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BA 275 Quantitative Business Methods Agenda Summarizing Quantitative Data The Empirical Rule Experiencing Random Behavior Binomial Experiment Binomial Probability Distribution Attention: Project 1 is due on Wednesday, 1/25/06. 1 Quiz #2 Given the data below, complete the following summary statistics table. (Data are in ascending order): 10.0, 10.5, 12.2, 13.9, 13.9, 14.1, 14.7, 14.7, 15.1, 15.3, 15.9, 17.7, 18.5 2.5 frequency Count Average Median Variance Standard deviation Minimum Maximum Range Lower quartile Upper quartile Interquartile range Sum 1 10 12 14 16 18 20 Variable_X 5 2 1 8 5 3 1.5 0 6 4 2 0.5 6 frequency frequency 3 Variable X 13 14.3462 14.7 5.94936 2.43913 10.0 18.5 8.5 13.9 15.3 1.4 186.5 4 3 2 1 0 8 10 12 14 16 Variable_X 18 20 0 9.99 11.99 13.99 15.99 17.99 Variable_X 19.99 21.99 2 Review Example Given the data below, complete the following summary statistics table. (Data are in ascending order): 10.0, 10.5, 12.2, 13.9, 13.9, 14.1, 14.7, 14.7, 15.1, 15.3, 15.9, 17.7, 18.5 Count Average Median Variance Standard deviation Minimum Maximum Range Lower quartile Upper quartile Interquartile range Sum Variable X 13 14.3462 14.7 5.94936 2.43913 10.0 18.5 8.5 13.9 15.3 1.4 186.5 Box-and-Whisker Plot 10 12 14 16 18 20 Variable X Lower invisible line: 11.8 Upper invisible line: 17.4 3 The Empirical Rule 99.7% 95% 68% 0.15% 2.35% 13.5% 34% 34% 13.5% 2.35% 0.15% x 3s 3 x 2s xs x xs x 2s 2 2 3 2 1 0 x 3s 3 1 2 3 4 Example A set of data whose histogram is bell shaped yields a sample mean and standard deviation of 50 and 4, respectively. Approximately what proportion of observations Are between 46 and 54? Are between 42 and 58? Are between 38 and 62? Are less than 46? Are less than 58? 5 Example: The Empirical Rule A manufacturer of automobile batteries claims that the average length of life for its grade A battery is 60 months with a standard deviation of 10 months. 3 cars in your family used this brand of batteries and none of them lasted for more than 30 months. What do you think about the manufacturer’s claim? 6 Example A manufacturer produces wires with a mean diameter of 1000 microns, and a standard deviation of 1 micron. Is a wire of diameter 1050 microns: Fairly likely? Pretty unlikely? Wildly implausible? 7 Example Suppose that the average hourly earnings of production workers over the past three years were reported to be $12.27, $12.85, and $13.39 with the standard deviations $0.15, $0.18, and $0.23, respectively. The average hourly earnings of the production workers in your company also continued to rise over the past three years from $12.72 in 2002, $13.35 in 2003, to $13.95 in 2004. Assuming the distribution of the hourly earnings for all production workers is mound-shaped, demonstrate quantitatively why the earnings in your company become less and less competitive. 8 Review Example Year Industry average Industry std. 2002 12.27 0.15 2003 12.85 0.18 4.73% 13.35 4.95% 2.77 2004 13.39 0.23 4.20% 13.95 4.50% 2.43 % increase Company average % increase 12.72 Z score 3 9 What statistical lesson can we learn? “Should we scare the opposition by announcing our mean height or lull them by announcing our median height?” 10 Coin-Tossing Example n = 10 p = 0.5 X = no. of tails in n trials Questions P(X = 5) P(X = 10) P(X < 4) P(X > 8) Business Applications? 11 Characteristics of a Binomial Experiment The experiment consists of n identical trials. There are only two possible outcomes on each trial. We will denote one outcome by S (for Success) and the other by F (for Failure). The probability of S remains the same from trial to trial. This probability is denoted by p, and the probability of F is denoted by 1 – p. The trials are independent. The binomial random variable X is the number of S’s in n trials. 12 Is it a Binomial Experiment? Flip a fair coin 50 times. Toss a fair die 20 times. A multiple-choice quiz has 15 questions. Each question has five possible answers, of which only one is correct. Time is running out and you quickly guess all 15 questions without reading them. You read every question carefully and answer them to the best of your knowledge. Among these questions, three of them are main questions each with 4 subsequent questions. If you don't know the answer to one question, you won't be able to answer the subsequent questions correctly. 13 Probability Distribution Binomial Distribution probability 0.25 Event prob.,Trials 0.5,10 0.2 0.15 0.1 0.05 0 0 2 4 6 8 10 The likelihood of observing a particular outcome No. of Successes All possible outcomes of an experiment 14 Binomial Formula and Distribution n x p( X x) p (1 p) n x x Binomial Distribution probability 0.25 Event prob.,Trials 0.5,10 0.2 0.15 0.1 0.05 0 0 2 4 6 8 10 No. of Successes 15 Example A sign at a gas station claims that one out of four cars needs to have oil added. If the claim is true, what is the probability of the following events? One out of the next four cars needs oil. Two out of the next eight cars need oil. 16