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Ch 12 實習

Ch 12 實習. Introduction. We shall develop techniques to estimate and test three population parameters. Population mean m Population variance s 2 Population proportion p. Inference About a Population Mean When the Population Standard Deviation Is Unknown.

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Ch 12 實習

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  1. Ch 12 實習

  2. Introduction • We shall develop techniques to estimate and test three population parameters. • Population mean m • Population variance s2 • Population proportion p Jia-Ying Chen

  3. Inference About a Population Mean When the Population Standard Deviation Is Unknown • Recall that when sis known we use the following • statistic to estimate and test a population mean • When sis unknown, we use its point estimator s, and the z-statistic is replaced then by the t-statistic Jia-Ying Chen

  4. The t - Statistic t s The “degrees of freedom”, (a function of the sample size) determine how spread the distribution is (compared to the normal distribution) The t distribution is mound-shaped, and symmetrical around zero. d.f. = v2 d.f. = v1 v1 < v2 0 Jia-Ying Chen

  5. 自由度 • 統計學上的自由度（degree of freedom, df），是指當以樣本的統計量來估計總體的參數時， 樣本中獨立或能自由變化的資料的個數，稱為該統計量的自由度 • Ex: Jia-Ying Chen

  6. How to calculus sample variance Jia-Ying Chen

  7. Testing m when s is unknown • Example 1 • In order to determine the number of workers required to meet demand, the productivity of newly hired trainees is studied. • It is believed that trainees can process and distribute more than 450 packages per hour within one week of hiring. • Can we conclude that this belief is correct, based on productivity observation of 50 trainees Jia-Ying Chen

  8. Testing m when s is unknown • Example 1 – Solution • The problem objective is to describe the population of the number of packages processed in one hour. • The data are interval. H0:m = 450 H1:m > 450 • The t statistic d.f. = n - 1 = 49 Jia-Ying Chen

  9. Testing m when s is unknown • Solution continued (solving by hand) • The rejection region is t > ta,n – 1ta,n - 1 = t.05,49 @ t.05,50 = 1.676. Jia-Ying Chen

  10. Testing m when s is unknown • The test statistic is Rejection region 1.676 1.89 • Since 1.89 > 1.676 we reject the null hypothesis in favor of the alternative. • There is sufficient evidence to infer that the mean productivity of trainees one week after being hired is greater than 450 packages at .05 significance level. Jia-Ying Chen

  11. Estimating m when s is unknown • Confidence interval estimator of m when sis unknown Jia-Ying Chen

  12. Estimating m when s is unknown • Example 2 • An investor is trying to estimate the return on investment in companies that won quality awards last year. • A random sample of 83 such companies is selected, and the return on investment is calculated had he invested in them. • Construct a 95% confidence interval for the mean return. Jia-Ying Chen

  13. Estimating m when s is unknown • Solution (solving by hand) • The problem objective is to describe the population of annual returns from buying shares of quality award-winners. • The data are interval. • Solving by hand • From the data we determine t.025,82@ t.025,80 Jia-Ying Chen

  14. Checking the required conditions • We need to check that the population is normally distributed, or at least not extremely nonnormal. • There are statistical methods to test for normality • From the sample histograms we see… Jia-Ying Chen

  15. A Histogram for Example 1 Packages A Histogram for Example 2 Returns Jia-Ying Chen

  16. Summary of Test Statistics to be Used in aHypothesis Test about a Population Mean Yes No n > 30 ? No Popul. approx. normal ? s known ? Yes Yes Use s to estimate s No s known ? No Use s to estimate s Yes Increase n to > 30 Jia-Ying Chen

  17. Example 1 • A federal agency responsible for enforcing laws governing weights and measures routinely inspects packages to determine whether the weight of the contents is at least as great as that advertised on the package. A random sample of 18 containers whose packaging states that the contents weigh 8 ounces was drawn. The contents were weighted and the results follows. Can we concluded at the 1% significance level that on average the containers are mislabeled? (Assume the random variable is normally distributed) • 7.80 7.91 7.93 7.99 7.94 7.75 7.97 7.95 7.79 8.06 7.82 7.89 7.92 7.87 7.92 7.98 8.05 7.91 Jia-Ying Chen

  18. Solution • H0:μ=8 H1:μ<8 There is enough evidence to conclude that the average container is mislabeled Jia-Ying Chen

  19. Example 2 • How much money do winners go home with from the television quiz show Feopardy? To determine an answer, a random sample of winners was drawn and the amount of money each won was recorded and is listed here. Estimate with 95% confidence the mean winnings for all show’s players (Assume the random variable is normally distributed) • 26650 6060 52820 8490 13660 25840 49840 23790 51480 18960 990 11450 41810 21060 7860 Jia-Ying Chen

  20. Solution Jia-Ying Chen

  21. Example 3 • A random sample of 10 college students was drawn from a large university. Their ages are 22, 17, 27, 20, 23, 19, 24, 18, 19, and 24 years. Assume the age is normal distributed. • a. Estimate the population mean with 90% confidence. • b. Test to determine if we can infer at the 5% significance level that the population mean is not equal to 20. • c. What is the required condition of the techniques used in the previous questions? What graphical device can you use to check to see if that required condition is satisfied? Jia-Ying Chen

  22. Solution • a. Thus, LCL = 19.446, and UCL = 23.154. • b. H0: μ = 20 vs. H1: μ ≠ 20 Rejection region: | t | > t0.025,9 = 2.262 Test statistic: t = 1.285 Conclusion: Don't reject H0. We can't infer at the 5% significance level that the population mean is not equal to 20. • The condition is that ages in the population are normally distributed. A histogram of the data can be used to check if the normality assumption is satisfied. Jia-Ying Chen

  23. Inference About a Population Variance • Sometimes we are interested in making inference about the variability of processes. • Examples: • The consistency of a production process for quality control purposes. • Investors use variance as a measure of risk. • To draw inference about variability, the parameter of interest is s2. Jia-Ying Chen

  24. Inference About a Population Variance • The sample variance s2 is an unbiased, consistent and efficient point estimator for s2. • The statistic has a distribution called Chi-squared, if the population is normally distributed. d.f. = 5 d.f. = 10 Jia-Ying Chen

  25. Testing the Population Variance • Example 3 (operation management application) • A container-filling machine is believed to fill 1 liter containers so consistently, that the variance of the filling will be less than 1 cc (.001 liter). • To test this belief a random sample of 25 1-liter fills was taken, and the results recorded • Do these data support the belief that the variance is less than 1cc at 5% significance level? Jia-Ying Chen

  26. Testing the Population Variance • Solution • The problem objective is to describe the population of 1-liter fills from a filling machine. • The data are interval, and we are interested in the variability of the fills. • The complete test is: H0:s2 = 1 H1: s2 <1 We want to know whether the process is consistent Jia-Ying Chen

  27. Solving by hand • Note that (n - 1)s2 = S(xi - x)2 = Sxi2 – (Sxi)2/n • From the sample, we can calculate Sxi = 24,996.4, and Sxi2 = 24,992,821.3 • Then (n - 1)s2 = 24,992,821.3-(24,996.4)2/25 =20.78 Testing the Population Variance There is insufficient evidence to reject the hypothesis that the variance is less than 1. Jia-Ying Chen

  28. Testing the Population Variance a = .05 1-a = .95 Rejection region 13.8484 20.8 Do not reject the null hypothesis Jia-Ying Chen

  29. Testing and Estimating a Population Variance • From the following probability statement P(c21-a/2 < c2 < c2a/2) = 1-awe have (by substituting c2 = [(n - 1)s2]/s2.) Jia-Ying Chen

  30. Example 4 • With gasoline prices increasing, drivers are becoming more concerned with their cars’ gasoline consumption. For the past 5 years, a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2=23 mpg2. Now that his car is 5 years old, he would like to know whether the variability of gas mileage has changed. He recorded the gas mileage from his last eight fill-ups; these are listed here. Conduct a test at a 10% significance level to infer whether the variability has changed. • 28 25 29 25 32 36 27 24 Jia-Ying Chen

  31. Solution • H0:σ2=23 H1:σ2≠23 Jia-Ying Chen

  32. Example 5 • During annual checkups physician routinely send their patients to medical laboratories to have various tests performed. One such test determines the cholesterol level in patients’ blood. However, not all tests are conducted in the same way. To acquire more information, a man was sent to 10 laboratories and in each had his cholesterol level measured. The results are listed here. Estimate with 95% confidence the variance of these measurements. • 4.70 4.83 4.65 4.60 4.75 4.88 4.68 4.75 4.80 4.90 Jia-Ying Chen

  33. Solution Jia-Ying Chen

  34. Example 6 • Which of the following conditions is needed regarding the chi-squared test statistic for the test of variance? • a. The population random variable must be normal. • b. The test statistic must be a non-negative number. • c. The test statistic must have a chi-squared distribution with n - 1 degrees of freedom. • d. All of these choices are true. Jia-Ying Chen

  35. Inference About a Population Proportion • When the population consists of nominal data, the only inference we can make is about the proportion of occurrence of a certain value. • The parameter p was used before to calculate these probabilities under the binomial distribution. Jia-Ying Chen

  36. Under certain conditions, [np > 5 and n(1-p) > 5], is approximately normally distributed, withm = p and s2 = p(1 - p)/n. Inference About a Population Proportion • Statistic and sampling distribution • the statistic used when making inference about p is: Jia-Ying Chen

  37. Testing and Estimating the Proportion • Test statistic for p • Interval estimator for p (1-a confidence level) Jia-Ying Chen

  38. Example 7 • A dean of a business school wanted to know whether the graduates of her school used a statistical inference technique during their first year of employment after graduation. She surveyed 314 graduates and asked about the use of statistical technique. After tallying up the responses, she found that 204 used statistical inference within one year of graduation. Estimate with 90% confidence the proportion of all business school graduates who use their statistical education within a year of graduation. Jia-Ying Chen

  39. Solution Jia-Ying Chen

  40. Example 8 • In some states the law requires drivers to turn on their headlights when driving in the rain. A highway patrol officer believes that less than one-quarter of all drivers follow this rule. As a test, he randomly samples 200 cars driving in the rain and counts the number whose headlights are turned on. H finds this number to be 41. Does the officer have enough evidence at the 10% significance level to support his belief? Jia-Ying Chen

  41. Solution There is enough evidence to support the officer’s belief Jia-Ying Chen

  42. Selecting the Sample Size to Estimate the Proportion • Recall: The confidence interval for the proportion is • Thus, to estimate the proportion to within W, we can write Jia-Ying Chen

  43. Selecting the Sample Size to Estimate the Proportion • The required sample size is Jia-Ying Chen

  44. Selecting the Sample Size Two methods – in each case we choose a value for then solve the equation for n. Method 1 : no knowledge of even a rough value of . This is a ‘worst case scenario’ so we substitute = .50 Method 2 : we have some idea about the value of . This is a better scenario and we substitute in our estimated value. Jia-Ying Chen 12.44

  45. Example 9 • As a manufacturer of golf clubs, a major corporation wants to estimate the proportion of golfers who are right-handed. How many golfers must be surveyed if they want to be within 0.02, with a 95% confidence? • a. Assume that there is no prior information that could be used as an estimate of . • b. Assume that the manufacturer has an estimate of found from a previous study, which suggests that 75% of golfers are right-handed. Jia-Ying Chen

  46. Solution • a. • b. 取n=1801 Jia-Ying Chen

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