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The idea of a -score is quite helpful, in that it tells us how far it is from the mean, relative to the size of the standard deviation (the average spread). If is very close to 0, then the score is not far from the mean. If it is very large, it is very far from the mean. A very useful theorem established by Russian mathematician, Lvovich Chebyshev, allows us to determine how large is very large. Chebyshev established the following theorem: Chebyshev’s Theorem For any , at least . / of the data values must be within (to the left and the right) standard deviations of the mean, for any. This works for any and all distributions. Example 3: A data value is 3 standard deviations above the mean. Is this an extreme value SOLUTION: Chebyshev‟s Theorem states that 89% of all data points in this distribution will lie between -3 and +3 standard deviations from the mean. Thus, there is, at most, an 11% chance of observing something higher than +3 standard deviations. This data value is fairly unlikely an might be considered a mild outlier. Homework Problems - 2.4 1. The Connecticut Agricultural Experiment Station conducted a study of the calorie content of different types of beer. The calorie content (calories per 100 mL) for 26 brands of light beer are: 29 28 33 31 30 33 30 28 27 41 39 31 29 23 32 31 32 19 40 22 34 31 42 35 29 43 a. Find the standard deviation. Explain the real-world meaning of this value. b. Find the interquartile range. Explain the real-world meaning of this value. c. Find the skewness. What type of shape does this distribution have 2. The UNICEF report “Progress for Children” (April, 2005) included the accompanying data on the percentage of primary-school-age children who were enrolled in school for 23 countries in Central Africa. Statistics for Decision-Making in Business © Milos Podmanik Page 78
58.3 34.6 35.5 45.4 38.6 63.8 53.9 61.9 69.9 43 85 63.4 58.4 61.9 40.9 73.9 34.8 74.4 97.4 61 66.7 79.6 98.9 a. Find the range, standard deviation, and interquartile range. Explain what these three values tell us about the shape of the distribution. b. Explain the real-world meaning of the standard deviation and the interquartile range. c. Produce descriptive statistics for this dataset with the Analysis ToolPak in Excel. d. Is the distribution skewed If so, in which direction e. Create a relative frequency histogram. Describe any trends in the data. f. Is an observation of 79.6 an outlier Use Chebyshev‟s Theorem to justify your answer. 3. The article “Determination of Most Representative Subdivision” (Journal of Energy Engineering [1993]: 43-55) gave data on various characteristics of subdivisions that could be used in deciding whether to provide electrical power using overhead lines or underground lines. Data on the variable x = total length of streets within a subdivision (in feet) are as follows: 1280 5320 4390 2100 1240 3060 4770 1050 360 3330 3380 340 1000 960 1320 530 3350 540 3870 1250 2400 960 1120 2120 450 2250 2320 2400 3150 5700 5220 500 1850 2460 5850 2700 2730 1670 100 5770 3150 1890 510 240 396 1419 2109 a. Find the range, standard deviation, and interquartile range. Explain what these three values tell us about the shape of the distribution. b. Explain the real-world meaning of the standard deviation and the interquartile range. c. Produce descriptive statistics for this dataset with the Analysis ToolPak in Excel. d. Is the distribution skewed If so, in which direction e. Find the -score for the observation 79.6. Explain what your answer means in real-world terms. f. Create a relative frequency histogram. Is an observation of 79.6 an outlier Use Chebyshev‟s Theorem to justify your answer. 4. Using the five class intervals 100 to 120, 120 to 140, . . ., 180 to 200, devise a frequency distribution based on 70 observations whose histogram could be described as follows: a. symmetric b. bimodal c. positively (right) skewed d. negatively (left) skewed Statistics for Decision-Making in Business © Milos Podmanik Page 79
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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
Page 27 and 28: We get: We can do the same for Old.
Page 29 and 30: Chapter 2 Visual Representations of
Page 31 and 32: We now have to enter a formula for
Page 33 and 34: We know from the data that this val
Page 35 and 36: Alternatively, it is possible to in
Page 37 and 38: 16 14 12 10 8 6 Series1 4 2 0 SD D
Page 39 and 40: Selected the copied graph. In Chart
Page 41 and 42: 2. The following data represents pe
Page 43 and 44: Frequency 2.2 Visualizing Quantitat
Page 45 and 46: You can either choose to have Excel
Page 47 and 48: If we had additional variables, the
Page 49 and 50: Select PivotChart and select the fi
Page 51 and 52: Frequency To make solid black lines
Page 53 and 54: Relative Frequency Histogram of Cal
Page 55 and 56: a. Using Excel, find the mean and r
Page 57 and 58: ̅ ̅ account for this discrepancy,
Page 59 and 60: 2.3.2 Percentile Another useful too
Page 61 and 62: 2.3.4 Rank What if, on the other ha
Page 63 and 64: Check the “Analysis ToolPak” an
Page 65 and 66: You‟ll immediately notice that a
Page 67 and 68: ̅ 2.4 Descriptive Statistics - Var
Page 69 and 70: √ ∑( ̅) √ √ This is what w
Page 71 and 72: This gives us a nice measure of how
Page 73 and 74: We can immediately see the mean and
Page 75 and 76: Lastly, a normal curve is the most
Page 77: Frequency ̅ Histogram of TV's Owne
Page 81 and 82: Chapter 3 Probability and Decision
Page 83 and 84: Proportion of Rainy Days 1.2 Propor
Page 85 and 86: This means that if samples of HFCS
Page 87 and 88: Are we confident in accusing a lung
Page 89 and 90: 3.2 Joint Probability In the previo
Page 91 and 92: Company 1 Choices Y Y Y Y Y Y Y Y N
Page 93 and 94: There is less than a 1% chance that
Page 95 and 96: We can now see that the situation i
Page 97 and 98: sequence, one ball is chose at rand
Page 99 and 100: 3.3 Probability of Unions Imagine t
Page 101 and 102: Physical .20 .10 .30 Mental .40 .30
Page 103 and 104: SOLUTION: We first arrange this inf
Page 105 and 106: Homework Problems - 3.3 1. A gaming
Page 107 and 108: 3.4 Conditional Probability In many
Page 109 and 110: ) : The Arizona Cardinals make it t
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
Page 125 and 126: Combination - Order Does NOT Matter
Page 127 and 128: ( ) 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
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Probability 0.7 0.6 0.5 0.4 0.3 0.2
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Statistics for Decision-Making in B
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For men, we see that the most frequ
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So, we had 10 trials and wanted to
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Example 1: A fair-two sided coin is
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( ) ( ) ( ) ( ) Example 2: A fair,
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Expected Value of a Binomial Random
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f. What is the probability that g.
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Probability For instance, we see th
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Density 0.25 Time Speng Waiting in
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Suppose we wish to find ( ), that i
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Density Resulting in a standard dev
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Density Without going into detail h
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a. What is the probability that the
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Statistics for Decision-Making in B
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As you can see, this is a difficult
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We wish to know the area of the sha
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We can easily find that the probabi
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In the second section, we can check
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Chapter 6 Sampling Distributions an
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Thus, the standard deviation would
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1.7 to 1.8 1.8 to 1.9 1.9 to 2 2 to
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2.4 to 2.5 2.5 to 2.6 2.6 to 2.7 2.
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The Central Limit Theorem (CLT) has
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can set this up in our applet by ha
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3) We cannot use this sample to cal
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Thus, we need to find the lower and
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We see that the standard deviation
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√ The lower limit is: √ And the
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following data was collected on the
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{ So, we have a set of twenty 1‟s
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DULY CAUTIONED: The assumptions her
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Homework Problems -6.2 1. In a samp
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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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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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