STATISTICS FOR BUSINESS Chapter 4 Discrete data and

STATISTICS FOR BUSINESS Chapter 4 : Discrete data and probability The shopping mall Derek L WALLER © 1

STATISTICS FOR BUSINESS (Discrete data and probability) The shopping mall Discrete data is composed of integer values, or whole numbers Information that is unconnected and comes from the counting process. . We could say: • 9 machines are shutdown • 29 bottles have been sold • 8 units are defective • 5 hotel rooms are vacant • 3 students are absent We could not say: • 9½ machines are shutdown • 29¾ bottles have been sold • 8½ units are defective • 5½ hotel rooms are empty • 3¼ students are absent With discrete data there is clear segregation • Data does not progress from one class to another. Derek L WALLER © 2

STATISTICS FOR BUSINESS (Discrete data and probability) • If discrete data occur in no special order, • No explanation of their distribution • Considered discrete random variables. The shopping mall Random means that, within the range of the possible data values, every item has an equal chance of occurring, or being selected • Value obtained by throwing a single die is random • Drawing of a card from a full pack is random Number of people arriving at a shopping mall in any particular day is random. • If we knew pattern it would help to better plan staff needs Number of cars on a particular stretch of road on any given day is random • Knowing pattern would help to decide on installation of stop signs, or signals Number of people seeking medical help at a hospital emergency is random • Understanding pattern helps in scheduling medical staff and making budgets. Derek L WALLER © 3

STATISTICS FOR BUSINESS (Discrete data and probability) Expected value of discrete random variable The shopping mall • x is specific value of the discrete random variable • P(x) is probability of obtaining x • E(X) is the expected, mean, or average value That is, a weighted average according to probabilities Variance Standard deviation Derek L WALLER © 4

STATISTICS FOR BUSINESS (Discrete data and probability) Store sells wine by case. Each case of wine generates € 7. 20 in profit Selling of wine is considered random The shopping mall Analysis of sales data for past 200 days (columns A & B) Probability of sale: Days amount sold/Total days in analysis (column C) Probability of a particular level of cases being sold is x. P(x) and ∑x. P(x) = µx A Cases sold per day, x 10 11 12 13 Total B Days this amount sold C Probability of sale P(x) (Relative frequency) 30 40 80 50 200 15. 00% 20. 00% 40. 00% 25. 00% 100. 00% Cases sold/day 10 Dispersion of data 11 12 13 Total (x - µx)2 3. 0625 0. 5625 0. 0625 1. 5625 5. 2500 D x. P(x) 1. 50 2. 20 4. 80 3. 25 11. 75 (x - µx)2*P(x) 0. 4594 0. 1125 0. 0250 0. 3906 0. 9875 Mean or expected value is 11. 75 Assumption is past data is representative of future Estimated future profit is, € 84. 60 (7. 20*11. 75) Here we are using weighting averages according to the probabilities Variance of data is 0. 9875 Standard deviation is 0. 9937 (√ 0. 9875) Derek L WALLER © 5

STATISTICS FOR BUSINESS (Discrete data and probability) Law of averages The shopping mall Mean, or expected value is not the value that will occur next, or even tomorrow • It is the value expected to be obtained over the long run • In the short term we really do not know what will happen In gambling when you play the slot machines you may win a few games. • If you continue playing, in the long run you will lose • Gambling casinos set machines so that the casino will be long-term winner • If, not they would go out of business! With probability, it is the law of averages that governs. • The average value obtained in the long-term will be close to expected value • Expected value is weighted outcome based on each probability of occurrence • In society, the “expected” behaviour is being honest, ethical, and abiding by rules. Derek L WALLER © 6

STATISTICS FOR BUSINESS (Discrete data and probability) Tossing a coin 1, 000 times: %age of heads or tails obtained is 50% The shopping mall Law of averages Derek L WALLER © 7

STATISTICS FOR BUSINESS (Discrete data and probability) Binomial distribution -characteristics The shopping mall • Each observation considered selected from an infinite population, without replacement. • Or, a finite population with replacement • Outcome of observation independent of any other observation Probability of “success”, p, constant over time • Probability of “failure”, q = (1 -p) • Tossing a coin: p = q = 0. 50 Each trial has only two outcomes • Success, or failure • Win, or lose • Good, or no good • In attendance, or absent • Late, or on time • Work, does not work • Binomial equation n is the number of trials p is the probability of success q is the probability of failure, or (1 -p) x is the number of successes desired Derek L WALLER © Mean µ = n*p Standard deviation = √(n*p*q) 8

STATISTICS FOR BUSINESS (Discrete data and probability) Binomial distribution –explanation of terms The shopping mall Gives how many sequences, arrangements, or combinations of the x successes out of n observations are possible (From the counting rules) Gives the probability of obtaining exactly x successes out of n observations in a particular sequence Derek L WALLER © 9

STATISTICS FOR BUSINESS (Discrete data and probability) Having 7 children: Probability of boy is 50%. Probability of girl is 50% The shopping mall 30% 27. 34% 28% 27. 34% 26% 24% 22% 20% 18% 16. 41% 16% Frequency of occurrence 14% 16. 41% Binomial distribution 12% 10% 8% 5. 47% 6% 5. 47% 4% 2% 0. 78% 0% 0 Derek L WALLER © 1 2 3 4 Number of boys 5 6 7 10

STATISTICS FOR BUSINESS (Discrete data and probability) Poisson Distribution The shopping mall • After the Frenchman Siméon Poisson • Used to describe waiting lines or queuing • Used in IT programs to manage these types of situations Equation for probability of occurrence x, P(x) = lxe-l/x! e, base of natural logarithm, 2. 71828 l, lambda is mean number of occurrences P(x) probability of exactly x occurrences x! Is x factorial Petrol pump: lambda = 8 15% 14% 13% 12% 11% 10% Frequency 9% of number 8% 7% using pump 6% 5% 4% 3% 2% 1% 0% 0 Derek L WALLER © 2 4 6 8 10 12 14 16 18 20 Number of customers using pump 11