Managerial Decision Making Facilitator: René Cintrón MBA / 510

Week 2 - Objectives Analyze data using descriptive statistics Apply basic probability concepts to facilitate business decision making Distinguish between discrete and continuous probability distributions Apply the normal distribution to facilitate business decision making.

Analyze data using descriptive statistics Population mean Sample mean Weighted mean Median Mode Variance and standard deviation Empirical rule

Mean, Median, Mode The mean is the usual average The median is the middle value The mode is the number that is repeated more often than any other

     68% 95% 99.7%

Empirical Rule Empirical Rule : For any symmetrical, bell- shaped distribution:  About 68% of the observations will lie within 1s the mean  About 95% of the observations will lie within 2s of the mean  Virtually all the observations will be within 3s of the mean 3- 6

Basic Probability Concepts What is a probability? Approaches to assigning probabilities Rules for computing probabilities Contingency tables

Classical The Classical definition applies when there are n equally likely outcomes. Empirical The Empirical definition applies when the number of times the event happens is divided by the number of observations. Subjective Subjective probability is based on whatever information is available. There are three definitions of probability: classical, empirical, and subjective.

Event An Event is the collection of one or more outcomes of an experiment. Outcome An Outcome is the particular result of an experiment. Experiment: A fair die is cast. Possible outcomes: The numbers 1, 2, 3, 4, 5, 6 One possible event: The occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.

Independent Events are Independent if the occurrence of one event does not affect the occurrence of another. Mutually Exclusive Events are Mutually Exclusive if the occurrence of any one event means that none of the others can occur at the same time. Mutually exclusive: Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 on the same roll. Independence: Rolling a 2 on the first throw does not influence the probability of a 3 on the next throw. It is still a one in 6 chance. Mutually Exclusive Events

Collectively Exhaustive Events are Collectively Exhaustive if at least one of the events must occur when an experiment is conducted.

Throughout her teaching career Professor Jones has awarded 186 A’s out of 1,200 students. What is the probability that a student in her section this semester will receive an A? This is an example of the empirical definition of probability.  To find the probability a selected student earned an A: Empirical Example

Examples of subjective probability are: estimating the probability the New Orleans Saints will win the Super Bowl this year. estimating the probability mortgage rates for home loans will top 8 percent.

P(A or B) = P(A) + P(B) If two events A and B are mutually exclusive, the Special Rule of Addition Special Rule of Addition states that the Probability of A or B occurring equals the sum of their respective probabilities.

If P(A) is the probability of event A and P(~A) is the complement of A, P(A) + P(~A) = 1 or P(A) = 1 - P(~A). Complement Rule The Complement Rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. The Complement Rule

 Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.  This rule is written: P(A and B) = P(A)P(B) Special Rule of Multiplication The Special Rule of Multiplication requires that two events A and B are independent.

Joint Probability A Joint Probability measures the likelihood that two or more events will happen concurrently. An example would be the event that a student has both a stereo and TV in his or her dorm room.

The probability of event A occurring given that the event B has occurred is written P(A|B). Conditional Probability A Conditional Probability is the probability of a particular event occurring, given that another event has occurred.

It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred. General Rule of Multiplication The General Rule of Multiplication is used to find the joint probability that two events will occur.

Discrete and Continuous Probability Distributions What is a probability distribution? Discrete probability distributions Random variables Discrete random variable Mean, variance, and standard deviation of a probability distribution Continuous probability distributions Continuous random variable

Discrete probability Distribution Discrete probability Distribution Can assume only certain outcomes Continuous Probability Distribution Continuous Probability Distribution Can assume an infinite number of values within a given range Types of Probability Distributions A listing of all possible outcomes of an experiment and the corresponding probability. Random variable A numerical value determined by the outcome of an experiment. Probability Distributions

Movie Continuous Probability Distribution

Discrete Probability Distribution Discrete Probability Distribution The sum of the probabilities of the various outcomes is The probability of a particular outcome is between 0 and The outcomes are mutually exclusive. The number of students in a class The number of children in a family The number of cars entering a carwash in a hour

Normal Distribution Family of normal probability distributions Standard normal distribution Empirical rule Finding areas under the normal curve

bell-shaped is bell-shaped and has a single peak at the center of the distribution. symmetrical  Is symmetrical about the mean. asymptotic  is asymptotic. That is the curve gets closer and closer to the X-axis but never actually touches it. mean,  standard deviation,   Has its mean, , to determine its location and its standard deviation, , to determine its dispersion. Normal The Normal probability distribution

x f ( x ral itrbuion:  =0,  = 1 Mean, median, and mode are equal Theoretically, curve extends to infinity a Characteristics of a Normal Distribution Normal curve is symmetrical

Variance and Standard Deviation

z-value A z-value is the distance between a selected value, designated X, and the population mean , divided by the population standard deviation, . The formula is: It is also called the z distribution. standard normal The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The Standard Normal Probability Distribution

Variance and Standard Deviation X is the value of an observation in the population m is the arithmetic mean of the population N is the number of observations in the population

Empirical Rule Empirical Rule : For any symmetrical, bell- shaped distribution:  About 68% of the observations will lie within 1s the mean  About 95% of the observations will lie within 2s of the mean  Virtually all the observations will be within 3s of the mean 3- 30

Areas Under the Normal Curve Practically all is within three standard deviations of the mean.  + 3  About 68 percent of the area under the normal curve is within one standard deviation of the mean.  + 1  About 95 percent is within two standard deviations of the mean.  + 2 

     68% 95% 99.7%

Next Week Apply inferential statistics in solving business problems Determine an appropriate sample size Apply confidence intervals in solving business problems. Problem Sets Team Business Problem Proposal