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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.

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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?

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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.

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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

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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

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Population Parameters

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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)

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Point Estimates and Sample Statistics

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A point estimate is a one-number estimate of the value of a population parameterA 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

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Measures of Central Tendency

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Mean, The average or expected value

Median, Md The value of the middle point of the ordered measurements

Mode, Mo The most frequent value

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The Mean

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Population X1, X2, …, XN

m

Population Mean

N

X

N

=1ii

Sample x1, x2, …, xn

Sample Mean

x

n

x x

n

=1ii

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The Sample Mean

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and is a point estimate of the population mean • It is the value to expect, on average and in the long run

n

xxx

n

xx n

n

ii

...211

For a sample of size n, the sample mean is defined as

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Example: Car Mileage Case

5554321

5

1 xxxxxx

x ii

26.315

3.156

5

1.326.311.307.318.30

x

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Example 3.1: Sample mean for first five car mileages from Table 2.4

30.8, 31.7, 30.1, 31.6, 32.1

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The Median

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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

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Example: Sample Median

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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

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The Mode

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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)

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Relationships Among Mean, Medianand Mode

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Notice tail to left

Notice tail to right

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Central Tendency By Itself Not Enough

Knowing the measures of central tendency is not enough

Both of the distributions shown below have identical measures of central tendency

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The Normal Curve

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Symmetrical and bell-shaped curve for a normally distributed populationThe height of the normal over any point represents the relative proportion of values near that point

Example 2.4, The Car Mileages Case

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The Empirical Rule forNormal Populations

2-18

If a population has mean m and standard deviation s and is described by a normal curve, then

68.26% of the population measurements lie within one standard deviation of the mean: [ - , + ]m s m s

95.44% of the population measurements lie within two standard deviations of the mean: [ -m 2 , +s m 2 ]s

99.73% of the population measurements lie within three standard deviations of the mean: [ -m 3 , +s m 3 ]s

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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

The z score is the number of standard deviations that x is from the meanA positive z score is for x above (greater

than) the meanA negative z score is for x below (less than)

the mean

deviation standard

mean

xz

2-19

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Measures of Variation (Spread)

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Range

Largest minus the smallest measurement

VarianceThe average of the squared deviations of all

the population measurements from the population mean

Standard Deviation

The square root of the variance

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The Range

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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)

Range = largest measurement - smallest measurement

The range measures the interval spanned by all the data

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Variance

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and is a point estimate for s2

For a population of size N, the population variance s2 is defined as

For a sample of size n, the sample variance s2 is defined as

N

xxx

N

xN

N

ii 22

22

11

2

2

11

222

211

2

2

n

xxxxxx

n

xxs n

n

ii

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The Standard Deviation

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Population Standard Deviation, s: 2

Sample Standard Deviation, s: 2ss

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Example: Population Varianceand Standard Deviation

%105

50

5

51215108

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Population of profit margins for five big American companies:

8%, 10%, 15%, 12%, 5%

6115

58

5

25425045

52502

5

105101210151010108

22222

222222

.

%40636112 ..

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Example: Sample Varianceand Standard Deviation

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5

1

2

2

i

i xxs

2-25

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 = 31.26

s2 = 2.572 4 = 0.643

8019.0643.2 ss

4

26311322631631263113026317312631830 22222 ..........

x

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Percentiles and Quartiles

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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

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Example: Quartiles

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20 customer satisfaction ratings:

1 3 5 5 7 8 8 8 8 8 8 9 9 9 9 9 10 10 10 10

Md = (8+8)/2 = 8

Q1 = (7+8)/2 = 7.5 Q3 = (9+9)/2 = 9

IQR = Q3 Q1 = 9 7.5 = 1.5

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Population and Sample Proportions

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Population X1, X2, …, XN

p

Population Proportion

Sample x1, x2, …, xn

Sample Proportion

n

x p

n

1=ii

ˆ

p ˆ

p is the point estimate of p^

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Example: Sample Proportion

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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

Sample Proportion: 570205

117.

n

Xp̂

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Getting Started with Minitab

• Course Home: Minitab• Tutorial• Download• Getting help with your Minitab

installation

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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

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Essential Questions and Problem Types for 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

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Closing

I will post a link to these charts where I hang out on the internet.

I call it the “Statcave.”

http://www.facebook.com/statcave

YOU 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.com

See you next week! Same Stat Time, Same Stat Channel.