Statistical Techniques in Business

1. SDUIS BUS 702
Text: Statistical Techniques in Business & Economics, by Lind, Marchal, Wathen;
McGraw Hill, 17th ed.
Homework Questions - Lesson One
Question No. 1
Although statistics is defined as numerical information or a collection of quantitative
data, what are the practical uses of statistics? Give three examples.
Answer 1.
Governmental Departments
Government departments use statistical techniques for arising on decisions regarding
crimes, finances, population, health, education, etc. Statistical departments conduct
research’s for seeking specific information on educational and other welfare matters to
check the progress and efficiency of spending in different sectors of the economy.
Government Departments use these techniques to forecast and predict future spending
and collections of funds.
Health and Medicine
Statistical techniques are being used in the medical industry to assess the reactions of
the human body while testing new developments. It accumulates the outcomes and
provides the information for future testing's and its practical uses. The techniques
available in statistics also assists the researcher in making decision making regarding
specific research's to carry on or to be stopped.
Corporate Sector
Large corporations use statistical techniques to forecast their sales based on available
data and evaluate market trends. Statistical techniques also provide a decision for
products to be manufactured or not or how much to produce.
Statistics uses in each and every sector, it is economics or welfare. Its uses cannot be
sum up, however, it is established that no field is complete without statistics. Statistics
is a tool that enhances productivity.
Question No. 2
What is the difference between qualitative and quantitative variables?
Answer 2.
By nature, variables can be of two types qualitative and quantitative. Qualitative
variables are descriptive variable, these are related to the quality of data. As red balls,
in it, red is the quality of the variable. Qualitative variables can influence the parameters
of research that can impact the experiment or research outcomes. Qualitative research
is to be inductive and specially used in social research. On the other side quantitative
variables is related to the quantity of any variable. These are numbers and figures. These
numbers and figures can be based on the quality of variables as red balls in the

2. population are qualitative variables and the number of balls is the quantity of data. The
quantitative variable can be the percentage of something, any measurement or any
number of the unit. It is used in economic data especially.
Question No. 3
What are the differences between a sample and a population?
Answer 3.
The population represents the complete set of data based on variables of persons, units,
objects and anything that is capable of being considered, based on definite assertions.
The population is the complete data which includes a complete amount of data
including all elements. However, a sample is the data from the population derived using
specific techniques to be assessed and evaluates on specific techniques and reach on the
outcome for representing the whole data.
Question No. 4
Hershey’s chocolate has been making Hershey’s chocolate bars and Hershey’s kisses
for more than 100 years. What data would they collect and how would they use the
data to determine if they should continue producing Hershey’s chocolates?
Answer 4.
Hershey supposed to keep and accumulate the data regarding sales over the past period
to simply forecast its future sales. However, if it is possible Hershey supposed to collect
the data regarding consumer choice and tastes to forecast numbers close to actual. It
also needs to collect the data regarding the buying power of the consumer and to assess
the potential number of customers. These specific data are required to reach at precise
points from the side of sales. From the production point of view, Hershey's go for
inflation rate over the past period and fluctuations in the cost of raw materials. However,
government policies couldn't be statistically measured regarding the specific nature of
businesses, but their estimations can be created.
These are some of the relevant data which need to be collected for deciding between
continuing or discontinuing the production process.
Question No. 5. How is data measured?
Answer 5.
Data has four different levels of measurement which are nominal, ordinal, interval and
ratio. In the nominal level of measurement, the numbers in the variables are used to
classify the data. In this level symbols, letters, signs or words can be used for
measurement. For example, gender data can be classified as M for Men F for Female
and T for transgender. The second level of measurement is the ordinal level. It creates

3. a relationship among data based on some order ascending or descending. For example,
grades for students in class reveals the level of standing in the class if ordered by the
highest to lowest. Interval level of measurement if the third level of measurement in
which intervals are created based on an equivalency, distances among the interval
values and measurement. The fourth level of measurement is the ratio level of
measurement in which data have equal intervals with properties of having zero value.
Among all these levels of measurements, the nominal level is simply used to classify
data, whereas the levels of measurement described by the interval level and the ratio level
are much more exact.
Question No. 6
Lesson Two
You are the manager of Joe’s Fast Gas Station. Joe has asked you to follow the
learning objectives for chapter two (LO 1 - LO 6) and make a recommendation to
increase sales), based on the number of customers who buy gas, and the number of
those customers who also buy coffee or a snack (make your own data).
Answer 6.
Name of Month Number of
Customer Visits
Number of events
when Fuel Sold
Number Customer
Buys Coffee or
Snack
January 150 138 85
February 146 140 80
March 161 152 70
April 168 162 68
May 158 155 66
June 149 140 55
July 171 142 52
August 158 148 60
September 144 136 65
October 140 126 72
November 128 116 78
December 118 106 90
Total During the
Year
1791 1661 841
The data of Joe's Fast Gas station depicts that customer who buys gas are strictly related
to the number of customers who visits its Gas Station. To enhance its sales of gas he
has to increase the number of customers by providing maximum capacity and
availability of gas filling machines. While looking at the data of customers who buy

4. coffee or snack reveals that the purchases are dependent on customer visits but these
are also influenced by the season. In the summer season, the sales of coffee are low
however in the winter season the coffee sales are high in relation to the number of
customer visits to enhance the number of coffee customers Joe should increase the
flavors of the coffee.
Learning Objectives
Learning Objective 1. Summarize qualitative variables with frequency and relative
frequency tables.
According to Joe’s Gas station data, data can be summarized in grouped form as follow:
Name of
Month
Number of
Customer
Visits
No of
Events
when Fuel
Sold
Hi-Octane Diesel
Number of
Coffee or
Snack Sold
January 150 138 40 98 85
February 146 140 45 95 80
March 161 152 44 108 70
April 168 162 49 113 68
May 158 155 48 107 66
June 149 140 44 96 55
July 171 142 43 99 52
August 158 148 48 100 60
September 144 136 39 97 65
October 140 126 36 90 72
November 128 116 36 80 78
December 118 106 28 78 90
Total
During the
Year
1791 1661 500 1161 841
Learning Objective 2. Display a frequency table using a bar or pie chart.
Answer:

5. Learning Objective 3. Summarize quantitative variables with frequency and relative
frequency distributions.
Answer:
Total Number of
Customers who visits Gas
Station Fuel Sold
Packs of Coffee or
Snacks Sold
X F
Relative
Frequency
F
Relative
Frequency
110-120 106 0.06 90 0.10702
120-130 116 0.07 78 0.09275
130-140 126 0.08 72 0.08561
140-150 554 0.33 285 0.33888
150-160 303 0.18 126 0.14982
160-170 314 0.19 138 0.16409
170-180 142 0.09 52 0.06183
1661 1 841 1
Learning Objective 4. Display a frequency distribution using a histogram or frequency
polygon
Answer
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Number of
Customer Visits
Events Fuel Sold Hi-Octane Sold Diesel Sold Number
Customer Buys
Coffee or Snack
Sales During the Year

6. Lesson Three
Using the data you created in Lesson Two, do LO 2,3,4, 5, and 10.
Learning Objectives of the Chapter 3 are as follows:
LO3-1 Compute and interpret the mean, the median, and the mode.
LO3-2 Compute a weighted mean.
LO3-3 Compute and interpret the geometric mean.
LO3-4 Compute and interpret the range, variance, and standard deviation.
LO3-5 Explain and apply Chebyshev’s theorem and the Empirical Rule.
Answer
Learning Objective 1
X F fx
January 118 106 12508
February 128 116 14848
March 140 126 17640
April 144 136 19584
May 146 140 20440
June 149 140 20860
July 150 138 20700
August 158 155 24490
September 158 148 23384

7. October 161 152 24472
November 168 162 27216
December 171 142 24282
1661 250424
Mean = x
̄ = ∑ fx/n
= 250424/1661
= 150.767
Mean computes the middle value in the data. However the 150.767 is the middle value
among the data.
X F
Cumulative Frequency
110-120 106 106
120-130 116 222
130-140 126 348
140-150 554 902
150-160 303 1205
160-170 314 1519
170-180 142 1661
Median = x ̃ = L+(n/2)-B/G x W
= 140+ (1661/2)-348 / 554 x 10
= 148.709
Median Value for the data is 148.709 slightly lower than mean valu which means the
dispersion is positively skewed.
Mode
X F
Cumulative Frequency
110-120 106 106
120-130 116 222
130-140 126 348
140-150 554 902
150-160 303 1205
160-170 314 1519
170-180 142 1661

8. Mode = l+h(fm – f1 ) / (2 fm – f1 – f2 )
= 140+10(554-126) / (2(554) – 126 – 303)
= 146.303
Mode of the data is 146.303 which depicts that the dispersion of the data is positively
skewed because it is lower than the mean value.
Learning Objective 2
Compute Weighted Mean
X
January 118
February 128
March 140
April 144
May 146
June 149
July 150
August 158
September 158
October 161
November 168
December 171
X
̄ w = ∑(wx)/∑w
= 118+128+140+144+146+149+150+2(158)+161+168+171 / 12
= 1791/12
= 149.25
LO3-3 Compute and interpret the geometric mean.
Geometric mean computes the rate of change over the period of time which could be
computed from other assumed data

9. Year Increase in Air Pollution in Gotham City
1991 2%
1992 1%
1993 3%
1994 4%
1995 5%
GM = n√(x1) (x2) (x3) (x4)……… (xn)
GM = 10√(1.02)(1.01)(1.03)(1.04)(1.05)
GM = 10√1.158728
GM = 1.014841
The GM depicts that about 1.48% increase in rated of pollution in Gotham City
averagely over the period of 5 years.
LO3-4 Compute and interpret the range, variance, and standard deviation
Range computes dispersion in the data so
Name of Month X
January 150
February 146
March 161
April 168
May 158
June 149
July 171
August 158
September 144
October 140
November 128
December 118
Range = Maximum value – Minimum Value
= 171-118
= 53

10. Range predicts that the data of visits of customer at Joes Fast Gas Station dispersed at
53. It shows the no. of customers visits can varies in total of 53 values in total.
Variance
Name of Month X x-µ
Squared Deviation
January 150 0.75 0.5625
February 146 -3.25 10.5625
March 161 11.75 138.0625
April 168 18.75 351.5625
May 158 8.75 76.5625
June 149 -0.25 0.0625
July 171 21.75 473.0625
August 158 8.75 76.5625
September 144 -5.25 27.5625
October 140 -9.25 85.5625
November 128 -21.25 451.5625
December 118 -31.25 976.5625
1791 2668.25
Variance = ∑(x-µ)/N
Variance = 2668.25/12
Variance = 222.345
Standard Deviation
Standard Deviation = √∑(x-µ)/n-1
Standard Deviation = √2668.25/11
Standard Deviation = 15.5746
Explain and apply Chebyshev’s theorem and the Empirical Rule.
The Russian mathematician P. L. Chebyshev (1821–1894) developed a theorem that
allows us to determine the minimum proportion of the values that lie within a specified
number of standard deviations of the mean. Chebyshev’s theorem, explains that at least
three out of every four, or 75%, of the values must lie between the mean plus two
standard deviations and the mean minus two standard deviations. This relationship
applies regardless of the shape of the distribution. Further, at least eight of nine values,

11. or 88.9%, will lie between plus three standard deviations and minus three standard
deviations of the mean. At least 24 of 25 values, or 96%, will lie between plus and
minus five standard deviations of the mean.
Chebyshev’s Theorem = 1 – 1/K2
At Joes Fast Gas Station has standard deviation of 15.5746 around mean no. of
customers visits 149.25. To determine at least what percent of customer visits lie around
the 3 standard deviations.
To Compute this, we use Chebyshev’s theorem
Chebyshev’s Theorem = 1 – 1/K2
Chebyshev’s Theorem = 1 – 1/(3)2
Chebyshev’s Theorem = 1 – 0.11111111
Chebyshev’s Theorem = 88.89% values
Empirical Rule
Under a frequency distribution, Empirical rule explains the standard deviation through
bell shaped graphical representation and scattered of data on the diagram that 68% of
the observations will lie within plus and minus one standard deviation of the mean value;
about 95% of the observations will lie within plus and minus two standard deviations
of the mean; and practically all (99.7%) will lie within plus and minus three standard
deviations of the mean.