61 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

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Transcript of 61 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)
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61 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE) Slide 2 62 Chapter 6 Confidence Intervals Slide 3 63 Using Statistics Confidence Interval for the Population Mean When the Population Standard Deviation is Known Confidence Intervals for When is Unknown  The t Distribution LargeSample Confidence Intervals for the Population Proportion p Confidence Intervals for the Population Variance Sample Size Determination The Templates Confidence Intervals 6 Slide 4 64 Explain confidence intervals Compute confidence intervals for population means Compute confidence intervals for population proportions Compute confidence intervals for population variances Compute minimum sample sizes needed for an estimation Compute confidence intervals for special types of sampling methods Use templates for all confidence interval and sample size computations LEARNING OBJECTIVES 6 After studying this chapter you should be able to: Slide 5 65 Consider the following statements: x = 550 A singlevalued 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. 61 Using Statistics Slide 6 66 Point Estimate A singlevalued 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. Confidence Interval or Interval Estimate An interval or range of values believed to include the unknown population parameter. confidence 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 Slide 7 67 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 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 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 Slide 8 68 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). 432101234 0.4 0.3 0.2 0.1 0.0 z f ( z ) Standard Normal Distribution: 95% Interval 62 Confidence Interval for When Is Known Slide 9 69 62 Confidence Interval for when is Known (Continued) Slide 10 610 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. 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. 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 A 95% Interval around the Population Mean Slide 11 611 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.) not * 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.) 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.) not * 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 * * Slide 12 612 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. For example, if:n = 25 = 20 = 122 A 95% confidence interval: The 95% Confidence Interval for The 95% Confidence Interval for Slide 13 613 A (1 )100% Confidence Interval for A (1 )100% Confidence Interval for Slide 14 614 Critical Values of z and Levels of Confidence 54321012345 0.4 0.3 0.2 0.1 0.0 Z f ( z ) Standard Normal Distribution Slide 15 615 When sampling from the same population, using a fixed sample size, the higher the confidence level, the wider the confidence interval. 54321012345 0.4 0.3 0.2 0.1 0.0 Z f ( z ) Standard Normal Distribution 54321012345 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 Slide 16 616 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 Slide 17 617 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 $15,056.37. A random sample of 30 companies obtains a sample mean of $10,672.87. Give a 95% and 90% confidence interval for the average Revenues. Example 61 Slide 18 618 Example 61 (continued)  Using the Template Note: The remaining part of the template display is shown on the next slide. Slide 19 619 Example 61 (continued)  Using the Template (Sigma) Slide 20 620 Example 61 (continued)  Using the Template when the Sample Data is Known Slide 21 621 The t is a family of bellshaped 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/(df2). 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 The t is a family of bellshaped 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/(df2). 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 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. Standard normal t, df = 20 t, df = 10 63 Confidence Interval or Interval Estimate for When Is Unknown  The t Distribution Slide 22 622 The t Distribution Template Slide 23 623 A (1 )100% confidence interval for when is not known (assuming a normally distributed population): where is the value of the t distribution with n1 degrees of freedom that cuts off a tail area of to its right. A (1 )100% confidence interval for when is not known (assuming a normally distributed population): where is the value of the t distribution with n1 degrees of freedom that cuts off a tail area of to its right. 63 Confidence Intervals for when is Unknown The t Distribution Slide 24 624 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 n1 degrees of freedom. Note, however, that for large degrees of freedom, the t distribution is approximated well by the Z distribution. The t Distribution Slide 25 625 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 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 righttail area of 0.025 is: The corresponding confidence interval or interval estimate is: Example 62 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...... Slide 26 626 Whenever is not known (and the population is assumed normal), the correct distribution to use is the t distribution with n1 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 Slide 27 627 Example 63: Example 63: An economist wants to estimate the average amount in checking accounts at banks in a given region. A random sample of 100 accounts gives xbar = $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 Slide 28 628 64 LargeSample Confidence Intervals for the Population Proportion, p Slide 29 629 64 LargeSample Confidence Intervals for the Population Proportion, p Slide 30 630 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 foreignmade products; the rest are users of domestic products. Give a 95% confidence interval for the share of foreign products in this market. Thus, the firm may be 95% confident that foreign manufacturers control anywhere from 24.72% to 43.28% of the market. Thus, the firm may be 95% confident that foreign manufacturers control anywhere from 24.72% to 43.28% of the market. LargeSample Confidence Interval for the Population Proportion, p (Example 64) Slide 31 631 LargeSample Confidence Interval for the Population Proportion, p (Example 64) Using the Template Slide 32 632 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 Confidence Larger Sample Size Reducing the Width of Confidence Intervals  The Value of Information Slide 33 633 The sample variance, s 2, is an unbiased estimator of the population variance, 2. Confidence intervals for the population variance are based on the chisquare ( 2 ) distribution. The chisquare distribution is the probability distribution of the sum of several independent, squared standard normal random variables. The mean of the chisquare distribution is equal to the degrees of freedom parameter, (E[ 2 ] = df). The variance of a chisquare is equal to twice the number of degrees of freedom, (V[ 2 ] = 2df). 65 Confidence Intervals for the Population Variance: The ChiSquare ( 2 ) Distribution Slide 34 634 The chisquare random variable cannot be negative, so it is bound by zero on the left. The chisquare distribution is skewed to the right. The chisquare 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 ) ChiSquare Distribution: df=10, df=30, df=50 df = 10 df = 30 df = 50 The ChiSquare ( 2 ) Distribution Slide 35 635 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 ChiSquare Distributions Slide 36 636 Template with Values and Probabilities of ChiSquare Distributions Slide 37 637 A (1 )100% confidence interval for the population variance * (where the population is assumed normal) is: where is the value of the chisquare 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 chisquare distribution is skewed, the confidence interval for the population variance is not symmetric Confidence Interval for the Population Variance Slide 38 638 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 65 Slide 39 639 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 65 (continued) Slide 40 640 Example 65 Using the Template Using Data Slide 41 641 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? 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 66 SampleSize Determination Slide 42 642 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 Slide 43 643 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 Slide 44 644 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? 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? SampleSize Determination: Example 66 Slide 45 645 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? SampleSize for Proportion: Example 67 Slide 46 646 67 The Templates Optimizing Population Mean Estimates Slide 47 647 67 The Templates Optimizing Population Proportion Estimates