Statistics for Decision- Making in Business - Maricopa Community ...

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1 1 18 22 25 27 30 18 21 2 3 19 20 32 20 25 29 32 33 40 29 25 29 24 23 29 29 26 27 1 31 32 31 31 35 33 30 32 31 33 19 20 22 21 20 20 19 22 22 9 23.46 9.250319 But, wait! Let‟s pretend that we actually don‟t have access to the entire population of values (yes, we clearly see them in the table above, but we normally do not have that luxury). Due to limited time and money, you are only able to sample 30 of these values. After taking a random sample, here is what you have chosen: 32 31 31 35 19 20 22 21 20 20 20 25 29 32 33 19 19 19 18 22 25 27 30 18 21 33 30 32 31 33 ̅ 25.56667 5.870342 Again, at this point, we would have no way of telling how close we are to the actual mean of 23.46. To get a good estimate of , we will come up with a confidence interval. A confidence interval is a range of values such that there is an probability that the true population mean, , is between those values. How do we calculate this Here is our motivation for what is to come: There are two ways to think about inferential statistics: 1) Use theoretical results and make conclusions using them 2) Build a sampling distribution for the statistic of choice ( ̅ or ̂) using the Bootstrap Method and make conclusions using this empirical data. We will draw parallels between the two regularly. Here is the basic idea of Bootstrap Sampling: 1) From the population, take a random sample, preferably of size 30 or greater. The larger the random sample, the more power we have in making inferences about the population. 2) If this is a truly representative sample, then we can think of it as a “mini” population that acts and behaves according to the population as a whole. This is a key ingredient! Statistics for Decision-Making in Business © Milos Podmanik Page 192
3) We cannot use this sample to calculate the corresponding parameter because of sampling variability. However, if this sample behaves like the population, then we can resample from it and get an idea of the overall variability. That is, draw a sample of the same sample size from this “mini” population, but do so with replacement. This is the same idea as rolling a die a fixed number of times – we are sampling with replacement from the population 1,2,3,4,5, 6. What will this do It will account for sampling variability, if repeated. 4) Calculate the statistic from this sample and record it. 5) Repeat steps 3) and 4) 1,000 to 10,000 times. We now have a sampling distribution and can make estimates about the true population parameter. And, guess what this distribution will look like You guessed it – it will be approximately normal, by the Central Limit Theorem. Below is a diagrammatic representation of steps 1) – 5): Sample 1 Sample 2 Sample 3 Population Random Sample, Sample 4 . . . Sample 10,000 Some of the assumptions we make are indeed dangerous. For example, do we really have a mini population If the answer is “no,” then theoretical results are equally worthless since they, too, assume that the sample is representative. Statistics for Decision-Making in Business © Milos Podmanik Page 193
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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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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
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3.4 Conditional Probability In many
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) : The Arizona Cardinals make it t
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We could just as well have written,
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That is, the probability that the c
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This can happen in one of two ways:
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f. All numerical red cards are remo
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3.5 Combinations and Permutations R
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1 st Bushel 2 nd Bushel U1 U1 U2 U2
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Which, in its final state gives: Th
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Combination - Order Does NOT Matter
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( ) There is only a .9% chance that
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1. Are repeats/replacements allowed
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Object 4 Object 5 Also notice that
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e. 2. Your classmate was absent whe
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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: , - (
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
Page 145 and 146: Statistics for Decision-Making in B
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
Page 157 and 158: f. What is the probability that g.
Page 159 and 160: Probability For instance, we see th
Page 161 and 162: Density 0.25 Time Speng Waiting in
Page 163 and 164: Suppose we wish to find ( ), that i
Page 165 and 166: Density Resulting in a standard dev
Page 167 and 168: Density Without going into detail h
Page 169 and 170: a. What is the probability that the
Page 171 and 172: Statistics for Decision-Making in B
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
Page 187 and 188: 2.4 to 2.5 2.5 to 2.6 2.6 to 2.7 2.
Page 189 and 190: The Central Limit Theorem (CLT) has
Page 191: can set this up in our applet by ha
Page 195 and 196: Thus, we need to find the lower and
Page 197 and 198: We see that the standard deviation
Page 199 and 200: √ The lower limit is: √ And the
Page 201 and 202: following data was collected on the
Page 203 and 204: { So, we have a set of twenty 1‟s
Page 205 and 206: DULY CAUTIONED: The assumptions her
Page 207 and 208: Homework Problems -6.2 1. In a samp
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Page 211 and 212: Example 3: Many older homes have el
Page 213 and 214: Hypothesis Test Conclusion Test Say
Page 215 and 216: Hypothesis Test Conclusion Since yo
Page 217 and 218: SOLUTION: d. A generic conclusion s
Page 219 and 220: 9. Based on the “Structure of a H
Page 221 and 222: 1.2 Descriptive VS. Inferential Sta
Page 223 and 224: Relative Frequency 35% 30% 25% 20%
Page 225 and 226: Symmetric: 35 30 25 20 15 10 5 0 10
Page 227 and 228: Repair Cost Mean 971 Standard Error
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Page 231 and 232: Probability Pizza Size Distribution
Page 233 and 234: e. ( ) f. ; the average response ti
Page 235 and 236: 2. a. The long-run proportion of al
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Page 239 and 240: e. The middle 50% score between abo
Page 241 and 242: 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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