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

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22.2666666666667 to 22.7666666666667 22.7666666666667 to 23.2666666666667 23.2666666666667 to 23.7666666666667 23.7666666666667 to 24.2666666666667 24.2666666666667 to 24.7666666666667 24.7666666666667 to 25.2666666666667 25.2666666666667 to 25.7666666666667 25.7666666666667 to 26.2666666666667 26.2666666666667 to 26.7666666666667 26.7666666666667 to 27.2666666666667 27.2666666666667 to 27.7666666666667 27.7666666666667 to 28.2666666666667 28.2666666666667 to 28.7666666666667 28.7666666666667 to 29.2666666666667 29.2666666666667 to 29.7666666666667 29.7666666666667> Now, back to our example… If we have truly collected a random sample, then we should be able to think about the sample as a small population. If this is a small population, then we should be able to sample from it. We will draw random samples of size from the small “population” which is also of size . Sounds strange, but we will sample with replacement, so it is possible to resample the same value multiple times. We will draw 1,000 samples of size from this “population” and, as you might have figured, we will calculate the mean of each and build the sampling distribution for ̅. 200 180 160 140 120 100 80 60 40 20 0 Sampling Distribution of x-bar As we should expect based on CLT, the distribution of these 1,000 means is approximately normal. Let‟s suppose that we want to have an interval within which there is a 95% probability that the true population mean, , lies. This is the same as looking for the middle 95% of means! Statistics for Decision-Making in Business © Milos Podmanik Page 194
Thus, we need to find the lower and upper limits for this interval by finding the 2.5 percentile and the 97.5 percentile. In Excel, we can do this by using the percentile() function. We get: Upper (97.5 percentile): 27.50 Lower (2.5 percentile): 23.60 Thus, we can say that we are 95% confident that the true population mean is between 23.6 years and 27.5 years. In other words, there is a 95% probability that we have “trapped” the population mean between our lower and upper limit. Said one other way, 95% of all sample means, when the variability from sample to sample is taken into account, are between these lower and upper limits. If this is representative of the population, then we should believe that 95% of the time, we will have means between these two values. What if we wanted to be 99% certain We would need to find lower and upper limits so that there is only 1% in the tails: Thus, we would like 0.01/2 = 0.005 (or .5%) in each of the two tails. To find the lower and upper limits, we would need to find the 0.005 percentile and the 1-0.005 = 0.995 percentile. We get: Upper (97.5 percentile): 28.17 Lower (2.5 percentile): 22.83 Thus, we are 99% confident that the true population mean age, , is between 22.83 years and 28.17 years. In other words, there is a 99% probability that the true mean age is between 22.83 and 28.17 years. If we want to be more confident, we need to expand our interval of values! Note that in only one of our confidence intervals (99%), we have captured the true mean within our range. This is very likely, since our confidence percentage is very high. BUT, keep in mind that we never know what the true mean is! Thus, we cannot say that it would have been better to stick with the wider 99% interval. After all, there is a 1% chance we might have made an error. The level of confidence that we desire depends on the situation and the allowable mean width we are willing to tolerate. More confidence means wider possibilities. In general, we never know Statistics for Decision-Making in Business © Milos Podmanik Page 195
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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: , - (
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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 and 192: can set this up in our applet by ha
Page 193: 3) We cannot use this sample to cal
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
Page 209 and 210: Since this is a mathematical questi
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
Page 229 and 230: el freq Nitrous Oxide (thous. Tons)
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
Page 237 and 238: e. 20% of all children are expected
Page 239 and 240: e. The middle 50% score between abo
Page 241 and 242: 5. b. This might indicate that the
Page 243 and 244: ̅ We have that ̅ and √ . So our
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6. This is the probability that we
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