Example Tartus Industries has seven production employees (considered the population). The hourly earnings of each employee are given in Table 8-2. TABLE 8-2 Hourly Earnings of the Production Employees of Tartus Industries Employee Hourly Earnings Employee Hourly Earnings Joe Jan Sam Art Sue 00 00 -4 Ted Bob 1. What is the population mean? 2. What is the sampling distribution of the sample mean for samples of size 2? 3. What is the mean of the sampling distribution? 4. What observations can be made about the population and the sampling distribution? Solution Here are the solutions to the questions. 1. The population mean is $7.71, found by: EX _$7 + $7 + $8 + $8 + $7 + $8 + $9 - - $7.71 N We identify the population mean with the Greek letter p. Our policy, stated in Chapters 1, 3, and 4, is to identify population parameters with Greek letters. 2. To arrive at the sampling distribution of the sample mean, we need to select all possible samples of 2 without replacement from the population, then compute the mean of each sample. There are 21 possible samples, found by using for- mula (5-10) in Section 5.8 on page 174. 7! Non - 2!(7 - 2)! 21 n!(W - n)! where / - 7 is the number of items in the population and n - 2 is the num- ber of items in the sample. The 21 sample means from all possible samples of 2 that can be drawn from the population are shown in Table 8-3. These 21 sample means are used to construct a probability distribution. This is the sampling distribution of the sample mean, and it is summarized in Table 8-4. TABLE 8-3 Sample Means for All Possible Samples of 2 Employees Hourly Hourty Sample Employees Earnings Sum Mean Sample Employees Earnings Sum Mean 1 Joe, Sam $7. 57 $14 $7.00 12 SUB, Bob $8, $8 $16 $8.OC Joe, Sue 7 8 15 7.50 13 Sue, Jan 8, 7 15 7.50 Joe, Bob 7, 8 15 7.50 14 SUB, Art 16 Joe, Jan 7, 7 14 7.00 15 Sun, Ted 8 9 17 8.50 Joe, Art 7, 8 15 7.50 16 Bob, Jan 8. 7 15 7.50 Joe, Ted 7, 9 16 8.00 17 Bob, Art 8 8 16 Sam, Sue 7, 8 15 7.50 18 Bob, Ted 8 9 17 8.50 Sam, Bob 7. 8 15 7.50 19 Jan, Art 7. 8 15 7.50 Sam, Jan 7, 7 14 7.00 20 Jan, Ted 7, 9 16 8.00 Sam, Art 7, B 15 7.50 21 Art, Ted 17 8.50 11 Sam, Ted 7. 9 16 8.00TABLE 8-4 Sampling Distribution of the Sample Mean for n - 2 Sample Mean Number of Means Probability 7100 1429 7.50 4285 8.00 2857 8.50 -1429 1200 00 3. The mean of the sampling distribution of the sample mean is obtained by sum- ming the various sample means and dividing the sum by the number of sam- ples. The mean of all the sample means is usually written up. The it reminds us that it is a population value because we have considered all possible samples. The subscript X indicates that it is the sampling distribution of the sample mean. Sum of all sample means $7.00 + $7.50 + ... + $8.50 Population mean is equal to the mean of Total number of samples 21 the sample means $162 - $7.71 21 4. Refer to Chart 8-1, which shows both the population distribution and the distri- bution of the sample mean. These observations can be made: a. The mean of the distribution of the sample mean ($7.71) is equal to the mean of the population: A - HI- b. The spread in the distribution of the sample mean is less than the spread in the population values. The sample mean ranges from $7.00 to $8.50, while the population values vary from $7.00 up to $9.00. Notice, as we increase the size of the sample, the spread of the distribution of the sample mean becomes smaller. c. The shape of the sampling distribution of the sample mean and the shape of the frequency distribution of the population values are different. The distri- bution of the sample mean tends to be more bell-shaped and to approximate the normal probability distribution. Population distribution Distribution of sample mean Probabilty Probabilty T 10 9 Hourly earnings 7 7.5 8 8.5 9 X Sample mean of hourly earnings CHART 8-1 Distributions of Population Values and Sample MeanFINDING THE Z VALUE OF X WHEN THE X POPULATION STANDARD DEVIATION IS KNOWN [8-2] 0/ Vn Example The Quality Assurance Department for Cola Inc. maintains records regarding the amount of cola in its Jumbo bottle. The actual amount of cola in each bottle is crit- ical, but varies a small amount from one bottle to the next. Cola Inc. does not wish to underfill the bottles, because it will have a problem with truth in labeling. On the other hand, it cannot overfill each bottle, because it would be giving cola away, hence reducing its profits. Its records indicate that the amount of cola follows the normal probability distribution. The mean amount per bottle is 31.2 ounces and the population standard deviation is 0.4 ounces. At 8 A.M. today the quality technician randomly selected 16 bottles from the filling line. The mean amount of cola con- tained in the bottles is 31.38 ounces. Is this an unlikely result? Is it likely the process is putting too much soda in the bottles? To put it another way, is the sampling error of 0.18 ounces unusual? Solution We can use the results of the previous section to find the likelihood that we could select a sample of 16 (n) bottles from a nomal population with a mean of 31.2 (1) ounces and a population standard deviation of 0.4 (o) ounces and find the sample mean to be 31.38(X). We use formula (8-2) to find the value of z. X - 1 31.38 - 31.20 0.4/V 16 - 1.80 The numerator of this equation, X - p - 31.38 - 31.20 - .18, is the sampling error. The denominator, @/Vn - 0.4/V16 - 0.1, is the standard error of the sam- pling distribution of the sample mean. So the z values express the sampling error in standard units-in other words, the standard error. Next, we compute the likelihood of a z value greater than 1.80. In Appendix B.1, locate the probability corresponding to a z value of 1.80. It is .4641. The likelihood of a z value greater than 1.80 is .0350, found by .5000 - .4641. What do we conclude? It is unlikely, less than a 4 percent chance, we could select a sample of 16 observations from a normal population with a mean of 31.2 ounces and a population standard deviation of 0.4 ounces and find the sample mean equal to or greater than 31.38 ounces. We conclude the process is putting too much cola in the bottles. The quality technician should see the production supervisor about reducing the amount of soda in each bottle. This information is summarized in Chart 8-6. 1359 4641 31.20 31.38 Ounces (X] 1.80 Z ValUB CHART 8-6 Sampling Distribution of the Mean Amount of Cola in a Jumbo BottleRefer to CIA data which reports demographic and economic information of 62 countries. Answer the following questions: 7. Select a random sample of 10 countries. For this sample calculate the mean GDP/capita. Repeat this sampling and calculation process five more times. Then find the mean and standard deviation of your six sample means. How do this mean and standard deviation compare with the mean and standard deviation of the original "population" of 62 countries? b. Suppose the population distribution is normal. For the first sample mean you computed, determine the likelihood of finding a sample mean of this large or larger from the population. Hint: To have an idea on how to answer question 7a-refer to Chapter 8, pages 276-277 example on Tartus Industries. To have an idea on how to answer question 7b-refer to Chapter 8, page 287 (formula 8-2) and page 288 (example on Quality Assurance) 8. Develop a 90 percent confidence interval for the mean percent of the population age 65 years and older. 9. Develop a 95 percent confidence interval for the mean exports. Interpret your answer.X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 Country Union SA G-20 Petroleu Area Pop 65kove Life Literacy GOP/c Labor Unemploys Imports Cell m (1000's) expectanc rate ap force yment phones Algeria 2 2E+06 31736 4.07 69.95 61.6 5.5 9.1 30 19.6 9.2 0.0034 Argentina 1 3E+06 37385 10.42 75.26 96.2 12.9 15 15 26.5 25.2 3 Australia BE+06 19357 12.5 79.87 100 23.2 9.5 6.4 69 77 5.4 0 0 - - Austria 0 83858 8150 15.38 77.84 98 25 3.7 5.4 63.2 65.6 4.5 Belgium 30510 10259 16.95 77.96 98 25.3 4.34 8.4 181.4 166 Bolivia 1098 3858 4.5 65.5 87.2 2.8 4.2 8 2.37 1.84 1.401 Brazil - -0 9E+06 174469 5.45 63.24 83.3 6.5 79 7.1 55.1 55.8 4.4 - - 6 0 0 0 0 0 0 - 60 0 0 Canada 1E+07 31592 12.77 79.56 97 24.8 16.1 6.8 272.3 238.2 4.2 Chile 757 15980 8 76.58 96.2 11.3 3.6 7.4 38 30.1 6.45 China 1E+07 E+06 7.11 71.62 81.5 3.6 700 10 232 197 65 Columbia 1139 42854 5.1 71.72 92.5 7.1 20.5 11.8 23.1 20.4 6.189 Cyprus 9 780 11.4 77.7 37.6 21.6 0.37 3.5 1.24 5.55 0.418 Czech Republic 79 10264 13.92 74.73 99.9 12.9 5.2 8.7 28.3 31.4 4.3 Denmark 43094 5352 14.85 76.72 100 25.5 2.9 5.3 50.8 43.6 1.4 Equador 0 283 13363 4.9 76.2 92.5 3.9 4.6 11.2 9.22 8.44 2.394 Estonia 45226 1332 16.8 71.77 99.8 16.4 0.67 9.2 7.44 9.19 0.881 Finland 0 337030 5175 15.03 77.58 100 22.9 2.6 9.8 44.4 32.7 2.2 France 0 547030 59551 16.13 78.9 99 24.4 25 9.7 325 320 11.1 French Guyana - 0 - 91 195 6.1 77 83 8.3 0.059 22 0.155 0.625 0.138 Germany 0 357021 83029 16.61 77.61 99 23.4 40.5 9.9 578 505 15.3 Greece 1 131940 10623 17.72 78.59 95 17.2 4.32 11.3 5.8 33.9 0.937 Guyana 0 215 765 5.1 65.5 98.8 3.9 0.418 9.1 0.587 0.682 0.087 Hungary 93030 10106 14.71 71.63 99 11.2 4.2 9.4 25.2 27.6 1.3 - -060 0 Iceland 0 103000 278 11.81 79.52 100 24.8 0.16 2.7 2 2.2 0.066 India 0 0 - - 0 0 0 0 0 6 0 -6 06 1-+ 3E+06 1E+06 4.68 62.88 52 2.2 43.1 60.8 2.93 Indonesia 2E+06 228437 4.63 68.27 83.8 2.9 99 17.5 64.7 40.4 Iran 2E+06 66129 4.65 69.95 72.1 6.3 17.3 14 25 15 0.265 Iraq 2 437072 23332 3.08 66.95 58 2.5 4.4 21.8 13.8 Ireland 70280 3840 11.35 76.99 98 21.6 1.82 4.1 73.5 45.7 2 Italy 0 301230 57680 18.35 79.14 98 22.1 23.4 10.4 241.1 231.4 20.5 Japan 0 377835 126771 17.35 80.8 88 24.9 67.7 4.7 450 355 63.9 Kuwait 0 17820 2041 2.42 76.27 78.6 15 1.3 1.8 23.2 7.6 0.21Kuwait 17820 2041 2.42 76.27 78.6 15 1.3 1.8 23.2 7.6 Latvia 0.21 64589 2290 16.1 71.05 99.8 12.8 11.11 8.8 5.75 ONOR 8.56 Libya 1.219 2E+06 5240 3.95 75.65 76.2 8.9 1.5 30 13.9 7.6 Lithuania 65200 3597 15.2 73.97 99.6 13.7 1.61 5.3 10.95 13.33 2.17 Luxembourg 2586 443 14.06 77.3 100 36.4 0.248 2.7 7.6 10 Mexico 0.215 2E+06 101879 4.4 71.76 89.6 9.1 39.8 2.2 168 176 Netherlands 2 41526 15981 13.72 78.43 99 24.4 7.2 2.6 210.3 New Zealand 201.2 4.1 D 286680 3864 11.53 77.99 99 17.7 1.88 6.3 Nigeria 14.6 14.3 0.6 2 923768 126635 2.82 51.07 57.1 0.95 66 28 22.2 10.7 Norway 0.027 1 324220 4503 15.1 78.79 100 27.7 2.4 3 59.2 35.2 Paraguay 0 0 - - 6 6 6 6 0 6 0 - 2 0 407 6348 4.8 74.9 94 4.9 2.7 16 3.13 3.83 Peru 1.77 1285 27926 5.2 69.3 87.7 6 9.1 8.7 15.95 Poland 12.15 2.908 0 312685 38634 12.44 73.42 99 8.5 17.2 12 28.4 42.7 Portugal 1.8 0 92391 10066 15.62 75.94 87.4 15.8 5 4.3 26.1 41 Qatar 3 11437 769 2.48 72.62 79 20.3 0.233 9.8 Russia 3.8 0.043 - -0 0 0 0 0 0 -0 0 0 0 0 0 0 0 0 0 -260 0 060 0 0 6 0 0 0 - - 0 - 0 0 0 0 -6 0 - 60 0 0 0 0 0 0 060 0 0 0 0 2E+07 145470 12.81 67.34 98 7.7 66 10.5 105.1 Saudi Arabia 44.2 2.5 2 2E+06 22757 2.68 68.09 62.8 10.5 7 81.2 30.1 Slovakia 48845 5431 11.9 74.5 99.6 15.7 2.62 11.5 32.39 Slovenia 24.48 3.68 0 20273 2011 15.4 76.14 99.7 20.9 0.92 9.8 18.53 South Africa 19.62 1.74 18+06 43586 4.88 48.09 81.1 8.5 17 30 30.8 South Korea 27.6 2 0 98480 47904 7.27 74.65 98 16.1 22 4.1 172.6 160.5 Spain 27 0 504782 40038 17.18 78.93 97 18 17 14 120.5 153.9 8.4 Suriname 163 438 6.2 69 88 4.7 0.104 17 0.881 0.75 Sweden 0.168 0 449964 8875 17.28 79.71 99 22.2 4.4 6 95.5 Switzerland 0 0 0 0 0 0 0 - 0 - 6 0 - - 80 3.8 0 41290 7283 15.3 79.73 99 28.6 3.9 1.9 91.3 91.6 Turkey 0 780580 66494 6.13 71.24 85 6.8 23 5.6 26.9 55.7 United Arab Emi 12.1 2 82880 2407 2.4 74.29 79.2 22.8 1.4 46 United Kingdom 34 1 1 244880 59648 15.7 77.82 99 22.8 29.2 5.5 282 324 United States 13 18+07 278059 12.61 77.26 97 36.2 140.9 4 776 1223 39 Uruguay O 176 3416 13.2 76.13 98 10 1.5 12 Venuzuela 3.55 3.54 0.652 2 912050 23917 4.72 73.31 91.1 6.2 9.9 14 32.8 14.7 2