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Welcome! Week 1 Live Lecture/Discussion Applied Managerial Statistics (GM533) Lecturer: Brent Heard Please note that I borrowed these charts from Joni Bynum and the textbook publisher. Thanks Joni!I will put my touch on them (in blue) as we go along. 1
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Tonight’s Agenda• Week 1 Terminal Course Objectives (TCOs)• Essential Questions and Problem Types• The Most Important Ideas in Statistics• Getting started with Minitab• Descriptive Statistics using Minitab• Questions? 2
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Week 1 Terminal Course Objectives (TCOs)• TCO A Descriptive Statistics: Given a managerial problem and accompanying data set, construct graphs (following principles of ethical data presentation), calculate and interpret numerical summaries appropriate for the situation. Use the graphs and numerical summaries as aids in determining a course of action relative to the problem at hand.• TCO F Statistics Software Competency: Students should be able to perform the necessary calculations for objectives A through E using technology, whether that be a computer statistical package or the TI-83, and be able to use the output to address a problem at hand. 3
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The Most Important Ideas in Statistics• Central tendency (measures of center) and dispersion (spread)• Quantitative (numbers) and qualitative (words and numbers with no meaning) variables• Description and inference• One variable versus two or more variables 4
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Selected Slides from the Text Book• The following slides from the text book are intended to complement the live demonstration and provide a bridge to Module 1 5
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Population Parameters A population parameter is a number calculated from all the population measurements that describes some aspect of the population (Remember “p” goes with “p”) The population mean, denoted , is a population parameter and is the average of the population measurements (Fancy letters are used for the population) 6
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Point Estimates and Sample Statistics A point estimate is a one-number estimate of the value of a population parameter A sample statistic is a number calculated using sample measurements that describes some aspect of the sample (“s” goes with “s”) Use sample statistics as point estimates of the population parameters The sample mean, denoted x, is a sample statistic and is the average of the sample measurements (Plain letters for the sample) The sample mean is a point estimate of the population mean 7
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Measures of Central Tendency Mean, The average or expected value Median, Md The value of the middle point of the ordered measurements Mode, Mo The most frequent value 8
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The Mean Population X1, X2, …, XN Sample x1, x2, …, xn x Population Mean Sample Mean N n Xi xi i=1 i=1 x N n 9
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The Sample Mean For a sample of size n, the sample mean is defined as n xi i 1 x1 x2 ... xn x n n and is a point estimate of the population mean • It is the value to expect, on average and in the long run 10
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Example: Car Mileage Case Example 3.1: Sample mean for first five car mileages from Table 2.4 30.8, 31.7, 30.1, 31.6, 32.1 5 xi i 1 x1 x2 x3 x4 x5x 5 5 30.8 31.7 30.1 31.6 32.1 156 .3x 31.26 5 5 11
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The Median The population or sample median Md is a value such that 50% of all measurements, after having been arranged in numerical order, lie above (or below) it. (The median is the “center.”) The median Md is found as follows: 1. If the number of measurements is odd, the median is the middlemost measurement in the ordered values 2. If the number of measurements is even, the median is the average of the two middlemost measurements in the ordered values 12
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Example: Sample Median Internist’s Yearly Salaries (x$1000) 127 132 138 141 144 146 152 154 165 171 177 192 241 (Note that the values are in ascending numerical order from left to right) Because n = 13 (odd,) then the median is the middlemost or 7th value of the ordered data, so Md=152 • An annual salary of $180,000 is in the high end, well above the median salary of $152,000 • In fact, $180,000 a very high and competitive salary 13
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The Mode The mode Mo of a population or sample of measurements is the measurement that occurs most frequently • Modes are the values that are observed “most typically” • Sometimes higher frequencies at two or more values • If there are two modes, the data is bimodal • If more than two modes, the data is multimodal • When data are in classes, the class with the highest frequency is the modal class • The tallest box in the histogram (The Tall Pole) 14
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Relationships Among Mean, Medianand Mode Notice tail to right Notice tail to left 15
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Central Tendency By Itself Not EnoughKnowing the measures of central tendency is not enoughBoth of the distributions shown below have identical measures of central tendency 16
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The Normal Curve Symmetrical and bell-shaped curve for a normally distributed population The height of the normal over any point represents the relative proportion of values near that point Example 2.4, The Car Mileages Case 17
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The Empirical Rule forNormal Populations If a population has mean and standard deviation and is described by a normal curve, then 68.26% of the population measurements lie within one standard deviation of the mean: [ 95.44% of the population measurements lie within two standard deviations of the mean: [ 2 2 99.73% of the population measurements lie within three standard deviations of the mean: [ 3 3 2-18 18
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z Scores (will be very important in our work with the Normal Distribution, beginning in Week 2 and for the entire course) For any x in a population or sample, the associated z score is x mean z standarddeviation The z score is the number of standard deviations that x is from the mean A positive z score is for x above (greater than) the mean A negative z score is for x below (less than) the mean 2-19 19
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Measures of Variation (Spread) Range Largest minus the smallest measurement Variance The average of the squared deviations of all the population measurements from the population mean Standard Deviation The square root of the variance 20
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The Range Range = largest measurement - smallest measurement The range measures the interval spanned by all the data Example: Internist’s Salaries (in thousands of dollars) 127 132 138 141 144 146 152 154 165 171 177 192 241 Range = 241 - 127 = 114 ($114,000) 21
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Variance For a population of size N, the population variance 2 is defined as N 2 xi 2 2 2 2 i 1 x1 x2 xN N N For a sample of size n, the sample variance s2 is defined as n 2 xi x 2 2 2 x1 x x2 x xn x s2 i 1 n 1 n 1 and is a point estimate for 2 22
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The Standard Deviation 2 Population Standard Deviation, : 2 Sample Standard Deviation, s: s s 23
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Example: Population Varianceand Standard Deviation Population of profit margins for five big American companies: 8%, 10%, 15%, 12%, 5% 8 10 15 12 5 50 10% 5 5 2 2 2 2 2 2 8 10 10 10 15 10 12 10 5 10 5 2 2 02 52 2 2 52 5 4 0 25 4 25 58 11 .6 5 5 2 11 .6 3.406 % 24
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Example: Sample Variance and Standard Deviation Example 3.7: Sample variance and standard deviation for first five car mileages from Table 2.4 30.8, 31.7, 30.1, 31.6, 32.1 so x = 31.26 5 2 xi xs2 i 1 5 1 30.8 31.26 2 31.7 31.26 2 30.1 31.26 2 31.6 31.26 2 32.1 31.26 2 4 s2 = 2.572 4 = 0.643 s s2 .643 0.8019 2-25 25
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Percentiles and Quartiles For a set of measurements arranged in increasing order, the pth percentile is a value such that p percent of the measurements fall at or below the value and (100-p) percent of the measurements fall at or above the value The first quartile Q1 is the 25th percentile The second quartile (or median) Md is the 50th percentile The third quartile Q3 is the 75th percentile The interquartile range IQR is Q3 - Q1 26
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Population and Sample Proportions Population X1, X2, …, XN Sample x1, x2, …, xn p ˆ p Sample Proportion Population Proportion n xi ˆ p i =1 n ^ p is the point estimate of p 28
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Example: Sample Proportion Marketing Ethics Case 117 out of 205 marketing researchers disapproved of action taken in a hypothetical scenario X = 117, number of researches who disapprove n = 205, number of researchers surveyed X 117 Sample Proportion: ˆ p 0.57 n 205 29
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Getting Started with Minitab• Course Home: Minitab• Tutorial• Download• Getting help with your Minitab installation 30
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Summary of Descriptive Statistics using Minitab(concluded)• Central tendency: mean, median, mode• Dispersion: Range, standard deviation, interquartile range• Stem – and - leaf display• Histogram and frequency distribution 31
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Essential Questions and Problem Typesfor the Week 1 Mastery Module• For a given data set, use Minitab to find numbers, pictures, and tables which show the central tendency, including: the mean, median, and mode, and the skewness• For a given data set, use Minitab to find numbers, pictures, and tables which show the variability, or dispersion, including: the range, the standard deviation the interquartile range, and the Empirical Rule 32
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ClosingI will post a link to these charts where I hang out on the internet.I call it the “Statcave.”http://www.facebook.com/statcaveYOU DO NOT HAVE TO BE A FACEBOOK PERSON TO SEE THE LINKS. I DO IT BECAUSE IT’S FREE AND FUN.In my spare time, I write a syndicated column (humor, life, feel goods, etc.) that appears in newspapers and magazines in the southeast. If you ever get bored, check it out at:http://www.cranksmytractor.comSee you next week! Same Stat Time, Same Stat Channel. 33
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