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Probability (Relative Frequency) Chapter 5 Continuous Probability Distributions Up until this point, we have only considered distribution that have discrete values – non-negative integers. There are many variables, however, that are continuous in nature. In fact, almost every variable you studied in algebra and calculus was continuous! Take, for example, heights of NBA basketball players, hourly wage, response time of a database server, temperature, depth of a lake, the value of a share of Intel stock, and the lifespan of a car engine, to name just a very few. These are all variables that can take on infinitely many values, even within a limited range. For example, the response time of a database could be 0 seconds and 1 second. It could be 0.01 seconds, 0.00001 seconds, or 0.98727495 seconds. 5.1 The Ideas Behind the Continuous Distribution 5.1.1 Conceptual Approach to Continuous Distributions Think back to a discrete distribution. The probability of a particular value was found by observing the height of the relative frequency bar. While relative frequency represents the percentage of observations found to have the value specified, it can also be thought of as a probability, if we feel that it accurately models predictions that we might use it for. Consider the example below showing the number of children in a classroom of 30 that are likely to likely to have the flu. Number of Children with Flu in a Class 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.4 0.2 0.14 0.16 0.1 0.1 0 1 2 3 4 5 Number of Children w/Flu Statistics for Decision-Making in Business © Milos Podmanik Page 158
Probability For instance, we see that the probability that any 2 children in a classroom have the flu is 0.2. Let‟s call this random variable # of children in a classroom of 30 that have the flu. Then, we will write the probability that any 2 children have the flu as: ( ) This reads, “the probability that the number of children that have the flue is 2” The output of this statement is: ( ) What would it mean to say ask: What is ( ) This is asking us to find the probability that 2 or fewer children have the flu. In other words, what is the probability that 0, 1, or 2 children have the flu. To answer this, we simply add the bar heights corresponding to . ( ) Thus, there is a 74% chance that 2 or fewer children in a class of 30 children have the flu. With continuous distributions, we cannot simply read the “height of the bar!” For instance consider the following continuous probability distribution that shows the likelihood of various wait times in line at a fast-food restaurant: 0.25 0.2 0.15 Time Speng Waiting in Line 0.1 0.05 0 0 1 2 3 4 5 Minutes Statistics for Decision-Making in Business © Milos Podmanik Page 159
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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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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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We know from the data that this val
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Alternatively, it is possible to in
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16 14 12 10 8 6 Series1 4 2 0 SD D
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Selected the copied graph. In Chart
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2. The following data represents pe
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Frequency 2.2 Visualizing Quantitat
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You can either choose to have Excel
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If we had additional variables, the
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Select PivotChart and select the fi
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Frequency To make solid black lines
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Relative Frequency Histogram of Cal
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a. Using Excel, find the mean and r
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̅ ̅ account for this discrepancy,
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2.3.2 Percentile Another useful too
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2.3.4 Rank What if, on the other ha
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Check the “Analysis ToolPak” an
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You‟ll immediately notice that a
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̅ 2.4 Descriptive Statistics - Var
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√ ∑( ̅) √ √ This is what w
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This gives us a nice measure of how
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We can immediately see the mean and
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Lastly, a normal curve is the most
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Frequency ̅ Histogram of TV's Owne
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58.3 34.6 35.5 45.4 38.6 63.8 53.9
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Chapter 3 Probability and Decision
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Proportion of Rainy Days 1.2 Propor
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This means that if samples of HFCS
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Are we confident in accusing a lung
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3.2 Joint Probability In the previo
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Company 1 Choices Y Y Y Y Y Y Y Y N
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There is less than a 1% chance that
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We can now see that the situation i
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sequence, one ball is chose at rand
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3.3 Probability of Unions Imagine t
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Physical .20 .10 .30 Mental .40 .30
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SOLUTION: We first arrange this inf
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Homework Problems - 3.3 1. A gaming
Page 107 and 108: 3.4 Conditional Probability In many
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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
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Page 139 and 140: Where The expected value is: , - (
Page 141 and 142: SOLUTION: We first note that there
Page 143 and 144: Probability 0.7 0.6 0.5 0.4 0.3 0.2
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Page 147 and 148: For men, we see that the most frequ
Page 149 and 150: So, we had 10 trials and wanted to
Page 151 and 152: Example 1: A fair-two sided coin is
Page 153 and 154: ( ) ( ) ( ) ( ) Example 2: A fair,
Page 155 and 156: Expected Value of a Binomial Random
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Page 163 and 164: Suppose we wish to find ( ), that i
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Page 169 and 170: a. What is the probability that the
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Page 173 and 174: As you can see, this is a difficult
Page 175 and 176: We wish to know the area of the sha
Page 177 and 178: We can easily find that the probabi
Page 179 and 180: In the second section, we can check
Page 181 and 182: Chapter 6 Sampling Distributions an
Page 183 and 184: Thus, the standard deviation would
Page 185 and 186: 1.7 to 1.8 1.8 to 1.9 1.9 to 2 2 to
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Page 189 and 190: The Central Limit Theorem (CLT) has
Page 191 and 192: can set this up in our applet by ha
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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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6. This is the probability that we
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