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Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer Chapter 6 Probability 1 Doing Statistics for Business Chapter 6 Objectives Basic Probability Rules Random Variables and Probability Distributions The Binomial Probability Distribution The Normal Probability Distribution 2 Doing Statistics for Business Probability is measure of how likely it is that something will occur. An Experiment is any action whose outcomes are recordable data. 3 Doing Statistics for Business The Sample Space is the set of all possible outcomes of an experiment. 4 Doing Statistics for Business TRY IT NOW! The Spinner Problem Writing out the Sample Space An experiment consists of spinning the different spinners pictured below: Write down the sample space for this experiment. 5 Doing Statistics for Business An event, A, is an outcome or a set of outcomes that are of interest to the experimenter. The probability of an event A, P(A), is a measure of the likelihood that an event A will occur. 6 Doing Statistics for Business TRY IT NOW! The Spinner Problem Classical Definition of Probability In the previous exercise you found the sample space for the spinner example to be: S = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C} Let A be the event that the first spinner lands on an odd number. Find P (A). Let B be the event that the second spinner is a vowel. Find P (B). 7 Doing Statistics for Business The complement of an event A, denoted A´, is the set of all outcomes in the sample space, S, that do not correspond to the event A. 8 Doing Statistics for Business The event A OR B describes when either A happens or B happens or they both happen. The event A and B is the event that A and B both occur. Two events A and B are said to be mutually exclusive if they have no outcomes in common. 9 Doing Statistics for Business TRY IT NOW! The Spinner Problem Calculating the Probability of A OR B The sample space for the experiment of spinning the two spinners is: S = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C} Let A be the event that the first spinner comes up a 1 and let B be the event that the first spinner comes up a 3. Find the probability that A OR B occurs using the sample space. 10 Doing Statistics for Business TRY IT NOW! The Spinner Problem Calculating the Probability of A OR B (con’t) Now find the same probability using the simple addition rule. Why are the two answers the same? 11 Doing Statistics for Business TRY IT NOW! Quality Problems Using the General Addition Rule The company that manufactures cardboard boxes collected data on the defect type and production shift. The data are summarized in the contingency table below: Shift Defect 1 2 3 Total Color 8 4 3 15 Printing 6 5 2 13 Skewness 0 2 0 2 Total 14 11 5 30 12 Doing Statistics for Business TRY IT NOW! Quality Problems Using the General Addition Rule (con’t) If a box has more than one defect, then it is classified by the more serious of the defects only. Suppose that a box from the sample is selected at random and examined more closely. What is the probability that the box has a color defect? What is the probability that the box was produced during the second shift? 13 Doing Statistics for Business TRY IT NOW! Quality Problems Using the General Addition Rule (con’t) Is it possible for the selected box to have a color defect and to have been produced on the second shift? If so, what is the probability? What is the probability that the selected box will have a color defect or will have been produced on the second shift? 14 Doing Statistics for Business The conditional probability of an event A given an event B is P(A|B) = P(A and B) P(B) 15 Doing Statistics for Business Two events are independent if the probability that one event occurs on any given trial of an experiment is not affected or changed by the occurrence of the other event. 16 Doing Statistics for Business A Random Variable, X, is a quantitative variable whose value varies according to the rules of probability. The Probability Distribution of a random variable, X, written as p (x), gives the probability that the random variable will take on each of its possible values. 17 Doing Statistics for Business TRY IT NOW! Defective Diskettes Finding Interval Probabilities A company that sells computer diskettes in bulk packages for a warehouse club outlet knows that the number of defective diskettes in a package is a random variable with the probability distribution given below: x 0 1 2 3 4 5 6 p(x) 0.30 0.21 0.12 0.10 0.10 0.09 0.08 18 Doing Statistics for Business TRY IT NOW! Defective Diskettes Finding Interval Probabilities (con’t) Find the probability that a package of the diskettes will contain at least 3 defective disks. Find the probability that the package will contain between 2 and 5 defective diskettes. Find the probability that the number of defective diskettes will be at most 2. 19 Doing Statistics for Business TRY IT NOW! Defective Diskettes Creating a Probability Histogram The company that sells computer diskettes in bulk packages for warehouse clubs would like to have a picture of how the number of defective diskettes in a package behaves. The probability distribution is given below: x 0 1 2 3 4 5 6 p(x) 0.30 0.21 0.12 0.10 0.10 0.09 0.08 20 Doing Statistics for Business TRY IT NOW! Defective Diskettes Creating a Probability Histogram (con’t) Create a probability histogram for the number of defective diskettes. Use the probability histogram to describe the distribution of the number of defective diskettes in a package. 21 Doing Statistics for Business A Binomial Random Variable is the number of successes in n trials or in a sample of size of n. 22 Doing Statistics for Business TRY IT NOW! Loan Defaults Recognizing a Binomial Random Variable While the Chamber of Commerce is concerned about the problems of small businesses, it must also be sensitive to the problems that the lending institutions have when issuing credit. One of the problems that banks have with small businesses is default on loan payments. It is estimated that approximately 20% of all small business with less than 50 employees are at least six months behind in loan payments. 23 Doing Statistics for Business TRY IT NOW! Loan Defaults Recognizing a Binomial Random Variable The Chamber of Commerce that surveyed the small businesses of a city wants to look at this problem in more detail. It finds that of the 1536 small businesses in the city, 965 have less than 50 employees. It randomly selects 25 of these small businesses, checks their credit histories and counts the number of companies in the sample of 25 that are at least six months behind in loan payments. Does this qualify as a binomial probability distribution? 24 Doing Statistics for Business TRY IT NOW! Loan Defaults Solving Binomial Probability Problems The Chamber of Commerce that is checking credit problems of small businesses estimated that 20% of all small businesses were at least six months behind in load payments. The Chamber of commerce took a random sample of 25 small businesses and counted that number of the businesses that were at least six months behind in loan payments. 25 Doing Statistics for Business TRY IT NOW! Loan Defaults Solving Binomial Probability Problems (con’t) Define a success for this problem. Describe the random variable, X, in words. Find the parameters of the binomial distribution for this problem. 26 Doing Statistics for Business TRY IT NOW! Loan Defaults Solving Binomial Probability Problems (con’t) Find the probability that in the sample of 25 businesses less than 6 were at least 6 months behind in loan payments. Find the probability that between 4 and 9 inclusive were at least 6 months behind in loan payments. 27 Doing Statistics for Business TRY IT NOW! Loan Defaults Calculating the Mean & Standard Deviation of a Binomial Random Variable The Chamber of Commerce that was looking at the loan defaults for small businesses wants to know the mean and standard deviation for the binomial random variable with n = 25 and = 0.20. Find the mean and standard deviation of the number of small businesses in 25 that will default on their loans. 28 Doing Statistics for Business Discovery Exercise 6.1 Exploring the Binomial Distribution Dear Mom and Dad: Send Cash According to USA Today, 70% of college students receive spending money from their parents when at school. For this exercise, you will need to simulate selecting 30 samples of 5 students from this population of college students and observe whether they receive spending money from their parents. Consider the successful outcome to be “receives money” with = 0.70 and the failure outcome to be “does not receive money.” 29 Doing Statistics for Business Discovery Exercise 6.1 Exploring the Binomial Distribution (con’t) If your instructor does not provide you with a method, you can take ten pieces (small) of paper and write an S on 7 of them and an F on 3 of them. Put the papers in a bag or other container and select one at random to simulate an observation. Note: Be sure to replace the the paper each time or will not always be 0.70. Record an S when you select a student who receives money from his/her parents and a F when you select a student who does not receive spending money from his/her parents. 30 Doing Statistics for Business Discovery Exercise 6.1 Exploring the Binomial Distribution (con’t) For each sample, record the number of successes you sampled. In the last outcome compute a running estimate of . Remember that is the probability of a successful outcome. In this case, is known to be 0.70. Let’s see how close the estimate gets to 0.70 as the sample size increases. So, after the first sample is selected your estimate of is simply the number of successes divided by 5. After the second sample is selected, your estimate of is the number of successes in both samples divided by 10 and so forth. 31 Doing Statistics for Business Binomial Distribtution n = 10 and = Binomial Distribution n = 10 and = 0.20 0.50 0.400 0.300 0.400 0.300 p(x) 0.200 p(x) 0.200 0.100 0.100 0.000 0.000 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Number of Successes Number of Successes Binomial Distribution n = 10 and = 0.80 0.400 0.300 p(x) 0.200 0.100 0.000 0 1 2 3 4 5 6 7 8 9 10 Number of Successes Figure 6.3 Effects of changing when n is fixed 32 Doing Statistics for Business Binomial Distribution n = 5 and = 0.20 Figure 6.4 Effects of Changing n for a 0.500 0.400 fixed value of p(x) 0.300 0.200 0.100 0.000 0 1 2 3 4 5 Number of Successes Binomial Distribution n = 25 and = Binomial Distribution n = 10 and = 0.20 0.20 0.450 0.400 0.400 0.350 0.300 0.300 0.250 p(x) 0.200 p(x) 0.200 0.150 0.100 0.100 0.050 0.000 0.000 0 2 4 6 8 10 12 14 16 18 20 22 24 0 1 2 3 4 5 6 7 8 9 10 Number of Statistics Number of Successes 33 Doing Statistics for Business Binomial Distribution for n = 10 and = 0.50 Binomial Distribution for n = 25 and = 0.50 0.25 1.60E-01 1.40E-01 0.2 1.20E-01 1.00E-01 0.15 8.00E-02 p(x) 0.1 p(x) 6.00E-02 4.00E-02 0.05 2.00E-02 0.00E+00 0 0 2 4 6 8 10 12 14 16 18 20 22 24 0 1 2 3 4 5 6 7 8 9 10 Number of Successes Number of Successes Binomial Distribution for n = 100 and = 0.50 Binomial Distribution for n = 50 and = 0.50 8.00E-02 1.20E-01 7.00E-02 1.00E-01 6.00E-02 8.00E-02 5.00E-02 p(x) 4.00E-02 6.00E-02 p(x) 3.00E-02 4.00E-02 2.00E-02 2.00E-02 1.00E-02 0.00E+00 0.00E+00 0 3 6 9 0 6 12 15 18 21 24 27 30 33 36 39 42 45 48 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 Number of Successes Number of Successes Figures 6.5 Binomial Distribution for large values of n 34 Doing Statistics for Business A Probability Density Function, f(x), is a smooth curve that represents the probability distribution of a continuous random variable. 35 Doing Statistics for Business Figure 6.6. Probability Distribution for a Continuous Random Probability Density Function for a Continuous Random Variable Variable. f(x) Values of X 36 Doing Statistics for Business Figure 6.7. Probability represented by an area under the curve P(x1 < X < x2) x1 x2 37 Doing Statistics for Business For a Normal Random Variable, the parameter is the mean of the normal random variable, X, and is the standard deviation. 38 Doing Statistics for Business Figure 6.8. Normal Probability Curve Normal Curve f(x) Values of X 39 Doing Statistics for Business Normal Curves with Equal Means and Different Standard Normal Curves with Different Means and Equal Standard Deviations Deviations 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Values of X 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Values of X Figure 6.9 Probability represented by an area under the curve 40 Doing Statistics for Business TRY IT NOW! Food Expenditures Looking at the Normal Curve The amount of money that a person working in a large city spends each week for lunch is a normally distributed random variable. For professional and management personnel the random variable has a mean of $35 and a standard deviation of $5. For hourly employees the mean is $30 with a standard deviation of $2. Sketch the normal curves for each of the two random variables on the same graph. 41 Doing Statistics for Business A Z Random Variable is normally distributed with a mean of 0 and a standard deviation of 1, Z ~ N(0,1). A Standard Normal Table is a table of probabilities for a Z random variable. 42 Doing Statistics for Business TRY IT NOW! Speed Reading Translating from X to Z The number of pages of a statistics textbook that a student can read in a given hour is a normally distributed random variable with a mean of 7 pages and a standard deviation of 1.5 pages. One of the professors who uses the book wants to know the probability that a randomly selected student can read more than 8.5 pages of the textbook in an hour. Draw a picture that depicts the problem to be solved and find the Z values necessary to solve the problem. 43 Doing Statistics for Business Figure 6.10 Probability given by the standard normal table 44 Doing Statistics for Business Second Decimal Place z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 Figure 6.11 The Standard Normal Table 45 Doing Statistics for Business Figure 6.12 Comparison of Upper and Lower Probabilities P(Z < -1) P( Z > 1) 46 Doing Statistics for Business Figure 6.13 Finding the area between two Z values 0.9772 0.0228 47 Doing Statistics for Business TRY IT NOW! The Standard Normal Table Using the Table to Find Probabilities For each of the following question, draw a picture of what you are trying to find BEFORE you use the table to find it. Find the probability that a Z random variable takes on a value that is less than 2.74. Find the probability that a Z random variable is greater than 0.85. 48 Doing Statistics for Business TRY IT NOW! The Standard Normal Table Using the Table to Find Probabilities (con’t) Find the probability that Z is between -1.36 and 1.87. 49 Doing Statistics for Business TRY IT NOW! Speed Reading Solving Normal Probability Problems The instructor who is interested in how many pages of the statistics textbook that students can read in an hour knows that the random variable is N(7, 1.5). Find the probability that a student could read more than 11.5 pages in an hour. 50 Doing Statistics for Business TRY IT NOW! Speed Reading Solving Normal Probability Problems (con’t) The instructor was worried about the percentage of students who could not finish reading a 5-page section in the given hour. What percentage of the students is this? 51 Doing Statistics for Business Top p% (p known) X Figure 6.14 The “inverse” normal probability problem 52 Doing Statistics for Business 10% or 0.1000 Figure 6.15 Bottom 10% of the normal distribution 53 Doing Statistics for Business z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 -1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 -1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 -1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 -1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 -1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 -1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 -0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 Figure 6.16 Normal Probability Table Containing the Probability 0.1000 54 Doing Statistics for Business TRY IT NOW! Speed Reading Solving the Inverse Problem The instructor who is interested in how fast students can read the statistics textbook would like to identify the bottom 25% of the class, in terms of the number of pages that they can read in an hour. Find the number of pages per hour that defines the bottom 25% of the students. 55 Doing Statistics for Business Calculating Binomial Probabilities in Excel 1. Position the cursor in an empty cell in the worksheet. 2. From the main toolbar, click on the Function Wizard icon. The Paste Function dialog box opens. Highlight Statistical for the function category and BINOMDIST for the function. Click OK and the dialog box for the BINOMDIST function opens. 3. Place the cursor in the textbox labeled Number_s. Type in the number “25.” 56 Doing Statistics for Business Calculating Binomial Probabilities in Excel (con’t) 4. Place the cursor in the textbox labeled Trials and enter “50.” 5. Place cursor in the textbox labeled Probability_s. Type in “0.45.” 6. The last textbox lets you indicate what kind of probability you want. In this case we want P(X = 25), so set this value to False. 7. Click on OK and the probability - 0.087330027 - will appear in the cell which you started. 57 Doing Statistics for Business Calculating Normal Probabilities in using KaddStat 1. Fill in the mean, , and the standard deviation, , of the random variable in the appropriate textboxes. 2. From the section labeled Range of Interest, select the type of probability you want. For left and right tail probabilities, you will enter one value of X in the textbox labeled Value. If you select Left and Right, either inside or outside,the box will change and you will enter two values of X. 58 Doing Statistics for Business Calculating Normal Probabilities in KaddStat (con’t) 3. To solve the inverse normal problem, select Inverse Value type in the left tail area in the box labeled Cumulative Probability. 4. Fill in where you want the output to go and click OK to obtain the results. 59 Doing Statistics for Business Generating Binomial Random Data in Excel 1. Select Binomial from the list of distributions and the dialog box will change to allow input of the appropriate parameters. 2. In the textbox for Number of Variables: type “1” and in the textbox for Number of Random Numbers: type “20.” 3. In the textbox labeled p Value enter the value for 0.70 & in the textbox for Number of Trials: enter “20.” 4. Specify where you want the output to appear and click OK. 60 Doing Statistics for Business Figure 6.25 The Binomial Random Variable Dialog Box 61 Doing Statistics for Business Figure 6.26 Random Data from Binomial Distribution 62 Doing Statistics for Business Chapter 6 Summary In this chapter you have learned: Probability is more than just flipping coins, spinning spinners, or gambling. It is important in the study and development of statistics. Probability is the bridge between Descriptive Statistics and Inferential Statistics. In Descriptive Statistics we use different techniques to describe sample data. 63 Doing Statistics for Business Chapter 6 Summary (con’t) In Inferential Statistics we will make test hypotheses about the populations from which these samples came. Probability is the tool that allows us to reconcile what happened (descriptive) with what we think is true by determining how likely the outcomes of the experiment we perform are. 64

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