1. Ch 12 實習

2. Jia-Ying Chen2 We shall develop techniques to estimate and test three population parameters. Population mean  Population variance  2 Population proportion p Introduction

3. Jia-Ying Chen3 Recall that when  is known we use the following statistic to estimate and test a population mean When  is unknown, we use its point estimator s, and the z-statistic is replaced then by the t-statistic Inference About a Population Mean When the Population Standard Deviation Is Unknown

4. Jia-Ying Chen4 The t - Statistic s 0 The t distribution is mound-shaped, and symmetrical around zero. The “degrees of freedom”, (a function of the sample size) determine how spread the distribution is (compared to the normal distribution) d.f. = v 2 d.f. = v 1 v 1 < v 2 t

5. Jia-Ying Chen5 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 統計學 Ex:

6. Jia-Ying Chen6 How to calculus sample variance

7. Jia-Ying Chen7 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 Testing  when  is unknown

8. Jia-Ying Chen8 Example 1 – Solution The problem objective is to describe the population of the number of packages processed in one hour. The data are interval. H 0 :  = 450 H 1 :  > 450 The t statistic d.f. = n - 1 = 49 Testing  when  is unknown

9. Jia-Ying Chen9 Solution continued (solving by hand) The rejection region is t > t ,n – 1 t ,n - 1 = t.05,49  t.05,50 = 1.676. Testing  when  is unknown

10. Jia-Ying Chen10 The test statistic is 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. 1.676 1.89 Rejection region Testing  when  is unknown

11. Jia-Ying Chen11 Estimating  when  is unknown Confidence interval estimator of  when  is unknown

12. Jia-Ying Chen12 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. Estimating  when  is unknown

13. Jia-Ying Chen13 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 Estimating  when  is unknown

14. Jia-Ying Chen14 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 …

15. Jia-Ying Chen15 A Histogram for Example 1 Packages A Histogram for Example 2 Returns

16. Jia-Ying Chen16 Summary of Test Statistics to be Used in a Hypothesis Test about a Population Mean n > 30 ?  known ?  known ? Popul. approx.normal ?  known ? Use s to estimate  Use s to estimate  Increase n to > 30 Yes Yes Yes Yes No No No No

17. Jia-Ying Chen17 Example 1 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

18. Jia-Ying Chen18 Solution

19. Jia-Ying Chen19 Example 2 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

20. Jia-Ying Chen20 Solution H 0 :μ=8 H 1 :μ<8 There is enough evidence to conclude that the average container is mislabeled

21. Jia-Ying Chen21 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  2.

22. Jia-Ying Chen22 The sample variance s 2 is an unbiased, consistent and efficient point estimator for  2. The statistic has a distribution called Chi-squared, if the population is normally distributed. d.f. = 5 d.f. = 10 Inference About a Population Variance

23. Jia-Ying Chen23 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? Testing the Population Variance

24. Jia-Ying Chen24 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: H 0 :  2 = 1 H 1 :  2 <1 We want to know whether the process is consistent Testing the Population Variance

25. Jia-Ying Chen25 There is insufficient evidence to reject the hypothesis that the variance is less than 1. There is insufficient evidence to reject the hypothesis that the variance is less than 1. Solving by hand –Note that (n - 1)s 2 =  (x i - x) 2 =  x i 2 – (  x i ) 2 /n –From the sample, we can calculate  x i = 24,996.4, and  x i 2 = 24,992,821.3 –Then (n - 1)s 2 = 24,992,821.3-(24,996.4) 2 /25 =20.78 Testing the Population Variance

26. Jia-Ying Chen26 13.848420.8 Rejection region  =.05 1-  =.95 Do not reject the null hypothesis Testing the Population Variance

27. Jia-Ying Chen27 Testing and Estimating a Population Variance From the following probability statement P(  2 1-  /2 <  2 <  2  /2 ) = 1-  we have (by substituting  2 = [(n - 1)s 2 ]/  2.)

28. Jia-Ying Chen28 Example 3 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 σ 2 =23 mpg 2. 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

29. Jia-Ying Chen29 Solution H 0 : σ 2 =23 H 1 : σ 2 ≠23

30. Jia-Ying Chen30 Example 4 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

31. Jia-Ying Chen31 Solution

32. Jia-Ying Chen32 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.

33. Jia-Ying Chen33 Statistic and sampling distribution the statistic used when making inference about p is: – Under certain conditions, [np > 5 and n(1-p) > 5], is approximately normally distributed, with  = p and  2 = p(1 - p)/n. Inference About a Population Proportion

34. Jia-Ying Chen34 Testing and Estimating the Proportion Test statistic for p Interval estimator for p (1-  confidence level)

35. Jia-Ying Chen35 Example 5 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.

36. Jia-Ying Chen36 Solution

37. Jia-Ying Chen37 Example 6 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?

38. Jia-Ying Chen38 Solution There is enough evidence to support the officer’s belief

39. Jia-Ying Chen39 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

40. Jia-Ying Chen40 Selecting the Sample Size to Estimate the Proportion The required sample size is

41. Jia-Ying Chen41 12.41 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.