Quantitative Analysis for Business: Survey, Histograms, Correlation and Regression Analysis

This article discusses the use of cross-sectional survey and non-probability sampling to identify the relationship between food expenditure and take home pay in Australia. It also includes histograms, numerical summary, correlation and regression analysis of the data collected.

QAB105 Quantitative Analysis for Business
Semester 2 2018
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Answer Part 1
(a) The researcher could use the cross-sectional survey to utilize the questionnaire to inquire
about food expenditure and take home pay in Australia, and to identify the relationship
between the two variables as a comparative study.
(b) The researcher could use non-probability sampling or convenience sampling for the
survey.
(c) The researcher might face problems related to biasness in the data collection procedure.
Answer Part 2
(a) The length of the intervals of Take Home Pay (THP) and Weekly Food Expenditure (WFE)
were considered approximately equal to the value of Range (RTHP = 985, RWFE = 329.16)
divided by number of intervals (N = 8). This subjective decision was taken by the researcher
based on the range of the variables such that each class does have at least 5 frequencies for
proper and significant distribution of the observations. The Sturges’ rule for k = 1+log2 n
could have been used for determining the number of subintervals (k = intervals, n =
observations) (Scott, 2015).
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(b) Histograms for both the variables have been provided in Figure 1 and Figure 2.
Figure 1: Histogram for Take Home Pay
Figure 2: Histogram for Weekly Food Expenditure
(c) Numerical summary for Take Home Pay (THP) and Weekly Food Expenditure (WFE) have
been provided in Table 1.
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Table 1: Descriptive Summary of THP and WFE
Descriptives Take-home pay Weekly food expenditure
Mean 501.59 197.99
Median 465.00 190.87
Standard Deviation 237.66 82.75
Variance 56481.72 6848.11
Smallest 105.00 44.31
Largest 1090.00 373.48
Range 985.00 329.16
First Quartile 315.00 130.43
Third Quartile 677.25 259.69
(d) The histogram for Take-home pay was found to be positively skewed with plenty of data in
the right tail. The mean of the distribution was less than the median, establishing the claim
of positive skewness.
(e) The histogram for Weekly food expenditure was found to be almost normally distributed
with approximately equal number of data in both the tails. The mean of the distribution was
almost equal to the median, establishing the claim of normal nature of the frequency
distribution.
Answer Part 3
(a) The most likely preference for independent variable was Take home pay, and Weekly food
expenditure for the dependent variable.
(b) The relation between the variables was established by two-way scatter plot. The relation
between the two variables was found to be highly positive.
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(c) The correlation between Take home pay and Weekly food expenditure was evaluated by
Spearman’s correlation coefficient (R = 0.8997) in MS Excel. The correlation coefficient
was positive and signified very high positive association between Take home pay and
Weekly food expenditure. Hence, people with high take home salary were spending more on
foodstuff.
(d) The regression summary has been presented in Table 2. From the summary table the
coefficient of Take home pay was found to be 0.31 and the slope was found to be 40.86.
Hence, the estimated linear regression line was calculated as,
From the slope a significant linear relationship (t = 25.07, p < 0.05) was visible between the
variables. The angle of the line implied an inclination of almost 17 degrees,
which indicated a practically significant linear relation. The intercept of 40.86 indicated food
purchase expenditure of $ 40.86 for zero income or take home pay.
Table 2: Summary Output of Regression Analysis
(e) Let the regression slope be denoted by .
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The null hypothesis for the linear relation was taken as H0: ( =0).
And the un-directional alternate hypothesis was considered as H1:
As the sample size was large enough (n>30), the distribution of the variables was considered
to be normal. Level of significance was considered at 5%.
The test statistic was calculated as where the estimated value of
was 0.3133 and the standard error of mean was 0.0125 (from Excel output).
P-value was calculated as at 149 degrees of freedom. It signified that
there was a statistically significant evidence for rejecting the null hypothesis.
The confidence interval was calculated as . The
estimated population parameter was outside the confidence interval, and it was an evidence
of the fact that null hypothesis should be rejected.
Hence, the alternate hypothesis was accepted at 5% level of significance. Therefore it was
concluded that Take home pay and Weekly food expenditure were linearly associated.
Answer Part 4
(a) A random sampling with a cross sectional survey was conducted to find the association
between weekly take home pay and weekly expenditure on food by people of Australia.
Overall 150 responses were collected from Australian people. Average weekly take home
salary (M = $ 501.59, SD = $ 237.66) was found to be right skewed, signifying that there
were some people who were getting high or very high salary. Average weekly expenditure
were (M = $ 197.79, SD = 82.75) found to be normally distributed, indicating balanced
expenditure pattern. There was a significant and very high correlation between weekly take
home pay and weekly expenditure. For one unit increase in take home pay, weekly food
expenditure was found to increase by 0.31 units. The relation was significantly linear, and a
31% growth in food expenditure was observed for increase in salary. The trend in the present
study was in line with the hypothetical models (Beck, 2018). In a similar study food
expenditure pattern was found to be positively and linearly associated with income of
Canadian people (Kirkpatrick, and Tarasuk, 2003). In China, a similar trend was observed for
expenditure for consumption of food outside home (Daniels, and Glorieux, 2015).
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References
Beck, U., 2018. What is globalization?. John Wiley & Sons.
Daniels, S. and Glorieux, I., 2015. Convenience, food and family lives. A socio-typological study
of household food expenditures in 21st-century Belgium. Appetite, 94, pp.54-61.
Kirkpatrick, S. and Tarasuk, V., 2003. The relationship between low income and household food
expenditure patterns in Canada. Public health nutrition, 6(6), pp.589-597.
Scott, D. W. (2015). Multivariate density estimation: theory, practice, and visualization. John
Wiley & Sons.
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