1

• Course review, syllabus, etc.

• Chapter 1 – Introduction

• Chapter 2 – Graphical Techniques

Quantitative Business Methods

A First Course

3-21-05

2

Population and Sample

Population Sample

Use parameters to summarize features

Use statistics to summarize features

Inference on the population from the sample

3

Some Important Definitions……

• A ___________________(or universe) is the whole collection of things under consideration.

• A ______________ is a portion of the population selected for analysis.

• A PARAMETER is a summary measure computed to describe a characteristic of the population. µ

• A STATISTIC is a summary measure computed to describe a characteristic of the sample.

Discuss examples…….. Ω

X

4

Statistical Methods

•Descriptive Statistics

•Inferential Statistics

Collecting and describing data.

Drawing conclusions and/or making decisions concerning a population based only on sample data.

5

• Collect Data– e.g. Survey

• Present Data– e.g. Tables and graphs

• Characterize Data– e.g. Sample Mean =

iX

n

Descriptive Statistics

6

Inferential Statistics•Estimation e.g. Estimate the population mean using the sample mean.

•Hypothesis Testing e.g. Test the claim that the population mean weight is 120 pounds.

Drawing conclusion and/or making decisions concerning a population based on sample results.

7

1. Write the following in scientific notation (ex. 4.63 x 102 or 4.63E02)(you may use as many significant figures as you wish)

A. 3864159831.025 B. 0.0000062514836

2. Write the following numbers in standard notation (ie. Not in scientific notation)

A. 4.3650217E10 B. 2.1097326 x 10 -6

3. Perform the following calculations, using only your calculator (try to enter it all in to your calculator).

?33

)25(6.

A ?

25

643.

B

4. Perform the following calculation without using your calculator.

?43

)45(2

Analytical Skills Inventory Exercise

8

___

X

Use the following information for problems 5-9.

= 4.6 n = 10

i Xi

1 3

2 5

3 2

4 6

5 10

6 4

7 5

8 3

9 7

10 1

?#5.1

n

iiX ?.6#

1

2

n

iiX

?1

.7#

2

niX i ?.8#

___

1

XXn

ii

?.9#1

___

n

ii XX

9

1. A. 3.864159831025 x 109 or 3.864E09

B. 6.2514836 x 10-6 or 6.251E-06

2. A. 43650217000

B. 0.0000021097326

3. A. 9B. 10

4. 10

5. 46

6. 274

7. 2116

8. 41.4

9. 0

10

Graphical Descriptive Techniques

Graphical Descriptive Techniques

Chapter 2Chapter 2

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2.1 Introduction

Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making.Descriptive statistics methods make use of graphical techniques numerical descriptive measures.

The methods presented apply to both the entire population the population sample

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2.2 Types of data and information

A variable - a characteristic of population or sample that is of interest for us. Cereal choice Capital expenditure The waiting time for medical services

Data - the actual values of variables Interval data are numerical observations Nominal data are categorical observations

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Types of data - examples

Interval data

Age - income55 7500042 68000

. .

. .

Age - income55 7500042 68000

. .

. .Weight gain+10+5..

Weight gain+10+5..

Nominal

Person Marital status1 married2 single3 single. .. .

Person Marital status1 married2 single3 single. .. .Computer Brand

1 IBM2 Dell3 IBM. .. .

Computer Brand1 IBM2 Dell3 IBM. .. .

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Types of data - examples

Interval data

Age - income55 7500042 68000

. .

. .

Age - income55 7500042 68000

. .

. .

Nominal data

With nominal data, all we can do is, calculate the proportion of data that falls into each category.

IBM Dell Compaq Other Total 25 11 8 6 50 50% 22% 16% 12%

IBM Dell Compaq Other Total 25 11 8 6 50 50% 22% 16% 12%

Weight gain+10+5..

Weight gain+10+5..

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2.3 Graphical Techniques forInterval Data

Example 2.1: The monthly bills of new subscribers in the first month after signing on with a telephone company. Collect data Prepare a frequency distribution Draw a histogram

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Largest observation

Collect dataBills42.1938.4529.2389.35118.04110.460.0072.8883.05

.

.

(There are 200 data points

Prepare a frequency distributionHow many classes to use?Number of observations Number of classes

Less then 50 5-750 - 200 7-9200 - 500 9-10500 - 1,000 10-111,000 – 5,000 11-135,000- 50,000 13-17More than 50,000 17-20

Class width = [Range] / [# of classes]

[119.63 - 0] / [8] = 14.95 15Largest observationLargest observation

Smallest observationSmallest observationSmallest observation_______ observation

_________observation

Example 2.1: Providing information

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0

20

40

60

80

15 30 45 60 75 90 105 120

Bills

Fre

qu

en

cy

Draw a HistogramBin Frequency

15 7130 3745 1360 975 1090 18

105 28120 14

Example 2.1: Providing information

18

0

20

40

60

8015 30 45 60 75 90 10

5

120

Bills

Fre

qu

ency

What information can we extract from this histogramAbout half of all the bills are small

71+37=108 13+9+10=32

A few bills are in the middle range

Relatively,large numberof large bills

18+28+14=60

Example 2.1: Providing information

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It is often preferable to show the relative frequency (proportion) of observations falling into each class, rather than the frequency itself.

Relative frequencies should be used when the population relative frequencies are studied comparing two or more histograms the number of observations of the samples studied are

different

Class relative frequency = Class relative frequency = Class frequency

Total number of observations

Class frequency

Total number of observations

Relative frequency

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There are four typical shape characteristics

Shapes of histograms

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______________ skewed

Negatively skewed

Shapes of histograms

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A modal class is the one with the largest number of observations.

A unimodal histogram

The modal class

Modal classes

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Modal classes

A bimodal histogram

A modal class A modal class

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Bell shaped histograms

“________________________”

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Example 2.3: Comparing students’ performance Students’ performance in two statistics classes. Different in their teaching emphasis

Class A – math analysis and development of theory. Class B – applications and computer based analysis.

The final mark was recorded. Draw histograms and interpret the results.

Interpreting histograms

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Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Fre

qu

en

cy

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Fre

qu

en

cy

Interpreting histograms

The mathematical emphasiscreates two groups, and a larger spread.

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Preliminary analysis.Original observations vs. histogram approach.Split each observation into two parts.There are several ways of doing that:

42.19 42.19

Stem Leaf 4219

Stem Leaf4 2

A stem and leaf display forExample 2.1 will use thismethod

Stem and Leaf Display

Observation:

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A stem and leaf display for Example 2.1 (See page 42 for ref)

Stem-and-Leaf Display for Bills: unit = 1.0 1|2 represents 12.0

52 0|0000000001111122222233333455555566666667788889999999 85 1|000001111233333334455555667889999 (23) 2|00001111123446667789999 92 3|001335589 83 4|12445589 75 5|33566 70 6|3458 66 7|022224556789 54 8|334457889999 42 9|00112222233344555999 22 10|001344446699 10 11|0124557889

The length of each linerepresents the _________ of the class defined by the stem.

Stem and Leaf DisplaySG Demo

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Ogives

Cumulative relative frequency

Bills

Cumulative relative frequency

Bills

Cumulative relative frequency for telephone billsCumulative Cum.Relative

Class Frequency frequency frquency0-15 71 71 71/200=.355

15-30 37 108 108/200=.54030-45 13 121 121/200=.60545-60 9 130 130/200=.65060-75 10 140 140/200=.70075-90 18 158 158/200=.79090-105 28 186 186/200=.930

105-200 14 200 200/200=1.000

Cumulative relative frequency for telephone billsCumulative Cum.Relative

Class Frequency frequency frquency0-15 71 71 71/200=.355

15-30 37 108 108/200=.54030-45 13 121 121/200=.60545-60 9 130 130/200=.65060-75 10 140 140/200=.70075-90 18 158 158/200=.79090-105 28 186 186/200=.930

105-200 14 200 200/200=1.000

15

.355

30

.540

45

.605

60

.650

75

.700

90

.790

105

.930

120

1.000

Ogives are cumulative relative frequency distributions.

Example 2.1 - continued

SG Demo: Freq Tab

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Frequency Tabulation for Bills

------------------------------------------------------------------------------------ Lower Upper Relative Cumulative Cum. Rel.Class Limit Limit Midpoint Frequency Frequency Frequency Frequency------------------------------------------------------------------------------------ at or below 0.0 8 0.0400 8 0.0400 1 0.0 15.0 7.5 63 0.3150 71 0.3550 2 15.0 30.0 22.5 37 0.1850 108 0.5400 3 30.0 45.0 37.5 13 0.0650 121 0.6050 4 45.0 60.0 52.5 9 0.0450 130 0.6500 5 60.0 75.0 67.5 10 0.0500 140 0.7000 6 75.0 90.0 82.5 18 0.0900 158 0.7900 7 90.0 105.0 97.5 28 0.1400 186 0.9300 8 105.0 120.0 112.5 14 0.0700 200 1.0000above 120.0 0 0.0000 200 1.0000------------------------------------------------------------------------------------Mean = 43.5876 Standard deviation = 38.9697

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2.4 Graphical Techniques for Nominal data

The only allowable calculation on nominal data is to count the frequency of each value of a variable.When the raw data can be naturally categorized in a meaningful manner, we can display frequencies by Bar charts – emphasize frequency of occurrences

of the different categories. Pie chart – emphasize the proportion of

occurrences of each category.

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Marketing25.3%

Finance20.6%

General management14.2%

Other11.1% Accounting

28.9%

(28.9 /100)(3600) = 1040

The Pie Chart Ex #2.4: The student placement office at a university wanted to determine the general areas of employment of last year school graduates.

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Rectangles represent each category.

The height of the rectangle represents the frequency.

The base of the rectangle is arbitrary

Bar Chart

0

10

20

30

40

50

60

70

80

1 2 3 4 5 More

Area

Fre

qu

en

cy

73

5236

64

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The Bar Chart

SG Demo: Desc-

Categ-Tab

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2.5 Describing the Relationship Between Two Variables

The relationship between two interval variables.

Example 2.7 A real estate agent wants to study the relationship

between house price and house size Twelve houses recently sold are sampled and the

size and price recorded Use graphical technique to describe the

relationship between size and price.

Size Price23 31524 22926 33527 261……………..……………..

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Solution The size (independent variable, X) affects

the price (dependent variable, Y) We use Excel to create a scatter diagram

2.5 Describing the Relationship Between Two Variables

0

100

200

300

400

0 10 20 30 40

Y

X

The greater the house siz

e,

the greater the price

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Typical Patterns of Scatter DiagramsPositive linear relationship Negative linear relationshipNo relationship

Negative nonlinear relationship

This is a weak linear relationship.A non linear relationship seems to fit the data better.

Nonlinear (concave) relationship

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Graphing the Relationship Between Two Nominal Variables

We create a contingency table.

This table lists the frequency for each combination of values of the two variables.

We can create a bar chart that represent the frequency of occurrence of each combination of values.

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Example 2.8 (Data: 2.8a)

To conduct an efficient advertisement campaign the relationship between occupation and newspapers readership is studied. The following table was created

Contingency table

Blue Collar White collar ProfessionalG&M 27 29 33Post 18 43 51Star 38 15 24Sun 37 21 18

Blue Collar White collar ProfessionalG&M 27 29 33Post 18 43 51Star 38 15 24Sun 37 21 18