BCOR 1020 Business Statistics

1. BCOR 1020Business Statistics Lecture 6 – February 5, 2007

2. Overview • Chapter 4 Example • Chapter 5 – Probability • Random Experiments • Probability

3. Chapter 4 - Example • Problem 4.22 list the rents paid by a random sample of 30 students who live off campus. The sorted data is below. • Using Excel, we can quickly calculate the sample average and standard deviation… • Using these, we can find standardized values (zi)… • Find the Quartiles and Construct a Boxplot…

4. Chapter 4 - Example • Ordered Data… • Median = 720 • Q1 = 660 • Q3 = 760 • IQR = 100

5. Chapter 5 - Probability

6. Chapter 5 – Random Experiments Sample Space: • A random experiment is an observational process whose results cannot be known in advance and whose outcomes will differ based on random chance. • The sample space (S) for the experiment is the set of all possible outcomes in the experiment. • A discrete sample space is one with a countable (but perhaps infinite) number of outcomes. • A continuous sample space is one where the outcomes fall on a continuous interval (often the result of a measurement).

7. Chapter 5 – Random Experiments Sample Space: • Discrete Sample Space Examples: • The sample space describing a Wal-Mart customer’s payment method is… S = {cash, debit card, credit card, check} • Continuous Sample Space Examples: • The sample space for the length of a randomly chosen cell phone call would be… S = {all X such that X > 0} or written as S = {X | X > 0}. • The sample space to describe a randomly chosen student’s GPA would be S = {X | 0.00 <X < 4.00}.

8. {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} S = Chapter 5 – Random Experiments Some sample spaces can be enumerated: • For Example: • For a single roll of a die, the sample space is: S = {1, 2, 3, 4, 5, 6}. • When two dice are rolled, the sample space is the following pairs:

9. Chapter 5 – Random Experiments Some sample spaces are not easily enumerated: • For Example: • Consider the sample space to describe a randomly chosen United Airlines employee by 6 home bases (major hubs), 2 genders, 21 job classifications, 4 education levels • There are: 6 x 22 x 21 x 4 = 1008 possible outcomes.

10. Chapter 5 – Random Experiments Events: • An event is any subset of outcomes in the sample space. • A simple event or elementary event, is a single outcome. • A discrete sample space S consists of all the simple events (Ei): S = {E1, E2, …, En} • A compound event consists of two or more simple events.

11. Chapter 5 – Random Experiments Example of a Simple Event: • Consider the random experiment of tossing a balanced coin. What is the sample space? S = {H, T} • What are the chances of observing a H or T? These two elementary events are equally likely.

12. Clickers When you buy a lottery ticket, the sample space S = {win, lose} has only two events. Are these two events equally likely to occur? A = Yes B = No

13. Chapter 5 – Random Experiments Example of Compound Events: • Recall: a compound event consists of two or more simple events. • For example, in a sample space of 6 simple events, we could define the compound events… A = {E1, E2} B = {E3, E5, E6} • These are displayed in a Venn diagram: • Many different compound events could be defined. • Compound events can be described by a rule.

14. {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} S = Clickers Recall our earlier example involving the roll of two dice where the sample space is given by… If we define the compound event A = “rolling a seven” on a roll of two dice, how many simple events does our compound event consist of? A = 4 B = 6 C = 7 D = 36

15. If P(A) = 1, then the event is certain to occur. If P(A) = 0, then the event cannot occur. Chapter 5 – Probability Definition: • The probability of an event is a number that measures the relative likelihood that the event will occur. • The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 <P(A) < 1

16. Probability Chapter 5 – Probability Definitions: • In a discrete sample space, the probabilities of all simple events must sum to unity: P(S) = P(E1) + P(E2) + … + P(En) = 1 • For example, if the following number of purchases were made by…

17. Chapter 5 – Probability • Businesses want to be able to quantify the uncertainty of future events. • For example, what are the chances that next month’s revenue will exceed last year’s average? • The study of probability helps us understand and quantify the uncertainty surrounding the future. • How can we increase the chance of positive future events and decrease the chance of negative future events?

18. number of defaults number of loans = Chapter 5 – Probability What is Probability? • There are three approaches to probability: • Empirical – Classical – Subjective Empirical Approach: • Use the empirical or relative frequency approach to assign probabilities by counting the frequency (fi) of observed outcomes defined on the experimental sample space. • For example, to estimate the default rate on student loans P(a student defaults) = f /n

19. Chapter 5 – Probability Empirical Approach: • Necessary when there is no prior knowledge of events. • As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate. Law of Large Numbers: • The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one.

20. Chapter 5 – Probability Example: Law of Large Numbers: • Flip a coin 50 times. We would expect the proportion of heads to be near .50. • However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.). • A large n may be needed to get close to .50. • Consider the results of simulating 10, 20, 50, and 500 coin flips…

21. Chapter 5 – Probability

22. Chapter 5 – Probability Classical Approach: • In this approach, we envision the entire sample space as a collection of equally likely outcomes. • Instead of performing the experiment, we can use deduction to determine P(A). • a priori refers to the process of assigning probabilities before the event is observed. • a priori probabilities are based on logic, not experience.

23. Chapter 5 – Probability Classical Approach: • For example, the two dice experiment has 36 equally likely simple events. The probability that the sum of the two dice is 7, P(7), is • The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:

24. Clickers Consider the Venn Diagram for the roll of two dice from the previous example: What is the probability that the two dice sum to 4, P(4)? A = 0.083 B = 0.111 C = 0.139 D = 0.167 E = 0.194

25. Chapter 5 – Probability Subjective Approach: • A subjective probability reflects someone’s personal belief about the likelihood of an event. • Used when there is no repeatable random experiment. • For example, • What is the probability that a new truck product program will show a return on investment of at least 10 percent? • What is the probability that the price of GM stock will rise within the next 30 days? • These probabilities rely on personal judgment or expert opinion. • Judgment is based on experience with similar events and knowledge of the underlying causal processes.