Probability and Statistics Assignment

This assignment covers fundamental probability and statistics concepts. Students are tasked with calculating probabilities from a given table, determining independence between events, and analyzing proportions within subcategories. It emphasizes understanding marginal distributions and applying statistical reasoning to real-world scenarios.

Running Head: STATISTICS FOR BUSINESS AND FINANCE
Statistics for Business and Finance
Name of the Student
Name of the University
Author Note

1STATISTICS FOR BUSINESS AND FINANCE
Table of Contents
TASK 1............................................................................................................................................3
Part A...........................................................................................................................................3
Part B...........................................................................................................................................3
Part C...........................................................................................................................................4
Part D...........................................................................................................................................5
TASK 2............................................................................................................................................6
Part A...........................................................................................................................................6
Part B...........................................................................................................................................6
Part C...........................................................................................................................................7
TASK 3............................................................................................................................................8
Part A...........................................................................................................................................8
Part B...........................................................................................................................................8
Part C...........................................................................................................................................8
Part D...........................................................................................................................................9
TASK 4..........................................................................................................................................10
Part A.........................................................................................................................................10
Part B.........................................................................................................................................10
Part C.........................................................................................................................................11
Part D.........................................................................................................................................11
Part E.........................................................................................................................................11
REFERENCES..............................................................................................................................12

2STATISTICS FOR BUSINESS AND FINANCE
TASK 1
Part A
A random sample of two hundred households has been collected in Excel. The sampling
method used in this case is simple random sampling. Simple random sampling is the sampling
method in which the sampling units are selected randomly with the probability of selecting each
unit being the same.
This is the best method of selecting the sample as with the help of this method samples
can be selected from a population with a minimum information about the population. Detailed
information about the population is not required. Classification errors are not present in this type
of sampling. Simple random sampling represents the whole population and is bias free.
Analyzing data becomes easier with the sample collected using this technique (Fowler, 2013).
This is a very simple technique and can be used easily. The sampling errors can also be assessed
very easily using this method.
Part B
Table 1.1: Descriptive Statistics
Alcohol Meals Fuel Phone
Mean 1242.895 1489.61 1557.85 1460.89
Standard Error 148.0106 111.569 117.7495 124.1273
Median 782 1200 1110 1020
Mode 0 1200 0 1200
Standard Deviation 2093.185 1577.824 1665.23 1755.425
Sample Variance 4381425 2489530 2772991 3081517
Kurtosis 78.86298 6.851153 10.89265 42.93626
Skewness 7.336376 2.28696 2.563535 5.391694
Range 24680 9600 12000 18000
Minimum 0 0 0 0
Maximum 24680 9600 12000 18000
Sum 248579 297922 311570 292178

3STATISTICS FOR BUSINESS AND FINANCE
Count 200 200 200 200
Alcohol Meals Fuel Phone
0
500
1000
1500
2000
2500
3000
Box and Whisker Plot
Annual Expenditure
Figure 1.1: Box and Whisker Plot showing the distribution of annual expenditure on Alcohol, Meals, Fuel and
Phone
Part C
The standard deviation of the annual expenses of alcohol consumption in the households
is 2093.185, for the consumption of meals the standard deviation is 1577.824, the standard
deviation for annual expenditures on consumption of fuel is 1665.23 and for phone is 1755.425.
Thus, the deviation is maximum in case of annual expense of alcohol consumption and minimum
in case of annual expense of consumption of meals. It is an appropriate measure of variability
because the standard deviation measures the distance of the values from the median. Thus, this
measure can show how much the values of the distribution are close to the mean or away from
the mean.

4STATISTICS FOR BUSINESS AND FINANCE
Part D
From the box and whisker plot in figure 1, it can be seen that, the annual expenditure on
alcohol is most variable followed by phone, fuel and meals. It can also be seen that the minimum
household expenditure on alcohol is zero. Thus, there are families who do not consume alcohol.
From the descriptive statistics, it can be seen that the mean of all the four variables are greater
than the median, which is again greater than the mode. Thus, the distributions of the expenses are
negatively skewed. This means that more families spend high on the consumption of these four
variables such as alcohol, meals, fuel and phone.

5STATISTICS FOR BUSINESS AND FINANCE
TASK 2
Part A
Table 2.1: Frequency Distribution Of Expenditures on Utilities
Annual Expenditure (in $) Frequencies Percentages Cumulative percentages
0-400 26 13 13
400-800 51 25.5 38.5
800-1200 59 29.5 68
1200-1600 35 17.5 85.5
1600-2000 13 6.5 92
2000-2400 8 4 96
2400-2800 3 1.5 97.5
2800-3200 2 1 98.5
More than 3200 3 1.5 100
Part B
 The percentage of households that spend on utilities at the most $ 1200 per annum is 68
percent.
 The percentage of households that spend on utilities between $1200 and $ 2400 per
annum is (17.5 + 6.5 + 4) % = 28 percent.
 The percentage of households that spend on utilities more than $ 2400 is (1.5 + 1 + 1.5) =
4 percent.

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Part C
0-400 400-800 800-1200 1200-
1600 1600-
2000 2000-
2400 2400-
2800 2800-
3200 More
than 3200
0
10
20
30
40
50
60
70
Histogram of Expenditures on Utilities
Annual Expenditure (in $)
Frequency
Figure 2.1: Distribution of Household Expenditures on Utilities
The utility expenditures are not normally distributed. Moreover, the distribution of the
annual income on utilities is positively skewed. Thus, it can be said that very less families spend
more on utilities. The distribution is not normally distributed because the normal distribution is
symmetric. From the histogram in figure 2.1, it is very clear that the distribution is not
symmetric.

7STATISTICS FOR BUSINESS AND FINANCE
TASK 3
Part A
The top 10 percent value of the household’s annual after tax income is the 90th percentile
and the bottom 10 percent value of the household’s annual after tax income is the 10th percentile.
For this data, the top 10 percent value of annual after tax income is $122673.5 and the bottom 10
percent value of the household’s annual after tax income is $19842. From this, it can be said that
10 percent of the population of households have an annual after tax income of $122,673 and
above. 10 percent of the household have an annual after tax income below $19,842.
Part B
The mean of the variable OwnHouse is found to be 0.67. The variable OwnHouse
contains two values 0 and 1. Here, 0 implies that the household does not own a house and 1
indicates that the household owns a house. The mean is found to be greater than 0.5. Thus it can
be said that most of the households own a house.
Part C
In the sample of 200 households, the number of households having a family size of 5 is
17. Thus, the probability that a randomly selected household will have a family size equal to 5 is
given by (17/200) = 0.085.

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Part D
8 8.5 9 9.5 10 10.5 11 11.5 12 12.5
0
2
4
6
8
10
12
14
Scatterplot
ln (Texp)
ln (ATanInc)
Figure 3.1: Scatter Plot of log of total expenditure and log of after tax income
Table 3.1: Correlation Table
ln(Texp)
ln(ATaxInc
)
ln(Texp) 1
ln(ATaxInc
)
0.11710
1 1
From the scatter diagram in figure 3.1, it can be seen clearly that the correlation between
the two variables such as the natural logarithm of total expenditure and natural log of annual
after tax income is very less (0.117). Nothing can be predicted about one variable from the value
of the other variable.

9STATISTICS FOR BUSINESS AND FINANCE
TASK 4
Part A
Table 4.1: Contingency Table
Highest Degree Gender Total
M F
P 16 23 39
S 19 23 42
I 19 26 45
B 21 12 33
M 15 26 41
Total 90 110 200
From the table, it can be seen that the number of males undergoing higher level of
education is (21 + 15) = 36 and the number of women undergoing higher level of education is
(12+26) = 38. These two values are more or less equal and thus can be said that male and female
heads of the households do not differ in their higher level of qualification. In this case higher
level of qualification has been considered as bachelor’s degree and master’s degree.
Part B
Table 4.2: Marginal Distribution Table
Highest Degree Gender Total
M F
P 0.08 0.115 0.195
S 0.095 0.115 0.21
I 0.095 0.13 0.225
B 0.105 0.06 0.165
M 0.075 0.13 0.205
Total 0.45 0.55 1
The probability that the head of the household is a female and her higher level of
education is intermediate is (26/200) = 0.13.

10STATISTICS FOR BUSINESS AND FINANCE
Part C
The probability that the head of the family is a male and has bachelor’s degree is (21/200)
= 0.105.
Part D
The proportion of having secondary as the highest degree from among females is
((23+23)/200) = 0.23.
Part E
The two events “gender of household head is male” and “having the master’s degree”
will be independent if the probability of intersection of the two events is equal to the product of
the probabilities of the two events separately. The probability of gender of a household head
being male is (90/200) = 0.45. The probability of having a master’s degree is (41/200) = 0.205.
The probability of a household head being male and having a master’s degree is (15/200) =
0.075. Now, (0.45*0.205) = 0.092 which is not equal to 0.075. Thus, the two events are
dependent.

11STATISTICS FOR BUSINESS AND FINANCE
REFERENCES
Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J.
(2014). Statistics for business & economics, revised. Cengage Learning.
Carlberg, C. (2014). Statistical analysis: microsoft excel 2013. Que Publishing.
De Finetti, B. (2017). Theory of probability: A critical introductory treatment(Vol. 6). John
Wiley & Sons.
Fowler Jr, F. J. (2013). Survey research methods. Sage publications.
Miller, I., & Miller, M. (2015). John E. Freund's mathematical statistics with applications.
Pearson.
Montgomery, D. C., Peck, E. A., & Vining, G. G. (2015). Introduction to linear regression
analysis. John Wiley & Sons.
Rodgers, K. A. (2016). Correlation Analysis with Excel Handout.
Triola, M. F. (2013). Elementary statistics using Excel. Pearson.