1. BCOR 1020 Business 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 Level of Measurement CharacteristicsExample Nominal Categories only Eye color (blue, brown, green, hazel) Ordinal Rank has meaning Bond ratings (Aaa, Aab, C, D, F, etc.) Interval Distance has meaning Temperature (57 o Celsius) Ratio Meaningful zero exists Accounts payable ($21.7 million)

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 Q 1, Q 2, Q 3 –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. and If A and B are independent: Conditional Probabilities:

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( x i )  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: x012345 P(x)0.050.400.250.150.100.05 a)Find E(X) b)Find the probability that a randomly-selected customer will submit at least 4 prescriptions. c)Find the probability that a randomly-selected customer will have at least one prescription.

17. Midterm Exam – Review x012345 P(x)0.050.400.250.150.100.05 a)E(X) = b)P(at least 4 prescriptions) = c)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  and .

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 . –This is often denoted X~b(n,  ).

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 > 0, then we say that X has the Poisson distribution with parameter.

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. 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( ,  ) –“Bell-shaped” Distribution –Domain is –  < X < +  –Defined by two parameters,  (the mean) and  (the standard deviation) To find probabilities or percentiles with a normal distribution… (i) Standard normal transformation: (ii) Cumulative Normal Tables ( Handouts – Included on Exam )