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# The Binomial Distribution

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Introduction to Bernoulli Trials, Binomial Distribution and Binomial Test in SPSS

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### The Binomial Distribution

1. 1. The Binomial Distribution: Objectives • To Introduce the notion of a ‘Bernoulli Trial’ • To introduce the Binomial Probability Distribution as the situation when a finite number of Bernoulli Trials is conducted • To recognise problems suitable for Binomial Probability modeling and calculate Binomial probabilities using a formula • To understand and apply the Binomial Distribution in hypothesis testing: Binomial Test 1 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Try this Have you got a coin? Toss it six times in a roll, each time counting the number of times the result, ‘heads’ is heads’ observed. 2 The Binomial Distribution. Max Chipulu, University of Southampton 2009
2. 2. Observed Probability Before a coin is tossed six times in a roll, what is the probability that in total there will be two ‘heads’ out of six? heads’ 3 The Binomial Distribution. Max Chipulu, University of Southampton 2009 A BERNOULLI TRIAL Tossing a coin is Bernoulli trial. A Bernoulli trial is a random experiment that has only two, mutually exclusive outcomes. Thus, when a coin is tossed, the two possible outcomes are revealing a ‘heads’ or revealing a ‘tails’. We want to heads’ tails’ see how many ‘heads’ are revealed. So revealing a heads’ ‘heads’ is a ‘success’. Revealing a ‘tails’ is a ‘failure’. heads’ success’ tails’ failure’ 4 The Binomial Distribution. Max Chipulu, University of Southampton 2009
3. 3. Properties of a BERNOULLI TRIAL Since you’re tossing the same coin in all six you’ Bernouli trials, the probability of a ‘heads’ or a heads’ success, p, is the same for each repeat of the Bernoulli trial. stationarity assumption But the coin has no memory: Bernoulli trials are independent. independence assumption 5 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Multiple Answer Question: Which of the following are Bernoulli Trials • A: The Lotto draw • B: The experiment: randomly select a company from all public limited companies in the UK. If the company is in Liquidation, the experiment is successful. If the company is in the Biomedical industry, the experiment is a failure. • C: Randomly selecting balls from a population containing white balls and black balls 6 The Binomial Distribution. Max Chipulu, University of Southampton 2009
4. 4. Multiple Answer Question: Which of the following are repeats of Bernoulli Trials • D: The game of joker’s challenge is played with joker’ the full deck of cards: The player is shown a card, and s/he calls whether the next card will be lower or higher, if the call is correct the game is repeated, and so on • E: A box contains 20 packs of chocolate, 8 of which are milk. Selecting a chocolate, checking if its milk, eating eat it, really enjoying it and then repeating the exercise. • F: The gender of a randomly selected CEO 7 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Binomial Probability Before a coin is tossed six times in a roll, what is the probability that in total there will be two ‘heads’ out of six? heads’ Method 1: The painstaking, torturous way: List all possible results that the arise from the six coin tosses. Count the total. Count how many times 2 'heads‘ are revealed. Calculate 'heads‘ the probability. 8 The Binomial Distribution. Max Chipulu, University of Southampton 2009
5. 5. Method 2: The easy Way. Use a formula. Somebody has already done all this before. There is a formula. It tells us how to obtain i things out of n. i n! It is : C n = i! ( n − i )! Where n! or ‘n factorial’ = n(n-1)(n-2)(n-3)…3*2*1; Thus, the total number of times 2 heads can be revealed by the six coin tosse s is : 6 * 5 * 4 * 3 * 2 *1 C 62 = = 15 ( 2 * 1 * ( 4 * 3 * 2 * 1)) 9 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Question: What is the probability that in total 2 coins out of six tossed will be heads? First, the Addition Rule We know there are 15 ways of selecting 2 successes out of six trials. So the probability we want is one these 15 combinations: P(x = 2) = P(HHTTTT or HTHTTT or HTTHTT or HTTTHT or HTTTTH or THHTTT etc) 10 The Binomial Distribution. Max Chipulu, University of Southampton 2009
6. 6. Mutual Exclusivity But we can only have one combination revealing two people at any given time: For example its either HTHTTT or HTTHTT, not HTTHTT, both They are mutually exclusive. So the Addition rule simplifies to: P(i = 2) = P(HHTTTT) +P(HTHTTT) + P(HTTHTT) + P(HTTTHT) + P(HTTTTH) + …. 11 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Second, the Multiplication Rule What is the probability that a sequence of results such as HHTTTT will be revealed? P(HHTTTT) = P(H)*P(HTTTT|H) We want to know: What is the probability that the sequence HTTTT will be revealed successively, if the first result is heads? But successive trials are independent: Whether or not the first result is heads does not affect the results of the next five trials. So we simplify: P(HHTTTT) = P(H)*P(HTTTT|H) = P(H)*P(HTTTT) 12 The Binomial Distribution. Max Chipulu, University of Southampton 2009
7. 7. The Multiplication Rule, Cont’d Cont’ In the same way, we can expand P(HTTTT) = P(H)*P(TTTT) And so on, and so forth, so that in the end: P(HHTTTT) = P(H)*P(H)*P(T)P*(T)*P(T)*P(T) P(HTTTTT) = P(H)2P(T)4 = 0.52*0.54 = 0.015625 13 The Binomial Distribution. Max Chipulu, University of Southampton 2009 The Multiplication Rule, Cont’d Cont’ A little thought should show that all the other combinations two heads out of six trials have the same probability: P(HHTTTT) =P(HTHTTT) = P(HTTHTT) =P(TTTHTT) =P(HTTTTH) =P(HTTTTH), etc.. Hence P(2 heads out of six trials) = 15 * 0.52*0.54 = 0.2344 14 The Binomial Distribution. Max Chipulu, University of Southampton 2009
8. 8. 2 4 Binomial Distribution successes failures We obtained these values as follows: P(2 Sufferers) = 15 * 0.52*0.54 C 62 = 15 P(failure) =1 - p = P(success) 0.5 = p = 0.5 15 The Binomial Distribution. Max Chipulu, University of Southampton 2009 We can generalise this expression so that we are able to calculate the probability for any number of successes i in n Bernoulli trials, for which the probability of success of each is p: P(x = i) = Cn p i (1 − p)n −i i Number of n – i failures. combinations of i i successes. P(failure) =1 - p out of n P(success) = p This is the binomial distribution 16 The Binomial Distribution. Max Chipulu, University of Southampton 2009
9. 9. We can derive the expected value and the variance for this distribution. They are: µ = np σ 2 = np(1 − p) 17 The Binomial Distribution. Max Chipulu, University of Southampton 2009 The Binomial Distribution Example: Mortgage Default Rates • It is estimated that 1.5% of home owners (with a mortgage) will default on their mortgage. Suppose a random sample of 12 home owners is taken. What is the probability that • (a) none of them will default • (b) at least one of them will default • Is the probability that all of them will default the same as none of them will default? • Suggested calculation steps • 1. This a Binomial Distribution- why? • 2. What is the trial, what is the success, what is the probability of a success? • 3. Calculate probabilities using the formula 18 The Binomial Distribution. Max Chipulu, University of Southampton 2009
10. 10. Solution 19 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Solution Continued 20 The Binomial Distribution. Max Chipulu, University of Southampton 2009
11. 11. Solution Continued 21 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Solution Continued 22 The Binomial Distribution. Max Chipulu, University of Southampton 2009
12. 12. Experiment: All Colas Taste Same Need Ten Volunteers Ten people that love colas 23 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Example: Cola Flavours in a Supermarket (1997/98 Exam) A product manager is in charge of a cola in a supermarket. His superior, the managing director, and the publicity department of the company, claim that this cola, Supercola, has a very distinctive flavour. The product manager is of the opinion that, without seeing the label, people are unable to discriminate between colas. In order to test his theory, the product manager conducts an experiment. He asks ten individuals to assess if the liquid in a cup is Supercola or one of the well-known brands: Pocacola and Colaloca. The results are as follows: 24 The Binomial Distribution. Max Chipulu, University of Southampton 2009
13. 13. Example: Cola Flavours in a Supermarket (1997/98 Exam) 1 Colaloca Supercola 2 Supercola Colaloca 3 Colaloca Pocacola 4 Pocacola Pocacola 5 Supercola Colaloca 6 Colaloca Supercola 7 Pocacola Pocacola 8 Pocacola Supercola 9 Supercola Supercola 10 Colaloca Pocacola 25 The Binomial Distribution. Max Chipulu, University of Southampton 2009 (a) What can the product manager conclude from the above experiment? Identify a probability model and use it to arrive at a conclusion. Explain your reasoning and give any relevant equations. (b) The managing director is of the opinion that ten trials is a very small number on which to base a marketing strategy, and insists that a larger experiment of 300 individuals should be conducted. You are requested to calculate the range of values within which it can be concluded that people cannot discriminate between colas. Explain your answer and give any relevant equations. 26 The Binomial Distribution. Max Chipulu, University of Southampton 2009
14. 14. Binomial Test Example: Are Greek SMEs Risk Averse? As part of her dissertation into the risk appetite of Greek SMEs (Small to Medium Enterprises) Margarita Georgousopoulou (2009 Risk Management Dissertation) asked respondents to rate their response on a Likert scale with categories 1 to 5, where by 1= ‘Very Risk Seeking’, 2 = ‘Risk Seeking’, 3 = ‘Risk Neutral’, 4 = ‘Risk Averse’ and 5 = ‘Very Risk Averse’. She tested several variables on this scale. For the ‘Machinery Investment’ Variable, the results were as in the table below: Can it be concluded that Greek SMEs are risk averse vis-à-vis ‘machinery investment’? 27 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Greek SMEs Solution • Define two events, a success and a failure, such that: • Event = ‘Success’ if response value is greater than 3, i.e. SME is Risk Averse • Event = ‘Failure’ if response value is 3 or less, i.e. SME is NOT risk averse • From the table, number of successes, i = 24. Number of trials = Sample size = 54. • Why is the binomial distribution a reasonable model for this situation? • Binomial Test question: is 24 successes out of 54 trials statistically significant? 28 The Binomial Distribution. Max Chipulu, University of Southampton 2009
15. 15. Greek SMEs: Binomial Test in SPSS 17 • Load the data file from blackboard called ‘Risk Appetite.xls’ into SPSS. • Select the worksheet called ‘raw data’ • From the ‘Analyze’ menu, select ‘nonparametric tests’ • Select ‘Binomial’ • Enter the variable ‘machinery investment’ into the ‘test variable box’ • In ‘Define Dichotomy’, select ‘cut point’. Enter ‘3’ in the cut point box. This tells SPSS that the values equal to or less than 3 are classed as one category, while values above are in the other category (hence the ‘dichotomy’, which refers to binary categories) 29 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Greek SMEs: Binomial Test in SPSS 17 Cont’d • In the box labeled ‘test proportion’, enter ‘0.4’:This will tell SPSS that under the null hypothesis, the expected proportion of values above 3 is 40% (this is because 2 of the five categories, i.e. 4 and 5 are higher than 3). Therefore the Binomial test is to see if the observed proportion (of 24/54) is significantly higher than 0.4 • Press ‘ok’ to run the model. 30 The Binomial Distribution. Max Chipulu, University of Southampton 2009
16. 16. Greek SMEs: Binomial Test in SPSS 17 Cont’d Or better still, press ‘paste’: This will paste the program syntax into a new syntax or program window. Whatever you do in SPSS, it is based on some program syntax. If you just use the menus, you cannot normally see the syntax. Yet, it is more efficient to run the syntax rather than use the menus, especially if you wish to repeat the analysis, since all you have to do is run the syntax again. Even better you can run tests on other variables simply by changing the variable name in the syntax. You could also change the test characteristics, such as the cut point and the test proportion. Furthermore, via the brilliant ‘copy’ and ‘paste’ keyboard controls, you can run all the analysis in one go, as one program. TRY THIS: you will feel better!! • To run the model, press the ‘run’ button in the syntax window, which looks like the ‘play’ button on a CD player. 31 The Binomial Distribution. Max Chipulu, University of Southampton 2009 Greek SMEs: SPSS Binomial Test Result • The significance of the binomial test is given under the column ‘Asymp. Sig.’ This is a one-tailed test, since we’re only testing whether the observed proportion is greater than 0.4, so that our test is only one side of 0.4. The test would be two-tailed if we were testing the hypothesis that the proportion is exactly equal to 0.4 • The value of significance is 0.015. What can we conclude from this? 32 The Binomial Distribution. Max Chipulu, University of Southampton 2009
17. 17. Greek SMEs: Binomial Test in SPSS 17 Exercise Please try this at home: Following the steps above, test whether or not the Greek SMEs are risk averse in the following variables. (i) ‘new product investment’; (ii) ‘new employee investment’; and (iii) ‘new market investment’ 33 The Binomial Distribution. Max Chipulu, University of Southampton 2009