BCOR 1020 Business Statistics

1. BCOR 1020Business Statistics Lecture 15 – March 6, 2008

2. Overview • Review for Midterm Exam… • Key Definitions • Visual Descriptions • Numerical Descriptions • Probability • Discrete Distributions • Continuous Distributions

3. Midterm Exam – Review • Statistics vs. statistics • Descriptive statistics refers to the collection, organization, presentation and summary of data • Inferential statistics refers to generalizing from a sample to a population, estimating unknown parameters, drawing conclusions, and making decisions. • Sample vs. Population: • statistics vs. parameters • Data Types: • Attribute • Discrete Numerical • Continuous Numerical

4. Midterm Exam – Review • Levels of Measurement: • Nominal – Ordinal • Interval – Ratio

5. Midterm Exam – Review • Time series data • Each observation in the sample represents a different equally spaced point in time (e.g., years, months, days). • Cross-sectional data • Each observation represents a different individual unit (e.g., person) at the same point in time. • Sampling Concepts & Methods: • When to sample vs. when to census • Probability samples (simple random, systematic, stratified, etc.) • Nonprobability samples (judgment, convenience)

6. Midterm Exam – Review • Visual Descriptions • Dot Plots • Frequency Distributions and Histograms – Including Modal Class, Symmetry & Skewness • Simple Line Charts – Time Series • Bar charts – Including Pareto Charts • Scatter Plots – Cross-sectional Data • Tables • Deceptive Graphs

7. Midterm Exam – Review • Numerical Description • Central Tendency – Mean, Median, Mode, Midrange, etc. • Skewness: • If Median/Mode > Mean, skewed left • If Median/Mode = Mean, symmetric • If Median/Mode < Mean, skewed right • Dispersion – Variance & Standard Deviation, Coefficient of Variation, etc. • The Empirical Rule (Symmetric Distributions): • Approximately 68.26% within • Approximately 95.44% within • Approximately 99.73% within

8. Midterm Exam – Review • Standardized Variables: • Unusual observations & outliers • Percentiles and Quartiles: • Find Q1, Q2, Q3 • Midhinge: • Midspread (IQR): • Coefficient of Quartile Variation (CQV): • Boxplots – Including Quartiles, Median, IQR, Unusual Observations, etc.

9. Midterm Exam – Review • Random Experiments: • Sample Space – discrete or continuous • Events – simple and compound • Probability: • Definition & Characteristics • Empirical, Classical, Subjective Approaches • Venn Diagrams

10. Midterm Exam – Review • Rules of Probability (illustrated w/ Venn Diagrams): • Compliments • Unions – Law of Addition • Mutually Exclusive Events • Conditional Probabilities, Independence, Intersection – Multiplication Rule • Independent vs. Mutually Exclusive • Contingency Tables and Trees as Tools (Example): • Marginal probabilities, conditional probabilities, etc.

11. Midterm Exam – Review and In general. For mutually exclusive events. Conditional Probabilities: and If A and B are independent: and

12. Midterm Exam – Review • Example: this table gives expense ratios by fund type for 21 bond funds and 23 stock funds. Find P(B), P(H), P(H|S),

13. Midterm Exam – Review • Random variable –a function or rule that assigns a numerical value to each outcome in the sample space of a random experiment. • Capital letters are used to represent random variables (e.g., X, Y). • Lower case letters are used to represent values of the random variable (e.g., x, y). • A discrete random variable has a countable number of distinct values. Values are integers. • A continuous random variable has an uncountable (infinite) number of distinct values. Values fall on an interval.

14. Midterm Exam – Review • Probability Distributions: • A discrete probability distribution is a rule (function) that assigns a probability to each value of a discrete random variable X. • To be a valid probability, each probability must be between 0  P(xi) 1 • and the sum of all the probabilities for the values of X must be equal to unity. • For a continuous random variable, the probability density function (PDF) is an equation that shows the height of the curve f(x) at each possible value of X over the range of X.

15. Midterm Exam – Review • The expected value, E(X), is a measure of central tendency. • For a discrete random variable, • For a continuous random variable, • The variance, V(X), is a measure of dispersion. • For a discrete random variable,

16. Midterm Exam – Review • Example: On any given day, the number of prescriptions submitted by a random customer at a pharmacy (X) is described by the probabilities in the following table: • Find E(X) • Find the probability that a randomly-selected customer will submit at least 4 prescriptions. • Find the probability that a randomly-selected customer will have at least one prescription.

17. Midterm Exam – Review • E(X) = • P(at least 4 prescriptions) = • P(> 1 prescription) =

18. Midterm Exam – Review • Common Distributions • Binomial • Poisson • Normal • Exponential • Be able to recognize the experimental conditions leading to these distributions. • For each of these distributions, be able to use the • PDF and/or CDF • formulas for m and s.

19. Midterm Exam – Review • If X denotes the number of “successes” observed in n Bernoulli trials, then we say that X has the Binomial distribution with parameters n and p. • This is often denoted X~b(n,p).

20. Midterm Exam – Review • If X denotes the number of “occurrences” of interest observed on a given interval of length 1 unit of a Poisson Process with parameter l > 0, then we say that Xhas the Poisson distribution with parameter l.

21. Midterm Exam – Review • If events per unit of time follow a Poisson distribution, the waiting time until the next event follows the Exponential distribution with parameter l. PDF:

22. Midterm Exam – Review • Generally, we will use the Normal random variable when we make the assumption that our population is normally distributed. • Denoted N(m, s) • “Bell-shaped” Distribution • Domain is – < X < +  • Defined by two parameters, m (the mean) and s (the standard deviation) • To find probabilities or percentiles with a normal distribution… (i) Standard normal transformation: (ii) Cumulative Normal Tables (Handouts – Included on Exam)