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conditions in the long-run. Measuring Probabilities While probability is considerably more complicated than we‟ll let on, the basic idea is that a probability can be calculated by considering the number of times some event occurs relative to the total number of “trials,” or observable situations. In simpler terms, it is the number of “successes” out of the total number of trials. Calculating Probability The probability that event occurs, denoted ( ), is the ratio (or fraction) of successes divided by the number of trials. Mathematically, we write the number of times occurs by ( ) and the total number of trials as ( ). That is, ( ) ( ) ( ) This formula works when all elements in the sample space are equiprobable, that is, each individual outcome in the sample space has the same probability of occurring as any other outcome. As a note the () notation stands for “the number of ways” the event in parenthesis can occur. The in the denominator stands for sample space or the total number of things/situations/trials being considered in the experiment. Example 2: In a 2009 study of high-fructose corn syrup (HFCS), a corn-based sweetener used in a wide variety of foods, beverages, and condiments, 20 samples of HFCS were analyzed. Of those, nine of them were found to contain mercury by researchers. Based on the results of this study, find the probability that a random sample of HFCS contains mercury and explain what this result means. SOURCE: http://www.washingtonpost.com/wpdyn/content/article/2009/01/26/AR2009012601831.html SOLUTION: The event in this scenario is that mercury is found. Out of the total 20 trials, nine of them contained mercury. Therefore, ( ) Statistics for Decision-Making in Business © Milos Podmanik Page 84
This means that if samples of HFCS were to be sampled randomly and repeatedly, it would be found that 45% of all samples would contain traces of mercury. This does not guarantee that exactly 45 samples out of 100 will contain mercury. Example 3: In July 2011, temperatures in Gilbert, Arizona were above 100 every day (SOURCE: www.weather.com). Based on this data, a researcher concludes that the probability of above 100 temperatures in Arizona is 100%. Comment on his findings. SOLUTION: Since temperatures in July 2011 were above 100 31 days of the 31 days in the month, it is fair to make the experimental observation that approximately 100% of all days in July 2011 have temperatures exceeding 100 , in the long-run (there have been days in the past when temperatures were below 100 ); However, because we know that temperatures are periodic, or that they go from low to high and back to low over the course of a year, 100% is not a good estimate for temperatures in Arizona, in general (temperatures are reasonably never above 100 in January!). This example truly stresses the importance of critical thinking when using probabilities. It is often that probabilities are used and abused in the media, education, and in politics, just to name a few. We want to make sure that we are as specific as possible. It will often be considerably helpful to display probabilities in a tabular form, that is, through the use of tables. This type of table is called a contingency table. This not only helps to organize data, but to simultaneously see the big picture. Let‟s consider an example. Example 4: In a 1950 study that considered 1,418 hospital patients in London (half of each) with and without lung cancer and whether or not they smoked over the course of their lives, the following was found: Smoker/Lung Cancer Yes No Yes 688 650 No 21 59 Assuming this data can be used as a representation of the entire population of London residents, analyze the data by discussing the following: a. What is the probability that a randomly selected participant within this study develops lung cancer b. Provided that a person was a smoker, what is the probability that he has lung cancer c. Provided that a person was not a smoker, what is the probability that he has lung cancer d. Given that a person has lung cancer, what is the probability that he smokes SOLUTION: Statistics for Decision-Making in Business © Milos Podmanik Page 85
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Statistics for Decision- Making in
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A Note to Students This book is far
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6.3 Confidence Interval for ̂ 202
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which has undergone no additional r
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of transporting such an animal. Not
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Although far from last, we will con
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Account Type Revenue ($) New $5,296
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For the present time, we‟ll proce
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Row Labels Average of Revenue ($) M
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(SOURCE: Essentials of Modern Busin
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1.3Statistics in Excel When conduct
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We can make it a bit more presentab
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For measures such as the range, Exc
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We get: We can do the same for Old.
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Chapter 2 Visual Representations of
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We now have to enter a formula for
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Page 35 and 36: Alternatively, it is possible to in
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Page 39 and 40: Selected the copied graph. In Chart
Page 41 and 42: 2. The following data represents pe
Page 43 and 44: Frequency 2.2 Visualizing Quantitat
Page 45 and 46: You can either choose to have Excel
Page 47 and 48: If we had additional variables, the
Page 49 and 50: Select PivotChart and select the fi
Page 51 and 52: Frequency To make solid black lines
Page 53 and 54: Relative Frequency Histogram of Cal
Page 55 and 56: a. Using Excel, find the mean and r
Page 57 and 58: ̅ ̅ account for this discrepancy,
Page 59 and 60: 2.3.2 Percentile Another useful too
Page 61 and 62: 2.3.4 Rank What if, on the other ha
Page 63 and 64: Check the “Analysis ToolPak” an
Page 65 and 66: You‟ll immediately notice that a
Page 67 and 68: ̅ 2.4 Descriptive Statistics - Var
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Page 71 and 72: This gives us a nice measure of how
Page 73 and 74: We can immediately see the mean and
Page 75 and 76: Lastly, a normal curve is the most
Page 77 and 78: Frequency ̅ Histogram of TV's Owne
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Page 81 and 82: Chapter 3 Probability and Decision
Page 83: Proportion of Rainy Days 1.2 Propor
Page 87 and 88: Are we confident in accusing a lung
Page 89 and 90: 3.2 Joint Probability In the previo
Page 91 and 92: Company 1 Choices Y Y Y Y Y Y Y Y N
Page 93 and 94: There is less than a 1% chance that
Page 95 and 96: We can now see that the situation i
Page 97 and 98: sequence, one ball is chose at rand
Page 99 and 100: 3.3 Probability of Unions Imagine t
Page 101 and 102: Physical .20 .10 .30 Mental .40 .30
Page 103 and 104: SOLUTION: We first arrange this inf
Page 105 and 106: Homework Problems - 3.3 1. A gaming
Page 107 and 108: 3.4 Conditional Probability In many
Page 109 and 110: ) : The Arizona Cardinals make it t
Page 111 and 112: We could just as well have written,
Page 113 and 114: That is, the probability that the c
Page 115 and 116: This can happen in one of two ways:
Page 117 and 118: f. All numerical red cards are remo
Page 119 and 120: 3.5 Combinations and Permutations R
Page 121 and 122: 1 st Bushel 2 nd Bushel U1 U1 U2 U2
Page 123 and 124: Which, in its final state gives: Th
Page 125 and 126: Combination - Order Does NOT Matter
Page 127 and 128: ( ) There is only a .9% chance that
Page 129 and 130: 1. Are repeats/replacements allowed
Page 131 and 132: Object 4 Object 5 Also notice that
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3.6 Expected Value Imagine that you
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An expected value is actually not s
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Where The expected value is: , - (
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SOLUTION: We first note that there
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Probability 0.7 0.6 0.5 0.4 0.3 0.2
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Statistics for Decision-Making in B
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For men, we see that the most frequ
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So, we had 10 trials and wanted to
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Example 1: A fair-two sided coin is
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( ) ( ) ( ) ( ) Example 2: A fair,
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Expected Value of a Binomial Random
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f. What is the probability that g.
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Probability For instance, we see th
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Density 0.25 Time Speng Waiting in
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Suppose we wish to find ( ), that i
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Density Resulting in a standard dev
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Density Without going into detail h
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a. What is the probability that the
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Statistics for Decision-Making in B
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As you can see, this is a difficult
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We wish to know the area of the sha
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We can easily find that the probabi
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In the second section, we can check
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Chapter 6 Sampling Distributions an
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Thus, the standard deviation would
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1.7 to 1.8 1.8 to 1.9 1.9 to 2 2 to
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2.4 to 2.5 2.5 to 2.6 2.6 to 2.7 2.
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The Central Limit Theorem (CLT) has
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can set this up in our applet by ha
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3) We cannot use this sample to cal
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Thus, we need to find the lower and
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We see that the standard deviation
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√ The lower limit is: √ And the
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following data was collected on the
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{ So, we have a set of twenty 1‟s
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DULY CAUTIONED: The assumptions her
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Homework Problems -6.2 1. In a samp
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Since this is a mathematical questi
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Example 3: Many older homes have el
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Hypothesis Test Conclusion Test Say
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Hypothesis Test Conclusion Since yo
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SOLUTION: d. A generic conclusion s
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9. Based on the “Structure of a H
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1.2 Descriptive VS. Inferential Sta
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Relative Frequency 35% 30% 25% 20%
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Symmetric: 35 30 25 20 15 10 5 0 10
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Repair Cost Mean 971 Standard Error
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el freq Nitrous Oxide (thous. Tons)
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Probability Pizza Size Distribution
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e. ( ) f. ; the average response ti
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2. a. The long-run proportion of al
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e. 20% of all children are expected
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e. The middle 50% score between abo
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5. b. This might indicate that the
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̅ We have that ̅ and √ . So our
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6. This is the probability that we
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