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Cell phone case

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Ankit Joshi
Ankit Joshi

This document discusses how a bank can use a random sample of 100 employees on a 500-minute cell phone plan to estimate the bank's overall cellular costs and determine if hiring a cellular management service could reduce costs. The bank has over 10,000 employees on various plans. A random sample of 100 employees is selected using random numbers from a table or software. The cellular usage of the 100 employees is recorded. This data will be used in the next chapter to estimate the bank's cellular cost per minute and assess if costs could be reduced by 12 cents per minute or more through automated management of calling plans.

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The Cell Phone Case: Estimating Cell Phone Costs Businesses and college students have at least two things in common！both find cellular phones to be nearly indispensable because of their convenience and mobility, and both often rack up unpleasantly high cell phone bills. Students¨ high bills are usually the result of overage！a student uses more minutes than his or her plan allows. Businesses also lose money due to overage, and, in addition, lose money due to underage when some employees do not use all of the (already paid-for) minutes allowed by their plans. Because cellular carriers offer more than 10,000 rate plans, it is nearly impossible for a business to intelligently choose calling plans that will meet its needs at a reasonable cost. Rising cell phone costs have forced companies having large numbers of cellular users to hire services to manage their cellular and other wireless resources. These cellular management services use sophisticated software and mathematical models to choose cost efficient cell phone plans for their clients. One such firm, MobileSense Inc. ofWestlake Village, California, specializes in automated wireless cost management. According to Doug L. Stevens, Vice President of Sales and Marketing at MobileSense, cell phone carriers count on overage and underage to deliver almost half of their revenues. As a result, a company¨s typical cost of cell phone use can easily exceed 25 cents per minute. However, Mr. Stevens explains that by using MobileSense automated cost management to select calling plans, this cost can be reduced to 12 cents per minute or less. In this case we will demonstrate how a bank can use a random sample of cell phone users to study its cellular phone costs. Based on this cost information, the bank will decide whether to hire a cellular management service to choose calling plans for the bank¨s employees. While the bank has over 10,000 employees on a variety of calling plans, the cellular management service suggests that by studying the calling patterns of cellular users on 500-minute plans, the bank can accurately assess whether its cell phone costs can be substantially reduced. The bank has 2,136 employees on a 500-minute-per-month plan with a monthly cost of $50. The overage charge is 40 cents per minute, and there are additional charges for long distance and roaming. The bank will estimate its cellular cost per minute for this plan by examining the number of minutes used last month by each of 100 randomly selected employees on this 500-minute plan. According to the cellular management service, if the cellular cost per minute for the random sample of 100 employees is over 18 cents per minute, the bank should benefit from automated cellular
TA B L E 1.1 Random Numbers (b) Software output of 100 different, four-digit random numbers between 1 and 2136 705 1131 169 1703 1709 609 1990 766 1286 1977 222 43 1007 1902 1209 2091 1742 1152 111 69 2049 1448 659 338 1732 1650 7 388 613 1477 838 272 1227 154 18 320 1053 1466 2087 265 2107 1992 582 1787 2098 1581 397 1099 757 1699 567 1255 1959 407 354 1567 1533 1097 1299 277 663 40 585 1486 1021 532 1629 182 372 1144 1569 1981 1332 1500 743 1262 1759 955 1832 378 728 1102 667 1885 514 1128 1046 116 1160 1333 831 2036 918 1535 660 928 1257 1468 503 468 management of its calling plans. In order to randomly select the sample of 100 cell phone users, the bank will make a numbered list of the 2,136 users on the 500-minute plan. This list is called a frame. The bank can then use a random number table, such as Table 1.1(a), to select the needed sample. To see how this is done, notice that any single-digit number in the table is assumed to have been randomly selected from the digits 0 to 9. Any two-digit number in the table is assumed to have been randomly selected from the numbers 00 to 99. Any three-digit number is assumed to have been randomly selected from the numbers 000 to 999, and so forth. Note that the table entries are segmented into groups of five to make the table easier to read. Because the total number of cell phone users on the 500-minute plan (2,136) is a four-digit number, we arbitrarily select any set of four digits in the table (we have circled these digits). This number, which is 0511, identifies the first randomly selected user. Then, moving in any direction from the 0511 (up, down, right, or left！it does not matter which), we select additional sets of four digits. These succeeding sets of digits identify additional randomly selected users. Here we arbitrarily move down from 0511 in the table. The first seven sets of four digits we obtain are 0511 7156 0285 4461 3990 4919 1915
(See Table 1.1(a)！these numbers are enclosed in a rectangle.) Since there are no users numbered 7156, 4461, 3990, or 4919 (remember only 2,136 users are on the 500-minute plan), we ignore these numbers. This implies that the first three randomly selected users are those numbered 0511, 0285, and 1915. Continuing this procedure, we can obtain the entire random sample of 100 users. Notice that, because we are sampling without replacement, we should ignore any set of four digits previously selected from the random number table. While using a random number table is one way to select a random sample, this approach has a disadvantage that is illustrated by the current situation. Specifically, since most four-digit random numbers are not between 0001 and 2136, obtaining 100 different, four-digit random numbers between 0001 and 2136 will require ignoring a large number of random numbers in the random number table, and we will in fact need to use a random number table that is larger than Table 1.1(a). Although larger random number tables are readily available in books of mathematical and statistical tables, a good alternative is to use a computer software package, which can generate random numbers that are between whatever values we specify. For example, Table 1.1(b) gives the MINITAB output of 100 different, four-digit random numbers that are between 0001 and 2136 (note that the ＾leading 0¨s￣ are not included in these four digit numbers). If used, the random numbers in Table 1.1(b) identify the 100 employees that should form the random sample. After the random sample of 100 employees is selected, the number of cellular minutes used by each employee during the month (the employee¨s cellular usage) is found and recorded. The 100 cellular-usage figures are given in Table 1.2. Looking at this table, we can see that there is substantial overage and underage！many employees used far more than 500 minutes, while many others failed to use all of the 500 minutes allowed by their plan. In Chapter 2 we will use these 100 usage figures to estimate the cellular cost per minute for the 500-minute plan. TA B L E 1.2 A Sample of Cellular Usages (in minutes) for 100 Randomly Selected Employees 75 485 37 547 753 93 897 694 797 477 654 578 504 670 490 225 509 247 597 173 496 553 0 198 507 157 672 296 774 479 0 822 705 814 20 513 546 801 721 273 879 433 420 521 648 41 528 359 367 948 511 704 535 585 341 530 216 512 491 0 542 562 49 505 461 496 241 624 885 259 571 338 503 529 737 444 372 555 290 830 719 120 468 730 853 18 479 144 24 513 482 683 212 418 399 376 323 173 669 611 Approximately random samples In general, to take a random sample we must have a list, or frame, of all the population units. This is needed because we must be able to number the population units in order to make random selections from them (by, for example, using a random number table). In Example 1.1, where we wished to study a population of 2,136 cell phone users who were on the bank¨s 500-minute cellular plan, we were able to produce a frame (list) of the population units. Therefore, we were able to select a random sample. Sometimes, however, it is not possible to list and thus number all the units in a population. In such a situation we often select a systematic sample, which approximates a random sample.

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Cell phone case

  • 1. The Cell Phone Case: Estimating Cell Phone Costs Businesses and college students have at least two things in common！both find cellular phones to be nearly indispensable because of their convenience and mobility, and both often rack up unpleasantly high cell phone bills. Students¨ high bills are usually the result of overage！a student uses more minutes than his or her plan allows. Businesses also lose money due to overage, and, in addition, lose money due to underage when some employees do not use all of the (already paid-for) minutes allowed by their plans. Because cellular carriers offer more than 10,000 rate plans, it is nearly impossible for a business to intelligently choose calling plans that will meet its needs at a reasonable cost. Rising cell phone costs have forced companies having large numbers of cellular users to hire services to manage their cellular and other wireless resources. These cellular management services use sophisticated software and mathematical models to choose cost efficient cell phone plans for their clients. One such firm, MobileSense Inc. ofWestlake Village, California, specializes in automated wireless cost management. According to Doug L. Stevens, Vice President of Sales and Marketing at MobileSense, cell phone carriers count on overage and underage to deliver almost half of their revenues. As a result, a company¨s typical cost of cell phone use can easily exceed 25 cents per minute. However, Mr. Stevens explains that by using MobileSense automated cost management to select calling plans, this cost can be reduced to 12 cents per minute or less. In this case we will demonstrate how a bank can use a random sample of cell phone users to study its cellular phone costs. Based on this cost information, the bank will decide whether to hire a cellular management service to choose calling plans for the bank¨s employees. While the bank has over 10,000 employees on a variety of calling plans, the cellular management service suggests that by studying the calling patterns of cellular users on 500-minute plans, the bank can accurately assess whether its cell phone costs can be substantially reduced. The bank has 2,136 employees on a 500-minute-per-month plan with a monthly cost of $50. The overage charge is 40 cents per minute, and there are additional charges for long distance and roaming. The bank will estimate its cellular cost per minute for this plan by examining the number of minutes used last month by each of 100 randomly selected employees on this 500-minute plan. According to the cellular management service, if the cellular cost per minute for the random sample of 100 employees is over 18 cents per minute, the bank should benefit from automated cellular
  • 2. TA B L E 1.1 Random Numbers (b) Software output of 100 different, four-digit random numbers between 1 and 2136 705 1131 169 1703 1709 609 1990 766 1286 1977 222 43 1007 1902 1209 2091 1742 1152 111 69 2049 1448 659 338 1732 1650 7 388 613 1477 838 272 1227 154 18 320 1053 1466 2087 265 2107 1992 582 1787 2098 1581 397 1099 757 1699 567 1255 1959 407 354 1567 1533 1097 1299 277 663 40 585 1486 1021 532 1629 182 372 1144 1569 1981 1332 1500 743 1262 1759 955 1832 378 728 1102 667 1885 514 1128 1046 116 1160 1333 831 2036 918 1535 660 928 1257 1468 503 468 management of its calling plans. In order to randomly select the sample of 100 cell phone users, the bank will make a numbered list of the 2,136 users on the 500-minute plan. This list is called a frame. The bank can then use a random number table, such as Table 1.1(a), to select the needed sample. To see how this is done, notice that any single-digit number in the table is assumed to have been randomly selected from the digits 0 to 9. Any two-digit number in the table is assumed to have been randomly selected from the numbers 00 to 99. Any three-digit number is assumed to have been randomly selected from the numbers 000 to 999, and so forth. Note that the table entries are segmented into groups of five to make the table easier to read. Because the total number of cell phone users on the 500-minute plan (2,136) is a four-digit number, we arbitrarily select any set of four digits in the table (we have circled these digits). This number, which is 0511, identifies the first randomly selected user. Then, moving in any direction from the 0511 (up, down, right, or left！it does not matter which), we select additional sets of four digits. These succeeding sets of digits identify additional randomly selected users. Here we arbitrarily move down from 0511 in the table. The first seven sets of four digits we obtain are 0511 7156 0285 4461 3990 4919 1915
  • 3. (See Table 1.1(a)！these numbers are enclosed in a rectangle.) Since there are no users numbered 7156, 4461, 3990, or 4919 (remember only 2,136 users are on the 500-minute plan), we ignore these numbers. This implies that the first three randomly selected users are those numbered 0511, 0285, and 1915. Continuing this procedure, we can obtain the entire random sample of 100 users. Notice that, because we are sampling without replacement, we should ignore any set of four digits previously selected from the random number table. While using a random number table is one way to select a random sample, this approach has a disadvantage that is illustrated by the current situation. Specifically, since most four-digit random numbers are not between 0001 and 2136, obtaining 100 different, four-digit random numbers between 0001 and 2136 will require ignoring a large number of random numbers in the random number table, and we will in fact need to use a random number table that is larger than Table 1.1(a). Although larger random number tables are readily available in books of mathematical and statistical tables, a good alternative is to use a computer software package, which can generate random numbers that are between whatever values we specify. For example, Table 1.1(b) gives the MINITAB output of 100 different, four-digit random numbers that are between 0001 and 2136 (note that the ＾leading 0¨s￣ are not included in these four digit numbers). If used, the random numbers in Table 1.1(b) identify the 100 employees that should form the random sample. After the random sample of 100 employees is selected, the number of cellular minutes used by each employee during the month (the employee¨s cellular usage) is found and recorded. The 100 cellular-usage figures are given in Table 1.2. Looking at this table, we can see that there is substantial overage and underage！many employees used far more than 500 minutes, while many others failed to use all of the 500 minutes allowed by their plan. In Chapter 2 we will use these 100 usage figures to estimate the cellular cost per minute for the 500-minute plan. TA B L E 1.2 A Sample of Cellular Usages (in minutes) for 100 Randomly Selected Employees 75 485 37 547 753 93 897 694 797 477 654 578 504 670 490 225 509 247 597 173 496 553 0 198 507 157 672 296 774 479 0 822 705 814 20 513 546 801 721 273 879 433 420 521 648 41 528 359 367 948 511 704 535 585 341 530 216 512 491 0 542 562 49 505 461 496 241 624 885 259 571 338 503 529 737 444 372 555 290 830 719 120 468 730 853 18 479 144 24 513 482 683 212 418 399 376 323 173 669 611 Approximately random samples In general, to take a random sample we must have a list, or frame, of all the population units. This is needed because we must be able to number the population units in order to make random selections from them (by, for example, using a random number table). In Example 1.1, where we wished to study a population of 2,136 cell phone users who were on the bank¨s 500-minute cellular plan, we were able to produce a frame (list) of the population units. Therefore, we were able to select a random sample. Sometimes, however, it is not possible to list and thus number all the units in a population. In such a situation we often select a systematic sample, which approximates a random sample.
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