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

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, ̅- √ For example, √ That is very close to the 0.52 we obtained! Similarly, for our sample of size 20, √ This one happens to be fairly spot-on! An finally, for our sample of size 30, This is again very close to our obtained 0.32! √ The reason for this difference is simply due to randomness, and estimates can be improved more (if desired) by increasing the number of “individuals rolling the die.” What we have observed here is formally known as the Central Limit Theorem. Central Limit Theorem Regardless of the distribution of a random variable, , if we take repeated random samples from this distribution of and compute the mean, ̅, for each sample, then the following will hold: 1.) The distribution of ̅ will be approximately normal 2.) , ̅- 3.) , ̅- √ (NOTE: A sample size of at least 30 is a rule-of-thumb and can vary slightly depending on the severity of skews and abnormalities in the distribution. For even severely skewed distributions, the approximate shape is typically normal.) 6.1.3 Why the Central Limit Theorem Statistics for Decision-Making in Business © Milos Podmanik Page 188
The Central Limit Theorem (CLT) has some very powerful, but subtle results. First of all, we do not need to understand the shape of the underlying distribution from which we are sampling. This is an amazing result in-and-of itself, since we usually have little to know information about the population itself (again, if we did, we wouldn‟t be wasting our time with any of this!). Secondly, since the resulting sampling distribution is approximately normally distributed, we can proceed to calculate probabilities using the normal distribution. This is also great, since we already have the background in that process! Example 1: After experimentation, researchers believe that the mean lifespan of a strain of bacteria is days with days. Due to the complexity of the bacteria, the shape of the distribution of bacteria lifespans is unknown. A sample of 60 bacteria strains is collected. a. Does the CLT apply here b. Calculate the probability that the sample mean lifespan, ̅, is less than 3 days. SOLUTION: a. Since the sample size is 60, we should be safe in assuming that the sampling distribution of all means is normally distributed with mean and standard deviation √ . b. We want ( ). Using our probability calculator Given the very small level of variability in the sampling distribution of lifespan means, we would consider observing an average smaller than 3 feasibly 0. 6.1.4 Limitations of the CLT Statistics for Decision-Making in Business © Milos Podmanik Page 189
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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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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
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
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Page 191 and 192: can set this up in our applet by ha
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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
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Page 207 and 208: Homework Problems -6.2 1. In a samp
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Page 213 and 214: Hypothesis Test Conclusion Test Say
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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%
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Page 227 and 228: Repair Cost Mean 971 Standard Error
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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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