STATISTICS FOR BUSINESS

1. STATISTICS FOR BUSINESS Chapter 4 : Discrete data and probability The shopping mall

2. 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 • .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.

3. 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.

4. 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

5. 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 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) Dispersion of data Here we are using weighting averages according to the probabilities Variance of data is 0.9875 Standard deviation is 0.9937 (√0.9875)

6. 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 averagesthat 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.

7. 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

8. 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 • 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 • Probability of “success”, p, constant over time • Probability of “failure”, q = (1-p) • Tossing a coin: p = q = 0.50 • 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 Mean µ = n*p Standard deviation = √(n*p*q)

9. 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

10. STATISTICS FOR BUSINESS (Discrete data and probability) The shopping mall Binomial distribution

11. 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