancient horses and the margin of error E. Use the T-Interval function (Data option). © 2012 Pearson Education, Inc. All rights reserved. 64 of 83

65. Section 8.2 Summary Interpreted the t-distribution and used a t-distribution table Constructed confidence intervals when n < 30, the population is normally distributed, and σ is unknown © 2012 Pearson Education, Inc. All rights reserved. 65 of 83

66. Section 8.3 Confidence Intervals for Population Proportions © 2012 Pearson Education, Inc. All rights reserved. 66 of 83

67. Section 8.3 Objectives Find a point estimate for the population proportion Construct a confidence interval for a population proportion Determine the minimum sample size required when estimating a population proportion © 2012 Pearson Education, Inc. All rights reserved. 67 of 83

68. Point Estimate for Population p © 2012 Pearson Education, Inc. All rigts reserved 68 of 83

69. Point Estimate for Population p Point Estimate for q, the population proportion of failures Denoted by Estimate Population Parameter… with Sample Statistic Proportion: p © 2012 Pearson Education, Inc. All rights reserved. 69 of 83

70. Example: Point Estimate for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal e-mail while at work. Find a point estimate for the population proportion of U.S. adults who say it is acceptable to check personal e-mail while at work. (Adapted from Liberty Mutual) Solution: n = 1000 and r = 662 © 2012 Pearson Education, Inc. All rights reserved. 70 of 83

71. Confidence Intervals for p A c-confidence interval for a population proportion p The probability that the confidence interval contains p is c. © 2012 Pearson Education, Inc. All rights reserved. 71 of 83

72. Constructing Confidence Intervals for p 1.Identify the sample statistics n and r. 2.Find the point estimate 3.Verify that the sampling distribution of can be approximated by a normal distribution. 4.Find the critical value z c that corresponds to the given level of confidence c. Use the Standard Normal Table or technology. In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 72 of 83

73. Constructing Confidence Intervals for p 5.Find the margin of error E. 6.Find the left and right endpoints and form the confidence interval. Left endpoint: Right endpoint: Interval: In WordsIn Symbols © 2012 Pearson Education, Inc. All rights reserved. 73 of 83

74. Exercise 1: Confidence Interval for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal e-mail while at work. Construct a 95% confidence interval for the population proportion of U.S. adults who say that it is acceptable to check personal e-mail while at work. Solution: Recall © 2012 Pearson Education, Inc. All rights reserved. 74 of 83

75. Solution: Confidence Interval for p Verify the sampling distribution of can be approximated by the normal distribution Margin of error: © 2012 Pearson Education, Inc. All rights reserved. 75 of 83

76. Solution: Confidence Interval for p Confidence interval: Left Endpoint:Right Endpoint: 0.633 < p < 0.691 © 2012 Pearson Education, Inc. All rights reserved. 76 of 83

77. Solution: Confidence Interval for p 0.633 < p < 0.691 With 95% confidence, you can say that the population proportion of U.S. adults who say that it is acceptable to check personal e-mail while at work is between 63.3% and 69.1%. Point estimate © 2012 Pearson Education, Inc. All rights reserved. 77 of 83

78. Exercise 2: Confidence Interval for p Suppose that 800 students were selected at random from a student body of 20,000 and given flu shots. All 800 students were exposed to the flu, and 600 of them did not get the flu. Let p represent the probability that the shot will be successful for any single student selected at random from the entire population. Find a 99% confidence interval for p. © 2012 Pearson Education, Inc. All rights reserved. 78 of 83

79. Solution: Confidence Interval for p Critical Values 800 600/800 = 0.75 1 – 0.75 = 0.25 800 (0.75) = 600 [exceeds 5] 800 (0.25) = 200 [exceeds 5] Confidence Interval Interpretive StatementWith 99% confidence, we can say that the probability that a flu shot will be effective for a student selected at random will be between 71% and 79%. © 2012 Pearson Education, Inc. All rights reserved. 79 of 83

80. Exercise 3: Confidence Interval for p via TI-4 Suppose that 800 students were selected at random from a student body of 20,000 and given flu shots. All 800 students were exposed to the flu, and 600 of them did not get the flu. Let p represent the probability that the shot will be successful for any single student selected at random from the entire population. Find a 99% confidence interval for p using the 1- PropZinterval function on the TI-84. © 2012 Pearson Education, Inc. All rights reserved. 80 of 83

81. Exercise 3: Confidence Interval for p via TI-84 1-PropZInterval Function © 2012 Pearson Education, Inc. All rights reserved. 81 of 83 Inputs Output(0.711,0.789) Margin of Error

82. Minimum Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate p is This formula assumes you have an estimate for and. If not, use and Round up to nearest integer using the CEILING function © 2012 Pearson Education, Inc. All rights reserved. 82 of 83

83. Minimum Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate p is If you have no preliminary estimate for p, use © 2012 Pearson Education, Inc. All rights reserved. 83 of 83

84. Exercise 4: Minimum Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if a)No preliminary estimate for p is available b)A preliminary estimate for p is 0.31 © 2012 Pearson Education, Inc. All rights reserved. 84 of 83

85. Solution: Minimum Sample Size: No Preliminary Estimate for p is available Critical Values 0.03 Apply Ceiling Function 1068 voters Interpretive Statement The sample size should be at least 1068 voters in order to be 95% confident that our estimate is within 3% of the true population proportion. © 2012 Pearson Education, Inc. All rights reserved. 85 of 83

86. Solution: Minimum Sample Size: Case: Preliminary Estimate for p is Available You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if a preliminary estimate gives Solution: Use the preliminary estimate © 2012 Pearson Education, Inc. All rights reserved. 86 of 83

87. Solution: Sample Size c = 0.95 z c = 1.96 E = 0.03 Round up to the nearest whole number. With a preliminary estimate of, the minimum sample size should be at least 914 voters. Need a larger sample size if no preliminary estimate is available. © 2012 Pearson Education, Inc. All rights reserved. 87 of 83

88. Exercise 5: Minimum Sample Size You wish to estimate, with 90% confidence and within 2% of the true population, the proportion of adults age 18 to 29 who have high blood pressure. Find the minimum sample size needed if a)No preliminary estimate for p is available b)A previous survey found that 4% of adults in this age group had high blood pressure. Solution: Because you do not have a preliminary estimate for use and © 2012 Pearson Education, Inc. All rights reserved. 88 of 83

89. Solution: Minimum Sample Size When Preliminary Estimate for p is Unavailable © 2012 Pearson Education, Inc. All rights reserved. 89 of 83 Critical Values 0.02 Apply Ceiling Function 1692 adults Interpretive StatementWith no preliminary estimate for p, the sample size should be at least 1692 adults in order to be 90% confident that our estimate is within 2% of the true population proportion.

90. Solution: Minimum Sample Size When Preliminary Estimate for p is Available © 2012 Pearson Education, Inc. All rights reserved. 90 of 83 Critical Values 0.02 Apply Ceiling Function 260 adults Interpretive StatementWith a preliminary estimate for p of 4%, the sample size should be at least 260 adults in order to be 90% confident that our estimate is within 2% of the true population proportion

91. Solution: Minimum Sample Size When Preliminary Estimate for p is known © 2012 Pearson Education, Inc. All rights reserved. 91 of 83

92. Section 8.3 Summary Found a point estimate for the population proportion Constructed a confidence interval for a population proportion Determined the minimum sample size required when estimating a population proportion © 2012 Pearson Education, Inc. All rights reserved. 92 of 83

93. Section 8.3 Summary Found a point estimate for the population proportion Constructed a confidence interval for a population proportion Determined the minimum sample size required when estimating a population proportion © 2012 Pearson Education, Inc. All rights reserved. 93 of 83

94. Section 8.4 Objectives © 2012 Pearson Education, Inc. All rights reserved. 94 of 83

95. Independent Samples Two samples are independent if sample data drawn from one population is completely unrelated to the selection of a sample from the other population. This situation occurs when we draw two random samples

96. Dependent Samples Two samples are dependent if each data value in one sample can be paired with a corresponding value in the other sample.  Occur naturally when taking the same measurement twice on one observation Example: your weight before and after the holiday season.

97. Confidence Intervals for μ 1 – μ 2 when σ 1, σ 2 known

100. Confidence Intervals for μ 1 – μ 2 when σ 1, σ 2 unknown If σ 1, σ 2 are unknown, we use the t distribution (just like the one-sample problem).

101. Confidence Intervals for μ 1 – μ 2 when σ 1, σ 2 unknown

102. What if σ 1 = σ 2 ? If the sample standard deviations s 1 and s 2 are sufficiently close, then it may be safe to assume that σ 1 = σ 2.  Use a pooled standard deviation.  See Section 8.4, problem 27.

103. What if σ 1 = σ 2 ?

104. Summarizing Intervals for Differences in Population Means

105. Estimating the Difference in Proportions We consider two independent binomial distributions. For distribution 1 and distribution 2, respectively, we have: n 1 p 1 q 1 r 1 n 2 p 2 q 2 r 2 We assume that all the following are greater than 5:

106. Estimating the Difference in Proportions

108. Critical Thinking