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

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One major oversight of our excitement with this idea is the notion that we would actually know the true population mean, , and the true population standard deviation, . If we have limited information about our population, then we certainly would not know these values. In the next parts of this chapter, we will learn how to use our sample to make these predictions about the population. Though similar in conceptual nature, it is not as straightforward as replacing with ̅ and with . Homework Problems – 6.1 1. In your own words, what does the Central Limit Theorem tell us 2. In your own words, why is the Central Limit Theorem a very powerful practical result 3. A sample of size 36 is taken from a population distribution of unknown shape, though the mean is believed to be 100 with a standard deviation of 18. What is the probability that the sample mean is: a. Greater than 102 b. Less than 98 c. Between 95 and 105 d. Between what two values will the middle 90% of means be 4. A stained glass company produces panes of glass with a mean thickness of 0.42 inches and a standard deviation of 0.04 inches, if produced properly. Suppose a random sample of windows reveals a sample mean of 0.43. a. What is the probability of this average, or a larger average b. Given the probability you have computed, what can be said about recent production standards 5. Promote Marketing has a research team to research new marketing tactics to propose to potential clients. A group of 40 clients have been invited for a conference to be put on by the marketing firm. The research team usually generates in revenues for each member of the team with . a. What will be the shape of the distribution of ̅ How do you know b. What is the probability that average sales will exceed $420,000 for this particular event c. How would your answer change if 100 clients were to show up d. If the team (300 people) have an average revenue that is in the 90 th percentile of revenues, they will earn 4-days of paid vacation. What average sales would be required for this 6. A computer simulation reveals that a distribution of average incomes in a sample of 500 has a standard deviation of $130. What is the standard deviation for the population of all incomes Interpret the result you get in real-world terms. 7. Use the Excel Sampling Distribution Applet to address this problem. In a population, it is found that 30% of homes have 5 rooms, 40% have 4 rooms, and 30% have 3 rooms. You Statistics for Decision-Making in Business © Milos Podmanik Page 190
can set this up in our applet by having a “die” with 10 values: three 5‟s, four 4‟s, and three 3‟s. a. What is the average number of rooms a home has in this population What is the standard deviation in the number of rooms in this population b. Now, suppose you take a sample of size 30 from this population. What shape will the distribution have and how do you know c. Take 1,000 random samples each of size and compute the 1,000 sample means. According to the applet, what is the average of the average rooms in the sample What is the standard deviation in the average number of rooms in a house Compare these two results to what the Central Limit Theorem says we should come up with. That is, find , ̅- and , ̅-. d. Take 1,000 random samples each of size and compute the 1,000 sample means. According to the applet, what is the average of the average rooms in the sample What is the standard deviation in the average number of rooms in a house Compare these two results to what the Central Limit Theorem says we should come up with. That is, find , ̅- and , ̅-. e. Take 1,000 random samples each of size and compute the 1,000 sample means. According to the applet, what is the average of the average rooms in the sample What is the standard deviation in the average number of rooms in a house Compare these two results to what the Central Limit Theorem says we should come up with. That is, find , ̅- and , ̅-. f. Why do the values in the population have the highest standard deviation when compared with the distribution of means in the last there parts g. What is the probability that, in a sample of 100 homes, the average number of rooms is greater than 5 h. Explain in practical terms why the standard deviation of any ̅ distribution decreases as the sample size increases. 6.2 Confidence Interval for ̅ 6.2.1 Confidence Interval for ̅ Using Sampling Distributions As discussed previously, our ultimate goal is to make inferences about the population parameter . Again, keep in mind that this is the only reason why we are spending time on this! Otherwise, we would have completed our semester early! When we generate our sampling distribution for ̅ we see very vividly that our sample means are subject to sampling variability, depending on which “die values” are “rolled” for each individual sample of size . Thus, we should be very skeptical of concluding that ̅ is representative of the true population mean. However if we have many, many “individuals roll the die,” we should get a fairly reasonable understanding of a range of values for the true value of . Let‟s consider an example. Suppose we want to better understand a population of ages of people in a town. Statistics for Decision-Making in Business © Milos Podmanik Page 191
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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
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
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: The Central Limit Theorem (CLT) has
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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 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
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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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