1 / 34

Probability (Part 2)

5B. Chapter. Probability (Part 2). Contingency Tables Counting Rules. McGraw-Hill/Irwin. © 2008 The McGraw-Hill Companies, Inc. All rights reserved. Variable 1. Col 1 Col 2 Col 3. Row 1 Row 2 Row 3 Row 4. Variable 2. Contingency Tables. What is a Contingency Table?.

london
Download Presentation

Probability (Part 2)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 5B Chapter Probability (Part 2) Contingency Tables Counting Rules McGraw-Hill/Irwin © 2008 The McGraw-Hill Companies, Inc. All rights reserved.

  2. Variable 1 Col 1 Col 2 Col 3 Row 1 Row 2 Row 3 Row 4 Variable 2 Contingency Tables • What is a Contingency Table? • A contingency table is a cross-tabulation of frequencies into rows and columns. Cell • A contingency table is like a frequency distribution for two variables.

  3. Contingency Tables • Example: Salary Gains and MBA Tuition • Consider the following cross-tabulation table for n = 67 top-tier MBA programs:

  4. The frequencies indicate that MBA graduates of high-tuition schools do tend to have large salary gains. Contingency Tables • Example: Salary Gains and MBA Tuition • Are large salary gains more likely to accrue to graduates of high-tuition MBA programs? • Also, most of the top-tier schools charge high tuition. • More precise interpretations of this data can be made using the concepts of probability.

  5. For example, find the marginal probability of a medium salary gain (P(S2)). Contingency Tables • Marginal Probabilities • The marginal probability of a single event is found by dividing a row or column total by the total sample size. P(S2) = 33/67 = .4925 • Conclude that about 49% of salary gains at the top-tier schools were between $50,000 and $100,000 (medium gain).

  6. Contingency Tables • Marginal Probabilities • Find the marginal probability of a low tuition P(T1). .2388 16/67 = P(T1) = • There is a 24% chance that a top-tier school’s MBA tuition is under $40.000.

  7. Contingency Tables • Joint Probabilities • A joint probability represents the intersection of two events in a cross-tabulation table. • Consider the joint event that the school has low tuition and large salary gains (denoted as P(T1S3)).

  8. Contingency Tables • Joint Probabilities • So, using the cross-tabulation table, P(T1S3) = 1/67 = .0149 • There is less than a 2% chance that a top-tier school has both low tuition and large salary gains.

  9. Contingency Tables • Conditional Probabilities • Found by restricting ourselves to a single row or column (the condition). • For example, knowing that a school’s MBA tuition is high (T3), we would restrict ourselves to the third row of the table.

  10. Contingency Tables • Conditional Probabilities • Find the probability that the salary gains are small (S1) given that the MBA tuition is large (T3). .1563 5/32 = P(S1|T3) = • What does this mean?

  11. Contingency Tables • Independence • To check for independent events in a contingency table, compare the conditional to the marginal probabilities. • For example, if large salary gains (S3) were independent of low tuition (T1), then P(S3 | T1) = P(S3). • What do you conclude about events S3 and T1?

  12. Contingency Tables • Relative Frequencies • Calculate the relative frequencies below for each cell of the cross-tabulation table to facilitate probability calculations. • Symbolic notation for relative frequencies:

  13. Contingency Tables • Relative Frequencies • Here are the resulting probabilities (relative frequencies). For example, P(T1 and S1) = 5/67 P(T2 and S2) = 11/67 P(T3 and S3) = 15/67 P(S1) = 17/67 P(T2) = 19/67

  14. Contingency Tables • Relative Frequencies • The nine joint probabilities sum to 1.0000 since these are all the possible intersections. • Summing the across a row or down a column gives marginal probabilities for the respective row or column.

  15. Contingency Tables • Example: Payment Method and Purchase Quantity • A small grocery store would like to know if the number of items purchased by a customer is independent of the type of payment method the customer chooses to use. • Why would this information be useful to the store manager? • The manager collected a random sample of 368 customer transactions.

  16. Contingency Tables • Example: Payment Method and Purchase Quantity • Here is the contingency table of frequencies:

  17. Calculate the marginal probability that a customer will use cash to make the payment. Contingency Tables • Example: Payment Method and Purchase Quantity • Let C be the event cash. P(C) = 126/368 = .3424 • Now, is this probability the same if we condition on number of items purchased?

  18. Contingency Tables • Example: Payment Method and Purchase Quantity P(C | 1-5) = 30/88 = .3409 P(C | 6-10) = 46/135 = .3407 P(C | 10-20) = 31/89 = .3483 P(C | 20+) = 19/56 = .3393 • P(C) = .3424, so what do you conclude about independence? • Based on this, the manager might decide to offer a cash-only lane that is not restricted to the number of items purchased.

  19. Contingency Tables • How Do We Get a Contingency Table? • Contingency tables require careful organization and are created from raw data. • Consider the data of salary gain and tuition for n = 67 top-tier MBA schools.

  20. Once coded, tabulate the frequency in each cell of the contingency table using MINITAB’s Stat | Tables | Cross Tabulation Contingency Tables • How Do We Get a Contingency Table? • The data should be coded so that the values can be placed into the contingency table.

  21. Counting Rules • Fundamental Rule of Counting • If event A can occur in n1 ways and event B can occur in n2 ways, then events A and B can occur in n1 x n2 ways. • In general, m events can occurn1 x n2 x … x nm ways.

  22. Counting Rules • Example: Stock-Keeping Labels • How many unique stock-keeping unit (SKU) labels can a hardware store create by using 2 letters (ranging from AA to ZZ) followed by four numbers (0 through 9)? • For example, AF1078: hex-head 6 cm bolts – box of 12RT4855: Lime-A-Way cleaner – 16 ounceLL3319: Rust-Oleum primer – gray 15 ounce

  23. Counting Rules • Example: Stock-Keeping Labels • View the problem as filling six empty boxes: • There are 26 ways to fill either the 1st or 2nd box and 10 ways to fill the 3rd through 6th. • Therefore, there are 26 x 26 x 10 x 10 x 10 x 10 = 6,760,000 unique inventory labels.

  24. Counting Rules • Example: Shirt Inventory • L.L. Bean men’s cotton chambray shirt comes in 6 colors (blue, stone, rust, green, plum, indigo), 5 sizes (S, M, L, XL, XXL) and two styles (short and long sleeves). • Their stock might include 6 x 5 x 2 = 60 possible shirts. • However, the number of each type of shirt to be stocked depends on prior demand.

  25. Counting Rules • Factorials • The number of ways that n items can be arranged in a particular order is nfactorial. • n factorial is the product of all integers from 1 to n. n! = n(n–1)(n–2)...1 • Factorials are useful for counting the possible arrangements of any n items. • There are n ways to choose the first, n-1 ways to choose the second, and so on.

  26. Counting Rules • Factorials • As illustrated below, there are n ways to choose the first item, n-1 ways to choose the second, n-2 ways to choose the third and so on.

  27. Counting Rules • Factorials • A home appliance service truck must make 3 stops (A, B, C). • In how many ways could the three stops be arranged? 3! = 3 x 2 x 1 = 6 • List all the possible arrangements: {ABC, ACB, BAC, BCA, CAB, CBA} • How many ways can you arrange 9 baseball players in batting order rotation? 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880

  28. Counting Rules • Permutations • A permutation is an arrangement in a particular order of r randomly sampled items from a group of n items and is denoted by nPr • In other words, how many ways can the r items be arranged, treating each arrangement as different (i.e., XYZ is different from ZYX)?

  29. Counting Rules • Example: Appliance Service Cans • n = 5 home appliance customers (A, B, C, D, E) need service calls, but the field technician can service only r = 3 of them before noon. • The order is important so each possible arrangement of the three service calls is different. • The number of possible permutations is:

  30. ABC, ACB, BAC, BCA, CAB, CBA Counting Rules • Example: Appliance Service Cans • The 60 permutations with r = 3 out of the n = 5 calls can be enumerated. • There are 10 distinct groups of 3 customers: • Each of these can be arranged in 6 distinct ways: ABCABDABEACD ACE ADEBCDBCEBDE CDE • Since there are 10 groups of 3 customers and 6 arrange-ments per group, there are 10 x 6 = 60 permutations.

  31. Counting Rules • Combinations • A combination is an arrangement of r items chosen at random from n items where the order of the selected items is not important (i.e., XYZ is the same as ZYX). • A combination is denoted nCr

  32. Counting Rules • Example: Appliance Service Calls Revisited • n = 5 home appliance customers (A, B, C, D, E) need service calls, but the field technician can service only r = 3 of them before noon. • This time order is not important. • Thus, ABC, ACB, BAC, BCA, CAB, CBA would all be considered the same event because they contain the same 3 customers. • The number of possible combinations is:

  33. Counting Rules • Example: Appliance Service Calls Revisited • 10 combinations is much smaller than the 60 permutations in the previous example. • The combinations are easily enumerated: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE

More Related