HI6007 Statistics and Research Methods Group Assignment Analysis

This assignment analyzes statistical data and research methods through three distinct questions. Question 1 examines Australian export data from 2004-05 and 2014-15, comparing export volumes and percentages across different countries using clustered bar charts and observations on trade shifts. Question 2 focuses on umbrella sales, presenting frequency, relative frequency, cumulative frequency, and cumulative relative frequency distributions, along with histograms and ogives to illustrate sales patterns. Question 3 explores economic data, including retail turnover per capita and final consumption expenditure, using time series plots, scatterplots, and a descriptive statistical summary to analyze their relationship. It employs Pearson's correlation coefficient and a simple linear regression model to quantify the correlation and predict final consumption expenditure based on retail turnover, providing regression statistics and conclusions on the significant positive linear association between the variables.

Statistics and Research Methods for Business Decision Making
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Question 1: Australian Exports
Table 1: Australia: Direction of exports Top 8 export markets for goods and services
Country A$ bn Percentage
2004-05 2014-15 2004-05 2014-15
China 15.9 90.3 15.39% 40.40%
Japan 28.2 46.6 27.30% 20.85%
United States 13.9 20.5 13.46% 9.17%
Republic of Korea 11 20.5 10.65% 9.17%
India 7.1 12.7 6.87% 5.68%
New Zealand 12.2 12.3 11.81% 5.50%
Singapore 5.8 12 5.61% 5.37%
United Kingdom 9.2 8.6 8.91% 3.85%
Subtotal 103.3 223.5 103.3 223.5
Source: DFAT and Austrade
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
Comparison of Australian exports (in A$ bn) in 2004-05 and 2014-15
2004-05
2014-15
Country of export destination
Exports (A$ bn)
Figure 1: Clustered Bar Chart for comparing Australian exports (in A$ bn) in 2004-05 and 2014-15
Observation: Comparison in exports is made for 2014-15 with respect to 2004-05 figures.
Exports have significantly increased to China, and then to Japan. Also, exports to United
States, Republic of Korea, India, and Singapore have increased considerably. A very small
increase in exports to New Zealand is noted, whereas, exports to United Kingdom has
decreased by 0.6 billion Australian dollars.
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China Japan United
States Republic
of Korea India New
Zealand Singapore United
Kingdom
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
40.00%
45.00%
15.39%
27.30%
13.46% 10.65%
6.87%
11.81%
5.61% 8.91%
40.40%
20.85%
9.17% 9.17% 5.68% 5.50% 5.37% 3.85%
Comparison of Australian exports (in A$ bn) in 2004-05 and 2014-15
2004-05
2014-15
Country of export destination
Percentage
Figure 2: Clustered Bar Chart for comparing percentage of Australian
Exports (in A$ bn) in 2004-05 and 2014-15
Observation: Comparison in percentage exports is made for 2014-15 with respect to
2004-05 figures. Interestingly, exports have increased by 25.01% to China, and the impact
is felt on the entire scenario. Exports have decreased by 6.45% to Japan. Also, exports to
United States, Republic of Korea, India, and Singapore have decreased respectively by
4.29%, 1.48%, 1.19%, and 0.24%. A heavy decrease in exports to New Zealand by 6.31%
is noted, and exports to United Kingdom have also decreased by 5.06% ( Hopkins, 2017,
pp.1-4).
Total export in 2004-2005 was 103.3 billion Australian dollars, which increased to 223.5
billion Australian dollars in 2014-15. Export volume in Australian dollars has considerably
increased in 2014-15 compared to 2004-05. From part A, it can be noted that export to
almost all the countries have increased in volume (except United Kingdom). Interestingly,
comparison in export percentages revealed that export to China has grown at a rapid rate,
whereas, percentage export to rest of the countries have decreased. This indicates that
Australian trade is favouring China compared to other countries for their export, which
eventually has a retarding impact on growth of export to other countries (Bluman, 2009).
Question 2: Umbrella sales
Frequency distribution of 40 days umbrella sales has been presented in Table 2, and
relative frequency distribution table has been presented in Table 3. Class limits, along with
classes and frequencies, have been included in both the tables.
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Table 2: Frequency distribution of 40 days umbrella sales
Class Limits Classes Frequency
30 40 30-40 2
40 50 40-50 4
50 60 50-60 8
60 70 60-70 11
70 80 70-80 8
80 90 80-90 5
90 100 90-100 2
Table 3: Relative frequency distribution of 40 days umbrella sales
Class Limits Classes Frequency Relative
Frequency
30 40 30-40 2 0.05
40 50 40-50 4 0.10
50 60 50-60 8 0.20
60 70 60-70 11 0.28
70 80 70-80 8 0.20
80 90 80-90 5 0.13
90 100 90-100 2 0.05
Cumulative frequency distribution of 40 days umbrella sales has been presented in Table
4, and cumulative relative frequency distribution table has been presented in Table 5.
Table 4: Cumulative frequency distribution of 40 days umbrella sales
Classes Frequency Relative
Frequency
Cumulative
Frequency
30-40 2 0.05 2
40-50 4 0.10 6
50-60 8 0.20 14
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60-70 11 0.28 25
70-80 8 0.20 33
80-90 5 0.13 38
90-100 2 0.05 40
Table 5: Cumulative relative frequency distribution of 40 days umbrella sales
Classe
s
Frequenc
y
Relative
Frequency
Cumulative
Frequency
Cumulative Relative
Frequency
30-40 2 0.05 2 0.05
40-50 4 0.10 6 0.15
50-60 8 0.20 14 0.35
60-70 11 0.28 25 0.625
70-80 8 0.20 33 0.825
80-90 5 0.13 38 0.95
90-100 2 0.05 40 1
Relative frequency histogram with class mid points in the x-axis and relative frequencies in
the data label of the bars of the histogram has been presented in Figure 3.
Table 6: Frequency Distribution of Umbrella Sales
Classe
s
Frequenc
y
Relative
Frequency
Cumulative
Frequency
Cumulative Relative
Frequency
30-40 2 0.05 2 0.05
40-50 4 0.10 6 0.15
50-60 8 0.20 14 0.35
60-70 11 0.28 25 0.625
70-80 8 0.20 33 0.825
80-90 5 0.13 38 0.95
90-100 2 0.05 40 1
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35 45 55 65 75 85 95
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.050
0.100
0.200
0.275
0.200
0.125
0.050
Histogram of Relative Frequency for Umbrella
Sales
Umbrella sales
Relative Frequency
Figure 3: Relative frequency Histogram of 40 days umbrella sales
With less than type cumulative frequency in Table 7, an Ogive has been plotted in Figure
4.
Table 7: Cumulative Frequency Table of Umbrella Sales
Umbrella Sales Cumulative
Frequency
29 0
40 2
50 6
60 14
70 25
80 33
90 38
100 40
20 30 40 50 60 70 80 90 100 110
0
5
10
15
20
25
30
35
40
45
Ogive for Umbrella Sales
Umbrella Sales
Cumulative Frequency
Figure 4: Ogive for of 40 days umbrella sales
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From the cumulative relative frequency distribution, 0.35 or 35.0% proportion of grades are
less than 60.
From the cumulative relative frequency distribution, 0.625 or 62.5% proportion of grades
are less than 70. Hence, 37.5% proportion of grades is more than 70.
Question 3: Economic consultancy firm
Sep-1983
Mar-1986
Sep-1988
Mar-1991
Sep-1993
Mar-1996
Sep-1998
Mar-2001
Sep-2003
Mar-2006
Sep-2008
Mar-2011
Sep-2013
Mar-2016
1200.0
1600.0
2000.0
2400.0
2800.0
3200.0
Time Series Plot of Retail turnover per capita ($)
Quarter
Retail turnover
per capita ($)
Figure 5: Time series plot of retail turnover per capita ($)
Sep-1983
Feb-1986
Jul-1988
Dec-1990
May-1993
Oct-1995
Mar-1998
Aug-2000
Jan-2003
Jun-2005
Nov-2007
Apr-2010
Sep-2012
Feb-2015
57000
107000
157000
207000
257000
Time Series Plot of Final
Consumption Expenditure ($
Millions)
Quarter
Final
Consumption
Expenditure ($
Millions)
Figure 6: Time series plot of final consumption expenditure ($ millions)
Retail turnovers per capita ($), and final consumption expenditure ($ million) are noted to
have a steady linear growth with time. Interestingly, similar increasing trend in both the
graphs are noted.
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1000.0 1500.0 2000.0 2500.0 3000.0 3500.0
0
50000
100000
150000
200000
250000
Final Consumption Expenditure on Retail Turnover
Per-capita
Retail turnover per capita ($)
Final Consumption Expenditure ($
Millions)
Figure 7: Scatterplot to investigate relationship between retail turnovers per capita (X-axis)
And final consumption expenditure (Y-axis)
A highly positive and significant linear association is observed between retail turnovers per
capita, and final consumption expenditure. The positive correlation is noted to be almost
perfect correlation (Pearson’s correlation coefficient equal to 1 is called a perfect positive
correlation). Both the variables are noted to increase at a similar ratio.
A descriptive numerical summary with mean, standard deviation, variance, coefficient of
variation, median, range, smallest and largest values, and the three quartiles, for retail
turnovers per capita, and final consumption expenditure has been presented in Table 5.
Table 8: Descriptive Statistics for Retail turnover per capita ($) and Final Consumption Expenditure ($ Millions)
Descriptive Statistics
Retail turnover per capita
($)
Final Consumption
Expenditure ($ Millions)
Mean 2205.76 146019.85
Median (Second
Quartile) 2180.20 139137
Mode 2852.80 #N/A
Standard Deviation 543.19 46904.33
Variance 295059.60 2200016261.88
Coefficient of Variation 24.63% 32.12%
Range 1558.70 151259.00
Minimum 1455.90 81889.00
Maximum 3014.60 233148.00
First Quartile 1652.95 103558.50
Third Quartile 2793.40 192800.50
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Average retail turnover per capita ($) was $ 2205.76 (SD = $ 543.19), which varied
between $ 1455.90 and $ 3014.60. Relative risk or coefficient of variation (CV) for retail
turnover per capita ($) was 24.63%. Bottom 25% observations of retail turnover per capita
($) were below $ 1652.95. The top 25% retail turnovers were above $ 2793.40.
Average final consumption expenditure was 146019.85 ($ Millions) (SD = 46904.33),
which varied between 81889 ($ Millions) and 233148 ($ Millions). CV for final consumption
expenditure ($ Millions) was 32.12%. Bottom 25% observations of retail turnover per
capita ($) were below $ 103558.50 ($ Millions).The top 25% retail turnovers were above
192800.50 ($ Millions) (Mishra et al., 2019, p.67).
The
relationship between retail turnover per capita ($) and final consumption expenditure ($
Millions) was found by Pearson’s correlation coefficient. The Excel output of correlation
has been presented in Table 6 (Schober, Boer, and Schwarte, 2018, pp.1763-1768).
Table 9: Pearson’s between retail turnover per capita ($) and final consumption expenditure ($ Millions)
Retail turnover per
capita ($)
Final Consumption Expenditure
($ Millions)
Retail turnover per capita ($) 1
Final Consumption Expenditure
($ Millions) 0.9877 1
A highly positive and significant linear association is observed between retail turnovers per
capita, and final consumption expenditure ( r=0 . 99 , p<0 . 05) . The positive correlation is
noted to be almost perfect correlation (Pearson’s correlation coefficient equal to 1 is called
a perfect positive correlation). Final consumption expenditure ($ Millions) is noted to
increase in equal ratio to that of retail turnovers per capita ($).
A simple linear regression model is constructed to estimate final consumption expenditure
($ Millions) on retail turnovers per capita ($). The regression summary and diagnostics
have presented in Table 7.
The linear regression equation is evaluated as,
Final consumptin expenditure ( $ Millions)=85 .287*Retail turnovers per capita ( $)−42102 .533
The regression coefficient of 85.29 for retail turnovers per capita ($) implied that for one
unit increase in retail turnovers per capita by one dollar will increase final consumption
expenditure approximately by 85.29 million dollars. The coefficient of retail turnovers per
capita ($) is noted to be statistically significant ( t=71. 734 , p<0 . 05 ) in establishing a strong
linear relationship.
Table 10: Simple linear regression to estimate final consumption expenditure on retail turnovers per capita
Regression Statistics
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Multiple R 0.988
R Square 0.976
Adjusted R Square 0.975
Standard Error 7363.225
Observations 131
ANOVA
df SS MS F
Significance
F
Regressio
n 1
279008109980.0
68
279008109980.0
68
5146.12
9 0.000
Residual
12
9 6994004064.176 54217085.769
Total
13
0
286002114044.2
44
Model
Coefficient
s Standard Error t Stat P-value
Lower
95% Upper 95%
Intercept -42102.533 2700.165
-
15.593 0.000 -47444.876 -36760.191
Retail turnover per
capita ($) 85.287 1.189 71.737 0.000 82.935 87.639
The coefficient of determination is R2=0 . 976 which implied that the retail turnover per
capita is able to explain and predict 97.6% proportion of variation in final consumption
expenditure.
Null hypothesis: There is no linear relationship between RETAIL TURNOVER PER
CAPITA and FINAL CONSUMPTION EXPENDITURE.
Alternate hypothesis: There is significant positive linear correlation between RETAIL
TURNOVER PER CAPITA and FINAL CONSUMPTION EXPENDITURE.
Significance Level: α=0 . 05
Test Statistics: Pearson’s correlation coefficient r ( 129 ) =0 . 988 with p < 0.05
Conclusion: The null hypothesis is rejected at 5% level of significance, implying that there
was a significant positive linear correlation between RETAIL TURNOVER PER CAPITA
and FINAL CONSUMPTION EXPENDITURE.
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The standard error of the regression estimate is SE=1 .189 . The predictor, retail turnovers
per capita is noted to be statistically significant in estimating final consumption expenditure
( t=71. 734 , p<0 . 05 ) , even at 1% level of significance. The relation between the variables is
also noted to be linear.
The regression model is noted to statistically significant ( F ( 1 , 129 ) =5146 .13 , p<0 . 05 ) . Values
of R2=0 . 976 and adjusted R2=0 . 975 are observed to be almost equal. Hence, the linear
regression model fitness is really strong (Renaud, and Victoria-Feser, 2010, pp.1852-
1862).
References
Bluman, A.G., 2009. Elementary statistics: A step by step approach. New York: McGraw-
Hill Higher Education.
Hopkins, W.G., 2017. Spreadsheets for analysis of controlled trials, crossovers and time
series. Sportscience, 21, pp.1-4.
Mishra, P., Pandey, C.M., Singh, U., Gupta, A., Sahu, C. and Keshri, A., 2019. Descriptive
statistics and normality tests for statistical data. Annals of cardiac anaesthesia, 22(1), p.67.
Renaud, O. and Victoria-Feser, M.P., 2010. A robust coefficient of determination for
regression. Journal of Statistical Planning and Inference, 140(7), pp.1852-1862.
Schober, P., Boer, C. and Schwarte, L.A., 2018. Correlation coefficients: appropriate use
and interpretation. Anesthesia & Analgesia, 126(5), pp.1763-1768.
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