2. Yandell – Econ 216 Chap 8-1 Chapter 8 Confidence Interval Estimation

3. Yandell – Econ 216 Chap 8-2 Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions Determine the required sample size to estimate a single population mean within a specified margin of error Form and interpret a confidence interval estimate for a single population proportion

4. Yandell – Econ 216 Chap 8-3 Confidence Intervals Content of this chapter Confidence Intervals for the Population Mean,  when Population Standard Deviation  is Known when Population Standard Deviation  is Unknown Determining the Required Sample Size Confidence Intervals for the Population Proportion, π

5. Yandell – Econ 216 Chap 8-4 Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

6. Yandell – Econ 216 Chap 8-5 Point Estimates We can estimate a Population Parameter … with a Sample Statistic (a Point Estimate) Mean Proportion p π X μ

7. Yandell – Econ 216 Chap 8-6 Sampling Error Sample Statistics are used to estimate Population Parameters ex: X is an estimate of the population mean, μ Problems: Different samples provide different estimates of the population parameter Sample results have potential variability, thus sampling error exits

8. Yandell – Econ 216 Chap 8-7 Calculating Sampling Error Sampling Error: The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population Example: (for the mean) where:

9. Yandell – Econ 216 Chap 8-8 Example If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is

10. Yandell – Econ 216 Chap 8-9 Sampling Errors Different samples will yield different sampling errors The sampling error may be positive or negative ( may be greater than or less than μ) The expected sampling error decreases as the sample size increases

11. Yandell – Econ 216 Chap 8-10 Confidence Intervals How much uncertainty is associated with a point estimate of a population parameter? An interval estimate provides more information about a population characteristic than does a point estimate Such interval estimates are called confidence intervals

12. Yandell – Econ 216 Chap 8-11 Confidence Interval Estimate An interval gives a range of values: Takes into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Gives information about closeness to unknown population parameters Stated in terms of level of confidence Never 100% sure

13. Yandell – Econ 216 Chap 8-12 Estimation Process (mean, μ, is unknown) Population Random Sample Mean X = 50 Sample I am 95% confident that μ is between 40 & 60.

14. Yandell – Econ 216 Chap 8-13 General Formula The general formula for all confidence intervals is: Point Estimate  (Critical Value)(Standard Error)

15. Yandell – Econ 216 Chap 8-14 Confidence Level Confidence in which the interval will contain the unknown population parameter A percentage (less than 100%)

16. Yandell – Econ 216 Chap 8-15 Confidence Level, (1-  ) Suppose confidence level = 95% Also written (1 -  ) =.95 A relative frequency interpretation: In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter A specific interval either will contain or will not contain the true parameter No probability involved in a specific interval (continued)

17. Yandell – Econ 216 Chap 8-16 Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

18. Yandell – Econ 216 Chap 8-17 Confidence Interval for μ ( σ Known) Assumptions Population standard deviation σ is known Population is normally distributed If population is not normal, use large sample Confidence interval estimate

19. Yandell – Econ 216 Chap 8-18 Finding the Critical Value Consider a 95% confidence interval: Z.025 = -1.96Z.025 = 1.96 Point Estimate Lower Confidence Limit Upper Confidence Limit Z units: X units: Point Estimate 0

20. Yandell – Econ 216 Chap 8-19 Common Levels of Confidence Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Confidence Coefficient, Z value, 1.28 1.645 1.96 2.33 2.57 3.08 3.27.80.90.95.98.99.998.999 80% 90% 95% 98% 99% 99.8% 99.9%

21. Yandell – Econ 216 Chap 8-20 Interval and Level of Confidence Confidence Intervals Intervals extend from to 100(1-  )% of intervals constructed contain μ ; 100  % do not. Sampling Distribution of the Mean x x1x1 x2x2

22. Yandell – Econ 216 Chap 8-21 Margin of Error Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval Example: Margin of error for estimating μ, σ known:

23. Yandell – Econ 216 Chap 8-22 Factors Affecting Margin of Error Data variation, σ : e as σ Sample size, n : e as n Level of confidence, 1 -  : e if 1 - 

24. Yandell – Econ 216 Chap 8-23 Example A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms. Determine a 95% confidence interval for the true mean resistance of the population.

25. Yandell – Econ 216 Chap 8-24 Example A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms. Solution: (continued)

26. Yandell – Econ 216 Chap 8-25 Interpretation We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean An incorrect interpretation is that there is 95% probability that this interval contains the true population mean. (This interval either does or does not contain the true mean, there is no probability for a single interval)

27. Yandell – Econ 216 Chap 8-26 Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

28. Yandell – Econ 216 Chap 8-27 If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to sample So we use the t distribution instead of the normal distribution Confidence Interval for μ ( σ Unknown)

29. Yandell – Econ 216 Chap 8-28 Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample Use Student’s t Distribution Confidence Interval Estimate Confidence Interval for μ ( σ Unknown) (continued)

30. Yandell – Econ 216 Chap 8-29 Student’s t Distribution The t is a family of distributions The t value depends on degrees of freedom (d.f.) Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1

31. Yandell – Econ 216 Chap 8-30 If the mean of these three values is 8.0, then X 3 must be 9 (i.e., X 3 is not free to vary) Degrees of Freedom (df) Here, n = 3, so degrees of freedom = n -1 = 3 – 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean) Idea: Number of observations that are free to vary after sample mean has been calculated Example: Suppose the mean of 3 numbers is 8.0 Let X 1 = 7 Let X 2 = 8 What is X 3 ?

32. Yandell – Econ 216 Chap 8-31 Student’s t Distribution t 0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df =  ) Note: t Z as n increases

33. Yandell – Econ 216 Chap 8-32 Student’s t Table Upper Tail Area df.25.10.05 11.0003.0786.314 2 0.8171.886 2.920 30.7651.6382.353 t 0 2.920 The body of the table contains t values, not probabilities Let: n = 3 df = n - 1 = 2  =.10  /2 =.05  /2 =.05

34. Yandell – Econ 216 Chap 8-33 t distribution values With comparison to the Z value Confidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____.80 1.372 1.325 1.310 1.28.90 1.812 1.725 1.697 1.645.95 2.228 2.086 2.042 1.96.99 3.169 2.845 2.750 2.57 Note: t Z as n increases

35. Yandell – Econ 216 Chap 8-34 Example A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ d.f. = n – 1 = 24, so The confidence interval is 46.698 …………….. 53.302

36. Yandell – Econ 216 Chap 8-35 Approximation for Large Samples Since t approaches Z as the sample size increases, an approximation is sometimes used when n  30: Technically correct Approximation for large n listen

37. Yandell – Econ 216 Chap 8-36 Determining Sample Size The required sample size can be found to reach a desired margin of error (e) and level of confidence (1 -  ) Required sample size, σ known:

38. Yandell – Econ 216 Chap 8-37 Required Sample Size Example If  = 45, what sample size is needed to be 90% confident of being correct within ± 5? (Always round up) So the required sample size is n = 220

39. Yandell – Econ 216 Chap 8-38 If σ is unknown If unknown, σ can be estimated when using the required sample size formula Use a value for σ that is expected to be at least as large as the true σ Select a pilot sample and estimate σ with the sample standard deviation, S

40. Yandell – Econ 216 Chap 8-39 PHStat demonstration Click here to watch demonstration part 1 Click below to watch PHStat demonstrations of confidence intervals for a population mean. Click here to watch demonstration part 2 Click here to watch demonstration part 3

41. Yandell – Econ 216 Chap 8-40 Confidence Intervals Population Mean σ Unknown Confidence Intervals Population Proportion σ Known

42. Yandell – Econ 216 Chap 8-41 Confidence Intervals for the Population Proportion, π An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p )

43. Yandell – Econ 216 Chap 8-42 Confidence Intervals for the Population Proportion, π Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation We will estimate this with sample data: (continued)

44. Yandell – Econ 216 Chap 8-43 Confidence interval endpoints Upper and lower confidence limits for the population proportion are calculated with the formula where Z is the standard normal value for the level of confidence desired p is the sample proportion n is the sample size listen

45. Yandell – Econ 216 Chap 8-44 Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers

46. Yandell – Econ 216 Chap 8-45 Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. 1. 2. 3. (continued)

47. Yandell – Econ 216 Chap 8-46 Interpretation We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

48. Yandell – Econ 216 Chap 8-47 PHStat Output Input Output

49. Yandell – Econ 216 Chap 8-48 Changing the sample size Increases in the sample size reduce the width of the confidence interval. Example: If the sample size in the above example is doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at.25, but the width shrinks to.19 …….31

50. Yandell – Econ 216 Chap 8-49 Finding the Required Sample Size for proportion problems Solve for n: Define the margin of error: π can be estimated with a pilot sample, if necessary (or conservatively use π =.50)

51. Yandell – Econ 216 Chap 8-50 What sample size...? How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p =.12)

52. Yandell – Econ 216 Chap 8-51 What sample size...? Solution: For 95% confidence, use Z = 1.96 e =.03 p =.12, so use this to estimate π So use n = 451 (continued)

53. Yandell – Econ 216 Chap 8-52 Using PHStat PHStat can be used for confidence intervals for the mean or proportion two options for the mean: known and unknown population standard deviation required sample size can also be found

54. Yandell – Econ 216 Chap 8-53 PHStat Interval Options options

55. Yandell – Econ 216 Chap 8-54 PHStat Sample Size Options

56. Yandell – Econ 216 Chap 8-55 Using PHStat (for μ, σ unknown) A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ

57. Yandell – Econ 216 Chap 8-56 Using PHStat (sample size for proportion) How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence? (Assume a pilot sample yields p =.12)

58. Yandell – Econ 216 Chap 8-57 PHStat Demonstration Click below to see how to use PHStat to get confidence intervals for a population proportion. Click here to watch demonstration part 1 Click here to watch demonstration part 2

59. Yandell – Econ 216 Chap 8-58 Summary Flowcharts Click below to see summary flowcharts for the creation of confidence intervals (for the mean and for the proportion) Click here to open the.pdf file

60. Yandell – Econ 216 Chap 8-59 Chapter Summary Illustrated estimation process Discussed point estimates Introduced interval estimates Discussed confidence interval estimation for the mean (σ known) Addressed determining sample size Discussed confidence interval estimation for the mean (σ unknown) Discussed confidence interval estimation for the proportion

61. Yandell – Econ 216 Chap 8-60 Practice Problem and PHStat demonstration After this slideshow, open the ch08-review.ppsx Slideshow for a practice problem (with solution)

62. Yandell – Econ 216 Chap 8-61 Practice Exercise 1 An auditor takes a sample of size 49 to estimate the mean value of daily scrap due to poor incoming quality. The sample mean value is $340 and the sample standard deviation is $230. Form a 90% confidence interval for the true mean scrap value. Suppose that the auditor wishes to be 98% confident that she has estimated the true value of daily spoilage within $35. How large is the necessary sample? What are the practical implications of this decision to be more accurate?

63. Yandell – Econ 216 Chap 8-62 Practice Exercise 2 1. Use M&M packages to do a simple problem... (We’ll do this one in class if time permits)

64. Yandell – Econ 216 Chap 8-63 Practice Exercise 3 To subsidize military housing for enlisted men and women, a study is done to determine the average monthly housing costs for personnel in San Diego. A randomly selected sample of enlisted men and women living off base gives the following information: Sample size = 100 Mean monthly expense = $820 Sample standard deviation = $120 Form a 95% confidence interval for the true mean housing expense for enlisted men and women in San Diego.

65. Yandell – Econ 216 Chap 8-64 Practice Exercise 4 Suppose that 28% of the drivers in a community are Blue and 72% are Red. A random sample of records from 81 traffic citations issued by the local police force show that 35% were issued to Blue people. Does this evidence suggest that Blue people are given more traffic citations than Red people? (Use a 90% level of confidence to frame your answer.)