Quantitative Analysis for Business: QAB105 Assignment 2018

This assignment solution addresses a quantitative analysis problem for a business course (QAB105). It begins with a discussion of survey techniques, specifically focusing on the administered questionnaire method and convenience sampling for collecting data on salary and food expenditure of Australian families. The solution then delves into frequency distributions, detailing how to determine interval lengths and presents frequency distributions and histograms for take-home pay and weekly food expenditure. Descriptive statistics, including mean, median, standard deviation, and variance, are provided for both variables. The analysis further explores the relationship between take-home pay and weekly food expenditure, using scatter plots, Pearson's correlation, and regression analysis. The regression analysis includes the least square regression equation, hypothesis testing for linearity, and residual plots. The conclusion summarizes the findings, highlighting the positive correlation between the variables and referencing relevant literature. The assignment utilizes Excel for calculations and graphical representations.

RUNNING HEAD: QUANTITATIVE ANALYSIS FOR BUSSINESS
Quantitative Analysis for Business
Course Code: QAB105
Semester: 2
Year: 2018

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QUANTITATIVE ANALYSIS FOR BUSSINESS
Answer
(a) The most suitable survey technique for collecting salary and food expenditure of the
Australian families, the researcher could have utilized administered questionnaire method
for survey. The survey could have been conducted with selected sample from the
population.
(b) The researcher could have used convenience sampling for collecting the specified fields
of data from the survey.
Figure 1 : Standard Normal Distribution Curve for the Take Home Pay
(c) The issue related to convenience sampling is the biasness of the data. The researcher
could have face problems of existence of probable biasness in the sample, as convenience
sampling helps in collecting information from a specific sector of the population.

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QUANTITATIVE ANALYSIS FOR BUSSINESS
Answer
(a) Interval length in a frequency distribution is generally decided based on the distribution of
data. The class interval length should be such that frequency in that particular class should
not go ideally below 5, for proper representation of that class. There are various ways to
decide number of intervals for a frequency distribution. Among the various existing
methods, the researcher could have adopted the Struge’s rule for selecting number of
intervals. The rule prescribes to select k-number of intervals, where
where was the number of data present (Olson, 2018).
(b) Histograms along with frequency distributions have been provided below for Take home pay
and weekly food expenditure for Australians.
Table 1: Frequency Distribution of Take Home Pay
Class Lower Limit Upper Limit Mid-Point Frequency
100-225 100 225 162.5 15
225-350 225 350 287.5 25
350-475 350 475 412.5 35
475-600 475 600 537.5 26
600-725 600 725 662.5 19
725-850 725 850 787.5 16
850-975 850 975 912.5 9
975-1000 975 1100 1037.5 5

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QUANTITATIVE ANALYSIS FOR BUSSINESS
Figure 2: Take Home Pay Distribution
Table 2: Frequency Distribution of Weekly Food Expenditure
Class Lower Limit Upper Limit Mid-Point Frequency
0-50 0 50 25 3
50-100 50 100 75 16
100-150 100 150 125 29
150-200 150 200 175 30
200-250 200 250 225 27
250-300 250 300 275 25
300-350 300 350 325 15
350-400 350 400 375 5
Figure 3: Weekly Food Expenditure Distribution

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QUANTITATIVE ANALYSIS FOR BUSSINESS
(c) In the following table the numerical/ descriptive summary for the variables Take home pay
and Weekly food expenditure have been presented.
Table 3: Descriptive Summary of Take Home Pay and Weekly Food Expenditure
Descriptive Statistics 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
Second Quartile 465.00 190.87
Third Quartile 677.25 259.69
(d) There was positive/ right skewness in the distribution of take home pay for the Australians.
The histogram presents a distribution with a comparatively long right tail. Hence, it was
prominent that some families were present with high income level, but with less frequency
(Wan, Wang, Liu, & Tong, 2014).
(e) The Weekly food expenditure distribution was found to be exhibit a normal distribution.
From the descriptive summary, the median was noted to be almost equal to mean, indicating
that the skewness of weekly food expenditure was almost equal to zero.

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QUANTITATIVE ANALYSIS FOR BUSSINESS
Answer
(a) The probable independent variable was Take home pay, as Weekly food expenditure was
supposed to depend on take home pay of a family (dependent variable).
(b) Two-way scatter plot was chosen to present the relation between Take home pay and
Weekly food expenditure graphically (Chambers, 2017). The trend of the data was positive
in nature, where weekly food expenditure was increasing with take home pay. A linear trend
line was fitted, and the equation was found as: Take home pay = 0.313*Weekly food
expenditure +40.85.
Figure 4: Scatter Plot for Weekly Food Expenditure for Take Home Pay
(c) Both the variables were continuous in nature, and the Pearson’s correlation was chosen to
measure the association between Take home pay and Weekly food expenditure. The value of
the correlation coefficient was found to be r = 0.89. The direction of the correlation was
positive and significant in nature. Hence, weekly food expenditure was found to be highly
affected take home pay in a family
(d) The regression summary for estimating weekly food expenditure by take home salary was
constructed in MS Excel and has been presented in the below table. The least square
regression equation was estimated as
The slope of regression for Take home pay was 0.313 and the intercept was found to be
40.86. The values signified that there was a strong positive linear relation existed between
the variables. For increase in take home pay by one unit would have positively increased
weekly food expenditure significantly (t = 25.07, p < 0.05) by 0.31 units. The intercept
indicated high expenditure on weekly food purchase expenditure even for zero income level.

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QUANTITATIVE ANALYSIS FOR BUSSINESS
Table 4: Regression Analysis Summary Output
Regression Summary Output
Multiple R 0.8997
R Square 0.8094
Adjusted R Square 0.8081
Standard Error 36.2474
Observations 150
ANOVA
df SS MS F Significance F
Regression 1.0000 825915.4606 825915.4606 628.6120 0.0000
Residual 148.0000 194453.0108 1313.8717
Total 149.0000 1020368.4714
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 40.8585 6.9309 5.8951 0.0000 27.1622 54.5548
Take-home pay 0.3133 0.0125 25.0721 0.0000 0.2886 0.3380
(e) For testing the linearity the following hypothesis testing was done.
The linear regression slope was considered as .
The null hypothesis H0: ( =0).
The two way alternate hypothesis HA:
The level of significance was 5% ( )
Choice of the test statistic was t-statistic, as population parameters were not present.
Calculated value of
The estimated value = 0.3133 and the standard error = 0.0125 from the regression output.
Calculated P-value for two tailed curve was for degrees of freedom
= 150 – 1 =149.
Hence, there existed significant evidence at 5% level of significance to reject the null
hypothesis.
The confidence interval for population mean was calculated as
.
The estimated population mean = 0 was not included in the confidence interval, and it
evident that the null hypothesis would be rejected.
Therefore it was established that Take home pay and Weekly food expenditure were linearly
associated. The residual plot below reflected that the assumption of normality was satisfied
for the regression model (Chatterjee, & Hadi, 2015).

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QUANTITATIVE ANALYSIS FOR BUSSINESS
Figure 5: Residual Plot of the Regression
Answer
(a) A non probability sampling with a questionnaire survey was used to find the relation of
weekly expenditure on food and take home pay of the Australian family. The researcher
intended to find the effect of take home pay with the expenses on weekly food purchases of
the families. The convenience sampling was conducted, and data from 150 families was
collected. The distribution of weekly take home salary (M = $ 501.59, SD = $ 237.66) was
right skewed, and the distribution weekly food expenditure (M = $ 197.79, SD = 82.75) was
found to be normally distributed. From the distribution of take home pay it was noted that
some of the families earning was higher compared to other families. The measure of
association between the variables was assessed by Pearson’s correlation, and two variables
were found to be having high positive and linear relation. The weekly food expenditure was
found to increase by a margin of 31.3% for 100% increase in take home pay.
The relation was significantly linear, and the result of the current study was in line with
previous literatures (Bellemare, 2015; Ravallion, 2017). The current relation indicated excess
spending tendencies towards food compared to the entire scenario.
.

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References
Bellemare, M. F. (2015). Rising food prices, food price volatility, and social unrest. American
Journal of Agricultural Economics, 97(1), 1-21.
Chambers, J. M. (2017). Graphical Methods for Data Analysis: 0. Chapman and Hall/CRC.
Chatterjee, S., & Hadi, A. S. (2015). Regression analysis by example. John Wiley & Sons.
Olson, E. (2018). A Study of the Effects of Histogram Binning on the Accuracy and Precision of
Particle Sizing Measurements. Pharmaceutical Technology, 29.
Ravallion, M. (2017). Poverty comparisons. Routledge.
Wan, X., Wang, W., Liu, J., & Tong, T. (2014). Estimating the sample mean and standard
deviation from the sample size, median, range and/or interquartile range. BMC medical
research methodology, 14(1), 135.