Holmes Institute HI6007: Statistics and Research Methods Assignment

This assignment analyzes the relationship between Retail Turnover per Capita and Final Consumption Expenditure using statistical methods. The analysis includes plotting time series data for both variables, creating a scatterplot to visualize their relationship, and calculating descriptive statistics such as mean, median, variance, and correlation. A simple linear regression model is developed to predict Final Consumption Expenditure based on Retail Turnover per Capita. The assignment explores the model's coefficients, standard error, and coefficient of determination (Adjusted R-squared) to assess its fit and explanatory power. Hypothesis testing is conducted to determine the significance of Retail Turnover per Capita in predicting Final Consumption Expenditure, concluding that it has a significant increasing effect. The findings suggest a strong positive correlation between the two variables, with the model explaining a substantial portion of the variance in Final Consumption Expenditure.

STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING

STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING
Question Three
Part a
From Figure 1: Plot of Retail Turnover per Capita against Time we observe that the
variable displays a generally increasing trend over time.
Sep-1983
Nov-1984
Jan-1986
Mar-1987
May-1988
Jul-1989
Sep-1990
Nov-1991
Jan-1993
Mar-1994
May-1995
Jul-1996
Sep-1997
Nov-1998
Jan-2000
Mar-2001
May-2002
Jul-2003
Sep-2004
Nov-2005
Jan-2007
Mar-2008
May-2009
Jul-2010
Sep-2011
Nov-2012
Jan-2014
Mar-2015
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
3500.0
Plot of Retail Turnover per Capita against Time
Time
Retail Turnover per Capita
Figure 1: Plot of Retail Turnover per Capita against Time
From Figure 2: Plot of Final Consumption Expenditure against Time we observe that the
variable also displays a generally increasing trend over time.
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STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING
Sep-1983
Nov-1984
Jan-1986
Mar-1987
May-1988
Jul-1989
Sep-1990
Nov-1991
Jan-1993
Mar-1994
May-1995
Jul-1996
Sep-1997
Nov-1998
Jan-2000
Mar-2001
May-2002
Jul-2003
Sep-2004
Nov-2005
Jan-2007
Mar-2008
May-2009
Jul-2010
Sep-2011
Nov-2012
Jan-2014
Mar-2015
0
50000
100000
150000
200000
250000
Plot of Final Consumption Expenditure against Time
Time
Finak Consumption Expenditure
Figure 2: Plot of Final Consumption Expenditure against Time
Part b
This study is interested in determining whether Retail Turnover per Capita is a good
predictor of Final Consumption Expenditure. Thus the Final Consumption Expenditure is
the dependent variable (y), hence on the Y axis, while Retail Turnover per Capita is the
independent variable (x), hence on the X axis on Figure 3: Scatterplot of Final
Consumption Expenditure Versus Retail Turnover per Capita.
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STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING
1200.0 1400.0 1600.0 1800.0 2000.0 2200.0 2400.0 2600.0 2800.0 3000.0 3200.0
0
50000
100000
150000
200000
250000
Scatterplot of Final Consumption Expenditure Versus Retail
Turnover per Capita
Retail Turnover per Capita
Final Consumption Expenditure
Figure 3: Scatterplot of Final Consumption Expenditure Versus Retail Turnover per Capita
Part cTable 1: Numerical Summary Descriptive Statistics
Retail Turnover per Capita Final Consumption
Expenditure
Mean 2205.761832 146019.855
Median 2180.2 139137
Variance 295059.5953 2200016262
Std.Dev 543.1938837 46904.33095
Coef.Var 24.62613487 32.12188573
Range 1558.7 151259
Min 1455.9 81889
Max 3014.6 233148
Q1 1641.8 103169
Q2 2180.2 139137
Q3 2799.7 192828
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STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING
Part d
From Table 2: Correlation we observe that the variables have a correlation = 1. This
implies that the two variables have a perfect positive association (very strong
correlation), an increase in the Retail Turnover per capital equally results in an increase
in the Final Consumption Expenditure.Table 2: Correlation
Correlation = 1
Part e
From Table 3: Regression Summary Output, the estimated simple linear regression
model is given by:
Fina lConsumptionExpenditure=−42102.5+85.3 RetailTurnover per Capita
When the value of the Retail Turnover per Capita = 0, the value of the Final
Consumption Expenditure = -42102.5 ($ millions). Also, a unit change in the Retail
Turnover per Capita results in an 85.3 ($) change in the Final Consumption
Expenditure.Table 3: Regression Summary Output
Multiple R 0.987697
R Square 0.975546
Adjusted R
Square
0.975356
Coefficient
s
Standard
Error
t Stat P-value
Intercept -42102.5 2700.165 -15.5926 1.82E-31
X Variable 1 85.28681 1.18889 71.73652 7.9E-106
Part f
The coefficient of determination refers to a statistical measure of the extent to which an
estimated model explains the association between the dependent variable(s) and the
independent variable(s) (Barbara & Susan, 2014; Freedman, 2009).
From Table 3: Regression Summary Output, the Adjusted R Squared = 0.9754. This
value of the coefficient of determination implies that the model explains up to 97.54% of
the association between the Final Consumption Expenditure and the Retail Turnover
per Capita.
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STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING
Part g
Hypothesis
H0: Retail Consumption per Capita is significant.
H1: Retail Consumption per Capita is not significant.
Test-Statistics
From Table 3: Regression Summary Output, the t statistic (t stat) and p-value for the X
Variable 1 are 71.74 and 7.9e-106.
Decision Rule
Considering 5% significance level, then α = 0.05. The p-value = 7.9e-106 < α = 0.05,
therefore we fail to reject H0 and conclude that Retail Consumption per Capita is
significant.
Also, since the coefficient for the Retail Consumption per Capita = +85.23, then we
conclude that it has a significant increasing effect on the Final Consumption
Expenditure.
Part h
From Table 3: Regression Summary Output, the estimate standard error (Se) = 1.189.
The coefficient of determination = 0.9754, hence the model explains up to 97.54% of
the association between the Final Consumption Expenditure and the Retail Turnover
per Capita. This value is significantly high (> 60%), thus the model can be described as
fit for the data.
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STATISTICS AND RESEARCH METHODS FOR BUSINESS DECISION MAKING
References
Barbara, I & Susan, D 2014, Introductory Statistics, 1st edn, OpenStax CNX, New York.
Freedman, DA 2009, Statistical Models: Theory and Practice, 1st edn, Cambridge
University Press, London.
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