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Business StatisticsChapter 1

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GOALS:

Explain what is meant by statistics Explain what is meant by descriptive statistics and

inferential statistics. Distinguish between a qualitative variable and a

quantitative variable; discrete variable and a continuous variable.

Define the terms mutually exclusive and exhaustive. Distinguish among the nominal, ordinal, interval, and ratio

levels of measurement.

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What is meant by Statistics?• Statistics is the science of data which

involves– collecting, – classifying – summarizing, – organizing, – analyzing, and interpreting numerical

information to assist in making effective decision

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Types of Statistics

Descriptive Statistics: Methods of organizing, summarizing, and presenting data in an informative way.

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Inferential Statistics: A decision, estimate, prediction, or generalization about a

population, based on a sample.

Descriptive Statistics Inferential Statistics

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Types of Statistics

Descriptive Statistics: Methods of organizing, summarizing, and presenting data in an informative way.

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Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample.

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Types of Statistics (examples of inferential statistics)

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Eg 1: TV networks constantly monitor the popularity of heir programs by hiring Nielsen and other organizations to sample the preferences of TV viewers.

Eg 2: The accounting department of a large firm will select a sample of the invoices to check for accuracy for all the invoices of the company.

Eg 3: Wine tasters sip a few drops of wine to make a decision with respect to all the wine waiting to be released for sale.

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Types of Statistics

A population is a collection of all possible individuals, objects, or measurements of interest. A parameter is a descriptive measure of the entire population of all observations of interest

A sample is a portion, or part, of the population of interest. A statistic describes a sample and serves as an estimate of the corresponding population parameter

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Types of Variables

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Qualitative or attribute(type of car owned)

Discrete(number of children)

Continuous(time taken for an exam)

Quantitative or Numerical

DATA

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Types of Variables For a Qualitative or Attribute variable the characteristic being studied is nonnumeric.

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Gender, religious affiliation, type of automobile owned, state of birth, eye color are examples.

In a Quantitative variable information is reported numerically.

balance in your checking account, minutes remaining in class, or number of children in a family.

Quantitative variables can be classified as either discrete or continuous.

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The pressure in a tire, the weight of a pork chop, or the height of students in a class.

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Discrete variables: can only assume certain values and there are usually “gaps” between values.

The number of bedrooms in a house, or the number of hammers sold at the local Home Depot (1,2,3,…,etc).

Continuous variable can assume any value within a specified range.

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Sources of Data

Primary data : Collected for specific purpose

Direct ObservationQuestionnairesInterviewing

Secondary Data : Collected for another purpose

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Who Uses Statistics?Statistical techniques are used extensively by marketing, accounting, quality control, consumers, hospital administrators, educators, politicians, physicians, etc...

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For Researching problems usually requires published data. Statistics on these problems can be found in published articles, journals, and magazines.

Published data is not always available on a given subject. In such cases, information will have to be collected and analyzed.

One way of collecting data is via questionnaires.

What are the other data collection methods?

Sources of Statistical Data

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Chapter Two

Describing Data: Describing Data:

Frequency Distributions and Frequency Distributions and Graphic PresentationGraphic Presentation

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Presentation of Data

Row data reveals very littleShows how a large data set can be organized

and managed to provide a quick visual interpretation of the massage the data convey.

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Methods of Data Presentation

i. Data Array

ii. Tabulation of Data

iii. Stem-and-Leaf display

iv. Frequency Distribution

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Data ArrayArrange data in systematic way(Ascending data array & Descending data Array)

1st Year 2nd Year

Physical 320 160

Bio 246 126

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Tabulation of Data

Arrange data in a tables (Rows & Columns)

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Frequency Distribution

A Frequency distribution is a grouping of data into mutually exclusive categories showing the number of observations in each class.

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Construction of a Frequency Distribution

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Frequency Distribution

Class midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits.

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Class frequency:

The number of observations in each class.

Class interval:

The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class.

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Eg 1: Dr. Tillman is Dean of the School of Business Socastee University. He wishes prepare to a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week.

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15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

Organize the data into a frequency distribution.

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Eg 1: continuedThere are 30 observationsTwo raised to the fifth power is 32.Therefore, we should have at least 5 classes. It turns out we will need 6.The range is 23.5 hrs, found by 33.8 hrs – 10.3 hrs.We choose an interval of 5 hrs.The lower limit of the first class is 7.5 hrs.26

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Hours studying Frequency, f 7.5 up to 12.5 1 12.5 up to 17.5 12 17.5 up to 22.5 10 22.5 up to 27.5 5 27.5 up to 32.5 1

32.5 up to 37.5 1

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Suggestions on Constructing a Frequency Distribution

The class intervals used in the frequency distribution should be equal.

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classes ofNumber

ue)Lowest val - lueHighest va(i

Determine a suggested class interval by using the formula:

Use the computed suggested class interval to construct the frequency distribution.

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Suggestions on Constructing a Frequency Distribution

Note: this is a suggested class interval; if the computed class interval is 97, it may be better to use 100.

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Count the number of values in each class.

Eg 1: A relative frequency distribution shows the percent of observations in each class.

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EXAMPLE Mr. Jayatissa wishes prepare to a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week.

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.Organize the data into a frequency distribution.

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Relative Frequency Distribution

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Hours f Relative Frequency

7.5 up to 12.5 1 1/30=.0333

12.5 up to 17.5 12 12/30=.400

17.5 up to 22.5 10 10/30=.333

22.5 up to 27.5 5 5/30=.1667

27.5 up to 32.5 1 1/30=.0333

32.5 up to 37.5 1 1/30=.0333 TOTAL 30 30/30=1

T

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Graphic Presentation of a Frequency Distribution

The three commonly used graphic forms are histograms, frequency polygons, and a cumulative frequency distribution.

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A Histogram is a graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis.

The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.

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Graphic Presentation of a Frequency Distribution

A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.

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A cumulative frequency distribution is used to determine how many or what proportion of the data values are below or above a certain value.

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Histogram for Hours Spent Studying

0

2

4

6

8

10

12

14

10 15 20 25 30 35

Hours spent studying

Fre

qu

ency

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Frequency Polygon for Hours Spent Studying

0

2

4

6

8

10

12

14

10 15 20 25 30 35

Hours spent studying

Fre

qu

ency

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Cumulative Frequency Distribution For Hours Studying

0

5

10

15

20

25

30

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10 15 20 25 30 35

Hours Spent Studying

Frequency

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Bar ChartA bar chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).

City Number of unemployed per 100,000 population

Atlanta, GA 7300 Boston, MA 5400 Chicago, IL 6700

Los Angeles, CA 8900 New York, NY 8200

Washington, D.C. 8900

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Eg 3: Construct a bar chart for the number of unemployed per 100,000 population for selected cities during 2001

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Bar Chart for the Unemployment Data

7300

5400

6700

89008200

8900

0

2000

4000

6000

8000

10000

1 2 3 4 5 6

Cities

# u

nem

plo

yed

/100

,000

AtlantaBostonChicagoLos AngelesNew YorkWashington

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Pie ChartA pie chart is useful for displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.

Type of shoes # of runners

Nike 92

Adidas 49

Reebook 37

Asics 13

Other 9 33

Eg 4: A sample of 200 runners were asked to indicate their favorite type of running shoe.

Draw a pie chart based on the following information.

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Pie Chart for Running Shoes

Nike

Adidas

ReebokAsics

Other

Nike

Adidas

Reebok

Asics

Other

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Chapter Three

Describing Data: Describing Data:

Measures of Central TendencyMeasures of Central Tendency

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Population Mean

For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:

N

X

where µ is the population mean. N is the total number of observations. X is a particular value. indicates the operation of adding.

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Eg 1: The Kiers family owns four cars. The following is the current mileage on each of the four cars:

56,000, 23,000, 42,000, 73,000

500,484

000,73...000,56

N

X

Find the mean mileage for the cars.

A Parameter is a measurable characteristic of a population.

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Sample Mean

• For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

Where n is the total number of values in the sample.

n

XX

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Eg 2: A sample of five executives received the following bonus last year ($000):

14.0, 15.0, 17.0, 16.0, 15.0

4.155

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5

0.15...0.14

n

XX

A statistic is a measurable characteristic of a sample.

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Weighted Mean

Eg 3: Consider the set of values: 3, 8, and 4. The mean is 5. Illustrating the fifth property:

)21

)2211

...(

...(

n

nnw

www

XwXwXwX

0)54()58()53()( XX

The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

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Eg 6: During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $1.10. Compute the weighted mean of the price of the drinks.

89.0$50

50.44$1515155

)15.1($15)90.0($15)75.0($15)50.0($5

wX

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The Median

• The Median is the midpoint of the values after they have been ordered from the smallest to the largest.

There areas many values above the median as below it in the data array.

For an even set of values, the median will be the arithmetic average of the two middle numbers.

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Eg 4 : The ages for a sample of five college students are:

21, 25, 19, 20, 22Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21.

Eg 5 : The heights of four basketball players, in inches, are:

76, 73, 80, 75

Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5

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The Mode

• The mode is the value of the observation that appears most frequently.

Eg 5: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode.

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Geometric Mean

• The geometric mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The formula is:

The geometric mean is used to average percents, indexes, and relatives.

GM X X X Xnn ( )( )( )...( )1 2 3

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• The interest rate on three bonds were 5, 21, and 4 percent.

• The geometric mean is

• The arithmetic mean is (5+21+4)/3 =10.0

• The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21percent.

49.7)4)(21)(5(3 GM

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Geometric Mean continued

• Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another.

1period) of beginningat (Value

period) of endat Value(nGM

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Eg 8 : The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%.

0127.1000,755

000,8358 GM

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Eg 9 : There are many flights from Houston to Little Rock, AK each day. The data below shows the number of minutes a flight was late (or early) in arriving in Little Rock for a sample of 5 flights. To explain, a positive number means the flight was late, a value of 0 indicates it arrived on time, and a negative number indicates it was early. So the first flight was 4 minutes late and the last flight 10 minutes early.

4 12 -9 6 -10

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a. Determine the mean amount flights were late (or early).

b. Determine the median amount flights were late (or early).

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50

60.05

3X

4Median

a. Determine the mean amount flights were late (or early).

b. Determine the median amount flights were late (or early).

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Eg10: Suppose your cousin started a management job with Ford Motor Company in 1990 at $30,000 per year. In the year 2002 her salary was $65,000. What was the geometric mean rate of increase per year for the period?

Eg 11: For a sample of 50 stocks traded yesterday on the American Stock Exchange, 10 showed a decline

of $1.00, 15 showed no change, and 25 increased by $2.00. Find the weighted mean.

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06655.00.106655.100.1000,30$

000,65$12 GM

80.0$251510

)00.2($25)0($15)00.1$(10

wX

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The Mean of Grouped Data

The mean of a sample of data organized in a frequency distribution is computed by the following formula:

6.610

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n

XX

n

XfX

Eg12: A sample of ten movie theaters in a large metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing.

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Eg 12 : continued

Movies showing

frequency f

class midpoint

X

(f)(X)

1 up to 3 1 2 2

3 up to 5 2 4 8

5 up to 7 3 6 18

7 up to 9 1 8 8

9 up to 11 3 10 30

Total 10 66

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The Median of Grouped Data

• The median of a sample of data organized in a frequency distribution is computed by:

)(2 if

CFn

LMedian

where L is the lower limit of the median class, CF is the cumulative frequency preceding the median class, f is the frequency of the median class, and i is the median class interval.

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Finding the Median Class

To determine the median class for grouped data:

– Construct a cumulative frequency distribution.

– Divide the total number of data values by 2.

– Determine which class will contain this value. For example, if n=50, 50/2 = 25, then determine which class will contain the 25th value.

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Eg13 :

33.6)2(3

32

10

5)(2

if

CFn

LMedian

Movies showing

Frequency Cumulative Frequency

1 up to 3 1 1

3 up to 5 2 3

5 up to 7 3 6

7 up to 9 1 7

9 up to 11 3 10

From the table, L=5, n=10, f=3, i=2, CF=3

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The Mode of Grouped Data

• The mode for grouped data is approximated by the midpoint of the class with the largest class frequency.

The modes in Eg 13 are 8 and 10. When two values occur a large number of times, the distribution is called bimodal, as in Eg 13.

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Eg14: The following frequency distribution reports the number of students enrolled in each of the 50 sections of various courses taught in the College of Business last summer.

Students Frequency0 up to 10 310 up to 20 820 up to 30 1630 up to 40 1040 up to 50 950 up to 60 4

Total 50

a. Determine the mean number of students per section.

b. Determine the median number of students per section.

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2.3050

1510X

75.281016

112520

Median

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End of the Chapter

• Thank you

• Questions

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