Holmes Institute HI6007: Statistics and Research Methods Group Report

This report presents a comprehensive analysis of statistical methods and research techniques applied to business decision-making. The assignment includes an analysis of Australian export data, comparing export volumes and percentages across various countries between 2004-05 and 2014-15, highlighting the impact of China's export growth. It also explores frequency distributions, cumulative frequency distributions, and histograms using umbrella sales data. The report further analyzes time series plots for retail turnover per capita and final consumption expenditure, along with a scatter plot and numerical summary. A regression model is developed to estimate final consumption expenditure based on retail turnover per capita, with hypothesis testing and an ANOVA model to assess the linear relationship. The report concludes with a discussion of the model's fit, including line fit and residual plots, and provides relevant references.

Decision Making from Graphical Measures and Linear Estimation
1

Answer 1
(a) Comparison of export to eight countries from Australia has been presented in Figure 1,
which clearly displays increase in export volume in 2014-15 compared to 2004-05,
except the United Kingdom.
China
Japan
United States
Republic of Korea
India
New Zealand
Singapore
United Kingdom
0 10 20 30 40 50 60 70 80 90 100
15.9
28.2
13.9
11
7.1
12.2
5.8
9.2
90.3
46.6
20.5
20.5
12.7
12.3
12
8.6
Export Figures for Goods and Services from Australia
2014-15
2004-05
Export (AUS $ Billion)
Figure 1: Australian Markets for Goods and Services export – Comparative Analysis
(b) Comparison of export to eight countries from Australia has been presented in Figure 2,
which clearly displays decrease in percentage export in 2014-15 compared to 2004-05,
except China (Fisher, Drucker, and König, 2012, pp.55-62).
2

China
Japan
United States
Republic of Korea
India
New Zealand
Singapore
United Kingdom
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0%
15.4%
27.3%
13.5%
10.6%
6.9%
11.8%
5.6%
8.9%
40.4%
20.9%
9.2%
9.2%
5.7%
5.5%
5.4%
3.8%
Percentage of Exports to Countries from Australia
2014-15
2004-05
Percentage
Figure 2: Australian export percentage in 2004-05 and 2014-15
(c) Exports from Australia to eight countries have increased in 2014-15 compared to 2004-
05 (exception was export to the United Kingdom). Unpredictably, export percentages to
all countries, except China, have decreased. Watchful observation reveals that export to
China has increased so drastically that it has affected export to other countries.
Therefore, instead of increase in volume, percentage wise exports have gone down.
Abrupt high export to China was noted to have regressive impact on percentage exports
to other countries.
3

a. The frequency distribution is presented below
Table 1: Frequency distribution of Umbrella Sales
The relative Frequency distribution is presented below
Table 2: Relative Frequency distribution of Umbrella Sales
b. The Cumulative frequency distribution is presented below
Table 3: Cumulative Frequency distribution of Umbrella Sales
4

Cumulative relative frequency distribution is presented below
Table 4: Cumulative Relative Frequency distribution of Umbrella Sales
c. Relative frequency histogram is presented below
30-40 40-50 50-60 60-70 70-80 80-90 90-100
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.05
0.10
0.20
0.28
0.20
0.13
0.05
Relative Frequency Histogram for 40 Days Umbrella Sales Figures
Class
Relative Frequency
Figure 3: Histogram of Umbrella Sales
d. Less than type Ogive has been constructed from the cumulative frequency
distribution of Umbrella Sales for 40 days.
5

20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
0
5
10
15
20
25
30
35
40
45
0 2
6
14
25
33
38 40
Less than Ogive for Umbrella Sales
Class (Sales)
Frequency
Figure 4: Less than type Ogive for Umbrella Sales
e. The less than 60 grades for umbrella sales per day have been using the cumulative
relative frequency. The cumulative relative frequency was 0.35 (Table 4). Therefore,
the “proportion of grades less than 60” is 0.35 or 35%.
f. The less than grade of 70 is 0.625. The cumulative relative frequency in Table 4
provides this value. Therefore, the “proportion of grades more than 70” is 0.375 or
37.5%.
6

a. The following time series plot indicates the trend of ““Retail Turnover per Capita”” ($) in
details. The period of the plot was from September 1983 to November 2014. “Retail
Turnover per Capita” ($) was noted to have an increasing trend. Temporary peaks are
noted on April 1999, December 2004, and October 2007.
Sep-1983
Jan-1985
May-1986
Sep-1987
Jan-1989
May-1990
Sep-1991
Jan-1993
May-1994
Sep-1995
Jan-1997
May-1998
Sep-1999
Jan-2001
May-2002
Sep-2003
Jan-2005
May-2006
Sep-2007
Jan-2009
May-2010
Sep-2011
Jan-2013
May-2014
Sep-2015
1200
1600
2000
2400
2800
3200
Time series plot of Retail turnover per capita ($)
Quarters
Retail turnover per capita ($)
Figure 5: Time series Plot for “Retail Turnover per Capita” ($)
A time series plot in Figure 6 has been drawn to estimate the trend of “Final
Consumption Expenditure” ($ Millions). The period of the plot was from September
1983 to November 2014. A strict increasing trend is prominent in Figure 6.
7

Sep-1983
Oct-1984
Nov-1985
Dec-1986
Jan-1988
Feb-1989
Mar-1990
Apr-1991
May-1992
Jun-1993
Jul-1994
Aug-1995
Sep-1996
Oct-1997
Nov-1998
Dec-1999
Jan-2001
Feb-2002
Mar-2003
Apr-2004
May-2005
Jun-2006
Jul-2007
Aug-2008
Sep-2009
Oct-2010
Nov-2011
Dec-2012
Jan-2014
Feb-2015
Mar-2016
80000
120000
160000
200000
240000
Time series plot of Final Consumption Expenditure ($ Millions)
Quarters
Final Consumption Expenditure ($ Millions)
Figure 6: The T-plot for “Final Consumption Expenditure”
b. “Retail Turnover per Capita”: Independent variable (X-axis)
“Final Consumption Expenditure”: Dependent variable (Y-axis)
It is known that the “Retail Turnover per Capita” contributes to the calculation of
“Final Consumption Expenditure” as a component. Choice of dependent and
independent variables was dependent on the casual relationship (Fornell, Rust, and
Dekimpe, 2010, pp.28-35).
1200 1600 2000 2400 2800 3200
80000
120000
160000
200000
240000
Final Consumption Expenditure on Retail Turnover per Capita
Retail turnover per capita ($)
Final Consumption Expenditure ($ Millions)
Correlation
= 0.988
Figure 7: The Scatter plot for “Final Consumption Expenditure” on “Retail Turnover per Capita”
c. Numerical Summary: Average “Retail Turnover per Capita” was evaluated to be $
2205.76 (SD = $ 543.19). The range of the variable was $ 1455.9 and $ 3014.6.
Median of “Retail Turnover per Capita” was $ 2180.2, implying the locational average.
The highest 25% values of Retail turnover were above $ 2793.4. The lowest 25%
Retail turnover were below $ 1652.95. The middle 50% observations of Retail turnover
ranged between $ 1652.95 and $ 2793.40. The C.V (coefficient of variation) was 25%.
8

Average “Final Consumption Expenditure” was $ 146019.85 (millions) (SD = $ 543.19
(millions)). The range was from $ 81889 (millions) to $ 233148 (millions). Median of
“Final Consumption Expenditure” was $ 103558.5 (millions). The highest 25% “Final
Consumption Expenditure” was above $ 192800.5 (millions). The lowest 25% “Final
Consumption Expenditure” was below $ 103558.5 (millions). The middle 50%
observations of the “Final Consumption Expenditure” ranged between $ 103558.5
(millions) and $192800.5 (millions), with a spread of $ 139137.5 (millions). The
coefficient of variation was 32%.
Table 5: Numerical Summary of Average Retail turnover and “Final Consumption Expenditure”
d. The Pearson’s correlation coefficient between “Retail Turnover per Capita” and “Final
Consumption Expenditure” was ( r=0. 988 ) . The measure of the correlation was positive,
strong, and statistically significant. There was almost a perfect positive association
between the two factors. “Final Consumption Expenditure” increased proportionately
with “Retail Turnover per Capita”.
e. A simple regression model was used to estimate “Final Consumption Expenditure” over
“Retail Turnover per Capita”. 131 pair of quarterly observations was used for this
estimation modelling. The regression equation for “Final Consumption Expenditure” was
evaluated as
Final Consumption Expenditure=-42102. 53+85 . 29*Retail Turnover per Capita .
The regression coefficient was β=85 . 29 , and “Retail Turnover per Capita” was a
statistically significant predictor ( t=71. 74 , p<0 . 05 ) .
Increase of 85.29 million dollars in “Final Consumption Expenditure” due to increase of
one dollar “Retail Turnover per Capita” was noted.
The ANOVA model was statistically significant (F = 5146.13, p < 0.05).
9

Table 6: Excel Regression output of “Final Consumption Expenditure” on “Retail Turnover per Capita”
f. The coefficient of determination was as R2=0 . 976 (From Table 6). The R2=0 . 976
implied that “Retail Turnover per Capita” was able to explain 97.6% variation of “Final
Consumption Expenditure”. As “Retail Turnover per Capita” was able to predict almost
100% variations of “Final Consumption Expenditure”, it can be considered as a very
good factor of the dependent variable.
g. Hypothesis test: “Retail Turnover per Capita” and “Final Consumption Expenditure”.
Null hypothesis: No linear relation between “Retail Turnover per Capita” and “Final
Consumption Expenditure” existed.
Alternate hypothesis: A positive and significantly linear relation exists between “Final
Consumption Expenditure” and “Retail Turnover per Capita”.
Significance level: 5%
Test statistic: t=71. 74 with p<0 . 05 (From Table 6).
Decision: The Null hypothesis is rejected and the alternate hypothesis is accepted.
“Retail Turnover per Capita” was significantly associated with “Final Consumption
Expenditure”.
h. Standard error of the estimate is se=1. 19 (From regression output).
The linear relationship between the factors was evident from conclusion of the
hypothesis testing.
A strong linear relation was evident from R2=0 . 976
The R2=0 . 976 described that 97.6% variation in “Final Consumption Expenditure”.
The ANOVA model was statistically significant ( F=5146 .13, p< 0. 05 ) .
So, it can be conclude that the linear fitting was really strong in this problem.
The line fit plot in Figure 8 reveals a strong fit of regression line (Allison, 2014, pp. 1-
13).
But, residual plot in Figure 9 indicates existence of a non-linear relationship.
10

1455.91512.31637.31685.51918.52183.82416.32706.42822.12878.23014.6
0
50000
100000
150000
200000
250000
Line Fit Plot
Retail Turnover per capita ($)
Final Consumption Expenditure ($ Millions)
Figure 8: Line fit plot (Blue lines: Actual, Orange lines: Predicted)
1200.01400.01600.01800.02000.02200.02400.02600.02800.03000.03200.0
-20000
-15000
-10000
-5000
0
5000
10000
15000
20000
25000
Residual Plot
Retail Turnover Per Capita ($)
Residuals
Figure 9: Residual plot of linear regression modelling
References
Allison, P.D., 2014, March. Measures of fit for logistic regression. In Proceedings of the
SAS Global Forum 2014 Conference (pp. 1-13).
Fisher, D., Drucker, S.M. and König, A.C., 2012. Exploratory visualization involving
incremental, approximate database queries and uncertainty. IEEE computer graphics
and applications, 32(4), pp.55-62.
Fornell, C., Rust, R.T. and Dekimpe, M.G., 2010. The effect of customer satisfaction on
consumer spending growth. Journal of Marketing Research, 47(1), pp.28-35.
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