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1 Business 260: Managerial Decision Analysis Professor David Mease Lecture 3 Agenda: 1) Reminder about Homework #1 (due Thursday 3/19) 2) Discuss Midterm Exam 1 (Thursday 3/19) 3) Some Important Discrete Probability Distributions (Stats Book P. 195) 4) The Normal Distribution and Other Continuous Distributions (Stats Book P. 231)

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2 Homework #1 Homework #1 will be due Thursday 3/19 We will have an exam that day after we review the solutions The homework is posted on the class web page: http://www.cob.sjsu.edu/mease_d/bus260/260homework.html The solutions are also posted so you can check your answers: http://www.cob.sjsu.edu/mease_d/bus260/260homework_solutions.html

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3 Midterm Exam #1 We will have the first midterm exam covering the first 3 lectures Thursday 3/19 after we go over the homework There are 36 questions (28 questions worth 3 percent each and 8 questions worth 2 percent each) It is worth 40 total points (40% of your grade) It is closed notes and closed book but you will have this: http://www.cob.sjsu.edu/mease_d/bus260/260formula_sheets.html You will have 2 hours to complete the exam Remember to bring a pocket calculator Some example questions are on the next 5 slides

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4 Midterm Example Question #1 Use the following sample of size n=7 for this question. 19 31 34 35 39 39 43 What is the third quartile (Q3)?

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5 Midterm Example Question #2 For a population with an approximately bell-shaped distribution with a mean of 12 and a standard deviation of 2 the empirical rule says about 95% of the data will fall where?

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6 Midterm Example Question #3 (In class exercise #31) The probability that a person has a certain disease is 0.03. A test is available to determine whether a person has the disease. If the person does actually have the disease, the test will indicate so with probability 0.90. If the person does not have the disease, the test will still indicate the person has the disease with probability 0.02. Suppose the test says you have the disease. What is the probability that you actually do?

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7 Midterm Example Question #4 Use the following discrete probability distribution: What is the variance of the number of accidents?

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8 Midterm Example Question #5 Phone calls are normally distributed with a mean of 15 minutes and a standard deviation of 2 minutes. What proportion of phone calls last between 18 and 18.5 minutes?

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9 Some Important Discrete Probability Distributions (Stats Book P. 195) Statistics for Managers Using Microsoft ® Excel 4 th Edition

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10 Chapter Goals After completing this chapter, you should be able to: Compute and interpret the mean and standard deviation for a discrete probability distribution Explain covariance and its application in finance Use the binomial probability distribution to find probabilities Describe when to apply the binomial distribution

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11 Introduction to Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable Ch. 5Ch. 6

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12 Discrete Probability Distributions A discrete probability distribution is given by a table listing all possible values for the random variable along with the corresponding probabilities. The appropriate chart to display it is a bar chart (which has gaps, unlike a histogram).

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13 Discrete Random Variable Summary Measures Expected Value (or mean) of a discrete distribution (Weighted Average)

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14 Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X X i = the ith outcome of X P(X i ) = Probability of the ith occurrence of X Discrete Random Variable Summary Measures (continued)

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15 In class exercise #44: A Bay Area software company is trying to hire as many qualified job candidates as possible. Next Monday they will interview 4 candidates. If the probability of each candidate being hired is 20%, give the probability distribution for the number of candidates they will hire that day assuming that the candidates are independent. Also compute the mean, variance and standard deviation for the number of candidates that will be hired that day.

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16 Binomial Probability Distribution A fixed number of observations, n e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse Two mutually exclusive and collectively exhaustive categories e.g., head or tail in each toss of a coin; defective or not defective light bulb Generally called “success” and “failure” Probability of success is p, probability of failure is 1 – p Constant probability for each observation e.g., Probability of getting a tail is the same each time we toss the coin Observations are independent The outcome of one observation does not affect the outcome of the other

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17 Examples of Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it

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18 P(X) = probability of X successes in n trials, with probability of success p on each trial X = number of ‘successes’ in sample, (X = 0, 1, 2,..., n) n = sample size (number of trials or observations) p = probability of “success” P(X) n X! nX p(1-p) X n X ! ()! Binomial Distribution Formulas Mean Variance and Standard Deviation

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19 In class exercise #45: Redo ICE #44 using the binomial formulas.

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20 Using Binomial Tables (like Table 6 in back of book) n = 10 x…p=.20p=.25p=.30p=.35p=.40p=.45p=.50 0 1 2 3 4 5 6 7 8 9 10 ………………………………………………………… 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0001 0.0000 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0000 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0001 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 0.2051 0.1172 0.0439 0.0098 0.0010 10 9 8 7 6 5 4 3 2 1 0 …p=.80p=.75p=.70p=.65p=.60p=.55p=.50x Examples: n = 10, p =.35, x = 3: P(x = 3) =.2522 n = 10, p =.75, x = 2: P(x = 2) =.0004

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21 In class exercise #46: Check your probabilities for ICE #44 using Table 6 in the back of the book.

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22 In class exercise #46: Check your probabilities for ICE #44 using Table 6 in the back of the book.

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23 In class exercise #47: An important part of the customer service responsibilities of a telephone company relates to the speed with which customer service troubles can be repaired. Suppose past data indicate that the likelihood is 0.70 that troubles in residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that a) all five will be repaired on the same day? b) at least three will be repaired on the same day?

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24 Table 6 again

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25 In class exercise #48: If I shoot three point baskets with 68% accuracy, what is the probability I make exactly 7 out of 10?

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26 Using Excel to get Binomial Probabilities Example: n = 10, p =.68, x = 7: then Excel tells us P(7) =.2644

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27 The Normal Distribution and Other Continuous Distributions (Stats Book P. 231) Statistics for Managers Using Microsoft ® Excel 4 th Edition

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28 Chapter Goals After completing this chapter, you should be able to: Describe the characteristics of the normal distribution Translate normal distribution problems into standardized normal distribution problems Find probabilities using a normal distribution table Define the concept of a sampling distribution Determine the mean and standard deviation for the sampling distribution of the sample mean Describe the Central Limit Theorem and its importance Apply the sampling distribution for the sample mean

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29 Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable Ch. 5Ch. 6

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30 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

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31 The Normal Distribution

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32 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to Mean = Median X f(X) μ σ

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33 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

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34 The formula for the normal probability density function is Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 μ = the population mean σ = the population standard deviation X = any value of the continuous variable The Normal Distribution

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35 The Normal Distribution You can obtain probabilities for the normal distribution using Table 2 in the back of the book You can look up Z (the number of standard deviations above or below the mean) on the left and top of this table and then the numbers inside the table will give you the probabilities to the LEFT of Z Z

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36 In class exercise #49: Use Table 2 to determine the probability that a normally distributed random variable is less than 1 standard deviation above the mean.

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37 In class exercise #50: Use Table 2 to determine the probability that a normally distributed random variable is greater than 1.6 standard deviations above the mean.

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38 In class exercise #51: Use Table 2 to determine the probability that a normally distributed random variable is greater than 1.63 standard deviations above the mean.

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39 In class exercise #52: Use Table 2 to determine the probability that a normally distributed random variable is greater than 2.34 standard deviations below the mean.

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40 In class exercise #53: Use Table 2 to determine the probability that a normally distributed random variable is less than two standard deviations above the mean but greater than two standard deviations below the mean. Where have you seen this probability before?

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41 In class exercise #54: Exam scores have a normal distribution with a mean of 70 and a standard deviation of 10. Find the percentage of students who score less than 85.

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42 The Normal Distribution The following formula is useful for finding Z (the number of standard deviations above or below the mean) Z itself has what is called a standard normal distribution (mean=zero and standard deviation=1)

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43 In class exercise #55: Exam scores have a normal distribution with a mean of 70 and a standard deviation of 10. a) Find the percentage of students who score less than 45 b) Find the percentage of students who score greater than 45 c) Find the percentage of students who score between 52 and 90 d) Find the percentage of students who score greater than 80 or less than 60. e) Find the percentage of students who score less than 5

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44 The Normal Distribution – Working Backwards Sometimes you have the probability and want to know the value of the normal random variable that corresponds to that probability For example, if I want to give 10% of students a failing grade on the exam, what should be the cutoff? In these case, the following formula is useful

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45 In class exercise #56: Exam scores have a normal distribution with a mean of 70 and a standard deviation of 10. a) Find the cutoff that will give 10% an F. b) Find the cutoff that will give 20% an A. c) Find the cutoffs that will give the middle 40% a C.

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46 Sampling Distributions A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population We will study the sampling distribution of the sample mean

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47 In class exercise #57: The data at http://www.cob.sjsu.edu/mease_d/newfreethrows.xls has updated free throw percentages for all NBA players. 1) Use Excel to compute the (population) mean and variance. 2) Draw some simple random samples of size n=4 and compute the sample means for each of these. 3) Compute the mean of these sample means. How does that compare that to the population mean? 4) Compute the variance of these sample means. How does that compare that to the population variance?

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48 The Sampling Distribution of the Sample Mean

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49 The Sampling Distribution of the Sample Mean If the original population has mean and standard deviation then the sampling distribution of the sample mean will have a mean of and a standard deviation of. Further, it will follow a normal distribution if 1) the original population has a normal distribution OR 2) the sample size n is large enough (bigger than 30) (Central Limit Theorem, top of page 270)

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50 In class exercise #58: (This is like #38 for your homework) Time spent using email per session is normally distributed with a mean of 8 minutes and a standard deviation of 2 minutes. If a random sample of 25 sessions is selected, what is the probability the mean will be between 7.8 and 8.2 minutes?

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