Statistics Study Material

This study material provides comprehensive coverage of Statistics, including topics such as Australian exports, umbrella sales, time series, scatter plots, numerical summary statistics, correlation coefficients, regression analysis, and more. It includes solved assignments, essays, and dissertations to help you improve your understanding of statistical concepts and techniques.

STATISTICS
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Question 1
(a) The given Australian exports values (FY2004/05 and FY2014/15) to selected destinations
is presented below.
(b) The given Australian exports percentage (FY2004/05 and FY2014/15) is as represented in
the pie chart below.
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(c) The key observations based on the above graphical representation are indicated as
follows.
 In 2004-2005, the top export destination was Japan but this changed dramatically in
2014-2015 as the top export destination emerged as China with more than 450%
increase in absolute exports during the given period. This led to reduced weightage
for all the destinations in 2014-2015 as compared to their weights in 2004-2005.
 From part(a), the comparison of the exports in 2004-2005 and 2014-2015 indicates
that Australian exports to most destinations have seen an improvement. Two
countries which seem to deviate from the above trend are New Zealand where
exports remained almost at same level and UK where the exports actually declined.
Question 2
(a) Umbrella sold frequency table
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Classes Frequency Relative Frequency
30-40 2 0.05
40-50 4 0.10
50-60 8 0.20
60-70 11 0.28
70-80 8 0.20
80-90 5 0.13
90-100 2 0.05
(b) Umbrella sold frequency table
Classes Cumulative Frequency Cumulative Relative Frequency
30-40 2 0.05
40-50 6 0.15
50-60 14 0.35
60-70 25 0.63
70-80 33 0.83
80-90 38 0.95
90-100 40 1.00
(c) Histogram
(d) Ogive
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(e) Proportion of umbrella sales lesser than 60
Proportion of umbrella sold < than 60 = 0.05 + 0.10 + 0.20 = 0.35
(f) Proportion of umbrella sales greater than 70
Proportion of umbrella sold > than 70 = 0.20 + 0.13 + 0.05 = 0.38
Question 3
(a) Time series
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(b) Scatter plot
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In the above scatterplot, it is noteworthy that the independent variable is the “Retail Turnover
per capita” while the “Final consumption expenditure” is the dependent variable. The reason
for this arrangement is based on the premise that changes in the retail turnover per capita
would influence the final consumption expenditure.
(c) Numerical summary statistics
(d) Coefficient of correlation
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A correlation coefficient has two key aspects namely magnitude (0.9877) and sign (+). The
strength of the linear relationship is said to be very strong is the magnitude is close to 1 (i.e.
greater than 0.9) which is the case here. Also, the + sign would imply that a directly
proportional relationship is evident between the given variables (Hastie, Tibshirani and
Friedman, 2014).
(e) Regression analysis
The above regression model has two key aspects which are intercept coefficient and slope
coefficient. Intercept coefficient is -42,102.53 which estimates the final consumption
expenditure that is expected in the event that per capital retail turnover turns zero. Further,
the slope is 85.29 which highlights the change in final consumption expenditure (in $ mn) if
the independent variable or “Retail turnover per capita” is altered by 1 unit or $1 (Lieberman
et. al., 2013).
(f) Coefficient of determination also termed as R2
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Comment: The above highlighted value of 0.9755 represents the proportion of change in final
consumption expenditure that the given simple regression model is capable of explaining
(Lind, Marchal and Wathen, 2016).
(g) Slope is statistically significant?
Null hypothesis H0: β = 0 Slope is insignificant.
Alternative hypothesis Ha : β ≠ 0 Slope is significant.
The relevant test statistic (t value) and the corresponding p-value for hypothesis testing is
indicated below.
Significance level (Alpha) = 0.05
As p-value fails to exceed significance level, hence rejection of H0 would happen. This would
indicate that the slope is statistically significant and cannot be reduced to zero.
(h) In the above output, standard error is marked as 7363.23. The standard error provides
indication about the model fit owing to the residuals being captured on a cumulative
basis. The standard error in current case is low which indicates the superior fit of the
obtained simple regression model (Medhi, 2016).
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References
Hastie, T., Tibshirani, R. and Friedman, J. (2014) The Elements of Statistical Learning. 4th
ed. New York: Springer Publications.
Lieberman, F. J., Nag, B., Hiller, F.S. and Basu, P. (2013) Introduction To Operations
Research. 5th ed.New Delhi: Tata McGraw Hill Publishers.
Lind, A.D., Marchal, G.W. and Wathen, A.S. (2016) Statistical Techniques in Business and
Economics. 15th ed. New York : McGraw-Hill/Irwin.
Medhi, J. (2016) Statistical Methods: An Introductory Text. 4th ed. Sydney: New Age
International.
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