1. Confidence Intervals Session 3

2. l Using Statistics. l Confidence Interval for the Population Mean When the Population Standard Deviation is Known. Confidence Intervals for  When  is Unknown - The t Distribution. l Large-Sample Confidence Intervals for the Population Proportion. Confidence Intervals

3. l The Finite-Population Correction Factor. l Confidence Intervals for the Population Variance. l Sample Size Determination. l One-Sided Confidence Intervals. l Using the Computer. l Summary and Review of Terms. Confidence Intervals (cont.)

4. Consider the following statements: – x = 550 A single-valued estimate that conveys little information about the actual value of the population mean. – We are 99% confident that  is in the interval [449,551] An interval estimate which locates the population mean within a narrow interval, with a high level of confidence. – We are 90% confident that  is in the interval [400,700] An interval estimate which locates the population mean within a broader interval, with a lower level of confidence. 3-1 Introduction

5. Point Estimate A single-valued estimate. A single element chosen from a sampling distribution. Conveys little information about the actual value of the population parameter, about the accuracy of the estimate. Types of Estimators

6. Confidence Interval or Interval Estimate An interval or range of values believed to include the unknown population parameter. Associated with the interval is a measure of the confidence we have that the interval does indeed contain the parameter of interest. Types of Estimators (cont.)

7. A confidence interval or interval estimate is a range or interval of numbers believed to include an unknown population parameter. Associated with the interval is a measure of the confidence we have that the interval does indeed contain the parameter of interest. A confidence interval or interval estimate has two components: A range or interval of values. An associated level of confidence. Confidence Interval or Interval Estimate

8. l If the population distribution is normal, the sampling distribution of the mean is normal. If the sample is sufficiently large, regardless of the shape of the population distribution, the sampling distribution is normal (Central Limit Theorem). 3-2 Confidence Interval for  when  Is Known

9. Confidence Interval for  when  Is Known 43210-1-2-3-4 0.4 0.3 0.2 0.1 0.0 z f ( z ) Standard Normal Distribution: 95% Interval

10. Confidence Interval for  when  Is Known (cont.)

11. 0.4 0.3 0.2 0.1 0.0 x f ( x ) Sampling Distribution of the Mean  x x x x x x x x 2.5% 95% 2.5% x 2.5% fall above the interval 2.5% fall below the interval 95% fall within the interval A 95% Interval around the Population Mean Approximately 95% of sample means can be expected to fall within the interval Conversely, about 2.5% can be expected to be above and 2.5% can be expected to be below: So 5% can be expected to fall outside the interval.

12. Approximately 95% of the intervals around the sample mean can be expected to include the actual value of the population mean, . (When the sample mean falls within the 95% interval around the population mean.) * 5% of such intervals around the sample mean can be expected not to include the actual value of the population mean. (When the sample mean falls outside the 95% interval around the population mean.) x x  x  95% Intervals around the Sample Mean 0.4 0.3 0.2 0.1 0.0 x f ( x ) Sampling Distribution of the Mean  x x x x x x x x 2.5% 95% 2.5% x x  x  * *

13. A 95% confidence interval for  when  is known and sampling is done from a normal population, or a large sample is used: The quantity is often called the margin of error or the sampling error. The 95% Confidence Interval for 

14. For example, if : n = 22  = 20 x = 122 A 95% confidence interval: The 95% Confidence Interval for 

15. We define as the z value that cuts off a right-tail area of under the standard normal curve. (1-  ) is called the confidence coefficient.  is called the error probability, and (1-  )100% is called the confidence level. Pzz Pzz Pzzz z n                                   2 2 22 2 2 2 1() (1-)100% Confidence Interval: x 543210-1-2-3-4-5 0.4 0.3 0.2 0.1 0.0 Z f ( z ) Standard Normal Distribution (1-  )100% Confidence Interval

16. Critical Values of z and Levels of Confidence 543210-1-2-3-4-5 0.4 0.3 0.2 0.1 0.0 Z f ( z ) Standard Normal Distribution

17. When sampling from the same population, using a fixed sample size, the higher the confidence level, the wider the confidence interval. 543210-1-2-3-4-5 0.4 0.3 0.2 0.1 0.0 Z f ( z ) Standard Normal Distribution 543210-1-2-3-4-5 0.4 0.3 0.2 0.1 0.0 Z f ( z ) Standard Normal Distribution The Level of Confidence and the Width of the Confidence Interval

18. The Sample Size and the Width of the Confidence Interval When sampling from the same population, using a fixed confidence level, the larger the sample size, n, the narrower the confidence interval. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 x f ( x ) Sampling Distribution of the Mean 95% Confidence Interval: n = 40 0.4 0.3 0.2 0.1 0.0 x f ( x ) Sampling Distribution of the Mean 95% Confidence Interval: n = 20

19. Population consists of the Fortune 500 Companies (Fortune Web Site), as ranked by Revenues. You are trying to to find out the average Revenues for the companies on the list. The population standard deviation is $15056.37. A random sample of 30 companies obtains a sample mean of $10672.87. Give a 95% and 90% confidence interval for the average Revenues. Example 3-1

20. Display shows the Excel function that is used. Example 3-1 (cont.) - Using Excel

21. Display shows the error value of 5387.75. Add and subtract this value to the sample mean to get the 95% confidence interval. Example 3-1 (cont.) - Using Excel

22. Display shows the Range of the 95% and 90% Confidence Intervals. Example 3-1 (cont.) - Using Excel

23. If the population standard deviation, , is not known, replace  with the sample standard deviation, s. If the population is normal, the resulting statistic: has a t distribution with (n - 1) degrees of freedom. 3.3 Confidence Interval or Interval Estimate for  when  Is Unknown - The t Distribution

24. The t is a family of bell-shaped and symmetric distributions, one for each number of degree of freedom. The expected value of t is 0. For df > 2, the variance of t is df/(df-2). This is greater than 1, but approaches 1 as the number of degrees of freedom increases. The t is flatter and has fatter tails than does the standard normal. The t distribution approaches a standard normal as the number of degrees of freedom increases. Standard normal t, df=20 t, df=10  3.3 Confidence Interval or Interval Estimate for  when  Is Unknown - The t Distribution

25. A (1-  )100% confidence interval for  when  is not known (assuming a normally distributed population): where is the value of the t distribution with n-1 degrees of freedom that cuts off a tail area of to its right. Confidence Intervals for  when  is Unknown- The t Distribution

26. df t 0.100 t 0.050 t 0.025 t 0.010 t 0.005 ------------------------------- 13.0786.31412.70631.82163.657 21.8862.9204.3036.9659.925 31.6382.3533.1824.5415.841 41.5332.1322.7763.7474.604 51.4762.0152.5713.3654.032 61.4401.9432.4473.1433.707 71.4151.8952.3652.9983.499 81.3971.8602.3062.8963.355 91.3831.8332.2622.8213.250 101.3721.812 2.2282.7643.169 111.3631.7962.2012.7183.106 121.3561.7822.1792.6813.055 131.3501.7712.1602.6503.012 141.3451.7612.1452.6242.977 151.3411.7532.1312.6022.947 161.3371.7462.1202.5832.921 171.3331.7402.1102.5672.898 181.3301.7342.1012.5522.878 191.3281.7292.0932.5392.861 201.3251.7252.0862.5282.845 211.3231.7212.0802.5182.831 221.3211.7172.0742.5082.819 231.3191.7142.0692.5002.807 241.3181.7112.0642.4922.797 251.3161.7082.0602.4852.787 261.3151.7062.0562.4792.779 271.3141.7032.0522.4732.771 281.3131.7012.0482.4672.763 291.3111.6992.0452.4622.756 301.3101.6972.0422.4572.750 401.3031.6842.0212.4232.704 601.2961.6712.0002.3902.660 1201.2891.6581.9802.3582.617 1.2821.6451.9602.3262.576 0 0.4 0.3 0.2 0.1 0.0 t f ( t ) t Distribution: df=10 Area = 0.10 } } Area = 0.025 } } 1.372 -1.372 2.228 -2.228 Whenever  is not known (and the population is assumed normal), the correct distribution to use is the t distribution with n-1 degrees of freedom. Note, however, that for large degrees of freedom, the t distribution is approximated well by the Z distribution. The t Distribution

27. A stock market analyst wants to estimate the average return on a certain stock. A random sample of 15 days yields an average (annualized) return of x=10.37% and a standard deviation of s = 3.5%. Assuming a normal population of returns, give a 95% confidence interval for the average return on this stock. The critical value of t for df = (n-1)=(15- 1)=14 and a right-tail area of 0.025 is: The corresponding confidence interval or interval estimate is: The t Distribution (Example 3-2) df t 0.100 t 0.050 t 0.025 t 0.010 t 0.005 ------------------------------- 13.0786.31412.70631.82163.657...... 131.3501.7712.1602.6503.012 141.3451.7612.1452.6242.977 151.3411.7532.1312.6022.947......

28. Note: The TINV function was used to determine the critical t value. The t Distribution (Example 3-2) Using Excel

29. Whenever  is not known (and the population is assumed normal), the correct distribution to use is the t distribution with n-1 degrees of freedom. Note, however, that for large degrees of freedom, the t distribution is approximated well by the Z distribution. df t 0.100 t 0.050 t 0.025 t 0.010 t 0.005 ------------------------------- 13.0786.31412.70631.82163.657...... 1201.2891.6581.9802.3582.617 1.2821.6451.9602.3262.576 Large Sample Confidence Intervals for the Population Mean

30. Example 3-3: An economist wants to estimate the average amount in checking accounts at banks in a given region. A random sample of 100 accounts gives x-bar = $357.60 and s = $140.00. Give a 95% confidence interval for , the average amount in any checking account at a bank in the given region. Large Sample Confidence Intervals for the Population Mean

31. 3-4 Large-Sample Confidence Intervals for the Population Proportion, p

33. A marketing research firm wants to estimate the share that foreign companies have in the American market for certain products. A random sample of 100 consumers is obtained, and it is found that 34 people in the sample are users of foreign-made products; the rest are users of domestic products. Give a 95% confidence interval for the share of foreign products in this market. Large-Sample Confidence Interval for the Population Proportion, p (Example 3-4)

34. Thus, the firm may be 95% confident that foreign manufacturers control anywhere from 24.72% to 43.28% of the market. Large-Sample Confidence Interval for the Population Proportion, p (Example 3-4)

35. The width of a confidence interval can be reduced only at the price of: a lower level of confidence, or a larger sample. 90% Confidence Interval Sample Size, n = 200 Lower Level of ConfidenceLarger Sample Size Reducing the Width of Confidence Intervals - The Value of Information

36. The sample variance, s 2, is an unbiased estimator of the population variance,  2. Confidence intervals for the population variance are based on the chi-square (  2 ) distribution. 3-5 Confidence Intervals for the Population Variance: The Chi- Square (  2 ) Distribution

37. The chi-square distribution is the probability distribution of the sum of several independent, squared standard normal random variables. The mean of the chi-square distribution is equal to the degrees of freedom parameter, (E[  2 ]=df). The variance of a chi-square is equal to twice the number of degrees of freedom, (V[  2 ]=2df). 3-5 Confidence Intervals for the Population Variance: The Chi- Square (  2 ) Distribution (cont.)

38. l The chi-square random variable cannot be negative, so it is bound by zero on the left. l The chi-square distribution is skewed to the right. l The chi-square distribution approaches a normal as the degrees of freedom increase.  100500 0.10 0.09 0.08 0.07 0.06 0.0 5 0.04 0.03 0.02 0.01 0.00  2 f ( 2 ) Chi-Square Distribution: df=10, df=30, df=50 df = 10 df = 30 df = 50 The Chi-Square (  2 ) Distribution

40. Area in Right Tail.995.990.975.950.900.100.050.025.010.005 Area in Left Tail df.005.010.025.050.100.900.950.975.990.995 10.00003930.0001570.0009820.0003930.01582.713.845.026.637.88 20.01000.02010.05060.1030.2114.615.997.389.2110.60 30.07170.1150.2160.3520.5846.257.819.3511.3412.84 40.2070.2970.4840.7111.067.789.4911.1413.2814.86 50.4120.5540.8311.151.619.2411.0712.8315.0916.75 60.6760.8721.241.642.2010.6412.5914.4516.8118.55 70.9891.241.692.172.8312.0214.0716.0118.4820.28 81.341.652.182.733.4913.3615.5117.5320.0921.95 91.732.092.703.334.1714.6816.9219.0221.6723.59 102.162.563.253.944.8715.9918.3120.4823.2125.19 112.603.053.824.575.5817.2819.6821.9224.7226.76 123.073.574.405.236.3018.5521.0323.3426.2228.30 133.574.115.015.897.0419.8122.3624.7427.6929.82 144.074.665.636.577.7921.0623.6826.1229.1431.32 154.605.236.267.268.5522.3125.0027.4930.5832.80 165.145.816.917.969.3123.5426.3028.8532.0034.27 175.706.417.568.6710.0924.7727.5930.1933.4135.72 186.267.018.239.3910.8625.9928.8731.5334.8137.16 196.847.638.9110.1211.6527.2030.1432.8536.1938.58 207.438.269.5910.8512.4428.4131.4134.1737.5740.00 218.038.9010.2811.5913.2429.6232.6735.4838.9341.40 228.649.5410.9812.3414.0430.8133.9236.7840.2942.80 239.2610.2011.6913.0914.8532.0135.1738.0841.6444.18 249.8910.8612.4013.8515.6633.2036.4239.3642.9845.56 2510.5211.5213.1214.6116.4734.3837.6540.6544.3146.93 2611.1612.2013.8415.3817.2935.5638.8941.9245.6448.29 2711.8112.8814.5716.1518.1136.7440.1143.1946.9649.65 2812.4613.5615.3116.9318.9437.9241.3444.4648.2850.99 2913.1214.2616.0517.7119.7739.0942.5645.7249.5952.34 3013.7914.9516.7918.4920.6040.2643.7746.9850.8953.67 Values and Probabilities of Chi-Square Distributions

41. A (1-  )100% confidence interval for the population variance * (where the population is assumed normal): where is the value of the chi-square distribution with n-1 degrees of freedom that cuts off an area to its right and is the value of the distribution that cuts off an area of to its left (equivalently, an area of to its right). * Note:Because the chi-square distribution is skewed, the confidence interval for the population variance is not symmetric. Confidence Interval for the Population Variance

42. In an automated process, a machine fills cans of coffee. If the average amount filled is different from what it should be, the machine may be adjusted to correct the mean. If the variance of the filling process is too high, however, the machine is out of control and needs to be repaired. Therefore, from time to time regular checks of the variance of the filling process are made. This is done by randomly sampling filled cans, measuring their amounts, and computing the sample variance. A random sample of 30 cans gives an estimate s 2 = 18,540. Give a 95% confidence interval for the population variance,  2. Confidence Interval for the Population Variance (Example 3-5)

43. Area in Right Tail df.995.990.975.950.900.100.050.025.010.005........... 2812.4613.5615.3116.9318.9437.9241.3444.4648.2850.99 2913.1214.2616.0517.7119.7739.0942.5645.7249.5952.34 3013.7914.9516.7918.4920.6040.2643.7746.9850.8953.67 Example 3-5 (continued)

44. How close do you want your sample estimate to be to the unknown parameter? (What is the desired bound, B?) What do you want the desired confidence level (1-  ) to be so that the distance between your estimate and the parameter is less than or equal to B? What is your estimate of the variance (or standard deviation) of the population in question? Before determining the necessary sample size, three questions must be answered: } Bound, B 3-6 Sample-Size Determination

45.  Standard error of statistic Sample size = n Sample size = 2n Standard error of statistic The sample size determines the bound of a statistic, since the standard error of a statistic shrinks as the sample size increases: Sample Size and Standard Error

46. Minimum required sample size in estimating the population mean, : Bound of estimate: B= z 2      n z B n  2 22 2 Minimum required sample size in estimating the population proportion, p  n zpq B   2 2 2 Minimum Sample Size: Mean and Proportion

47. A marketing research firm wants to conduct a survey to estimate the average amount spent on entertainment by each person visiting a popular resort. The people who plan the survey would like to determine the average amount spent by all people visiting the resort to within $120, with 95% confidence. From past operation of the resort, an estimate of the population standard deviation is s = $400. What is the minimum required sample size? Sample-Size Determination (Example 3-6)

48. The manufacturers of a sports car want to estimate the proportion of people in a given income bracket who are interested in the model. The company wants to know the population proportion, p, to within 0.01 with 99% confidence. Current company records indicate that the proportion p may be around 0.25. What is the minimum required sample size for this survey? Sample-Size for Proportion (Example 3-7)

49. The required sample sizes for various degrees of accuracy (express as interval half-widths) are displayed for 90%, 95%, and 99% confidence levels are shown on the next slide. The specific case requested in the example, where the desired half-width is $120 with a 95% confidence level, appears in the bordered cell. The NORMINV function was used to calculate the critical z values for each confidence interval and the ROUNDUP function was used to get the smallest integer value that is at least as great as the required sample size determined by the formulas in the text. Sample-Size for Proportion (Example 3-7) Using Excel

51. MTB > Set C1 DATA> 17.125 DATA> 17 DATA> 17.375 DATA> 17.25 DATA> 16.625 DATA> 16.875 DATA> 17 DATA> 17.25 DATA> 18 DATA> 18.125 DATA> 17.5 DATA> 17.375 DATA> 17.5 DATA> 17.25 DATA> end MTB > tinterval c1 Confidence Intervals Variable N Mean StDev SE Mean 95.0 % C.I. C1 15 17.283 0.397 0.102 ( 17.064, 17.503) MTB > zinterval 0.397 c1 Confidence Intervals The assumed sigma = 0.397 Variable N Mean StDev SE Mean 95.0 % C.I. C1 15 17.283 0.397 0.103 ( 17.082, 17.484) 3-7 Using the Computer: Confidence Intervals

52. MTB > CDF 2.00; SUBC> t df=24. Cumulative Distribution Function Student's t distribution with 24 d.f. x P( X <= x) 2.0000 0.9715 MTB > INVCDF 0.95; SUBC> T df=24. Inverse Cumulative Distribution Function Student's t distribution with 24 d.f. P( X <= x) x 0.9500 1.7109 MTB > CDF 35; SUBC> CHISQUARE 25. Cumulative Distribution Function Chisquare with 25 d.f. x P( X <= x) 35.0000 0.9118 MTB > INVCDF 0.99; SUBC> CHISQUARE 20. Inverse Cumulative Distribution Function Chisquare with 20 d.f. P( X <= x) x 0.9900 37.5662 Using the Computer: t and Chi-Square Probabilities

53. The DESCRIPTIVE STATISTICS option in the DATA ANALYSIS TOOLKIT (found in the TOOLS menu if the toolkit has been added) was used to describe the data, which appears on the left of the screen (see the next slide). The value appearing next to “Confidence Level (95%)” is the half-width for the 95% confidence interval on the mean. Adding or subtracting this value from the sample mean gives the upper and lower limits of the confidence interval, respectively. Using the Computer: Analysis of Data in Fig. 6- 11 Using Excel