Relationship between Food Expenditure and Take Home Pay in Australia

This study investigates the relationship between food expenditure and take home pay in Australia using a sample of 150 individuals. The report includes numerical summary measures, histograms, scatter plots, and hypothesis testing.

University
Quantitative Business Analysis
By
You’re Name
Date
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 <Your Name> 2018

Part 1
Problem Statement
A researcher wants to investigate the relationship between food expenditure and take home pay
in Australia by conducting a survey on a sample of individuals across the country. We are
required to use information provided about the researcher to answer a series of questions in
statistics.
Solutions
a. The researcher could use an interviewer administered survey either using personal
interview or telephone interview. The reason why the interviewer would opt for the above
method of survey is that it provides data of high quality, and would enable the researcher
to clear any misunderstandings between him and the respondent (Selvanathan,
Selvanathan, and Keller, 2017).
b. The researcher can use stratified random sampling. This method would help him divide
the population elements into various strata such as gender, income bracket, age bracket
and level of education, then select a simple random sample from each strata and
determine its expenditure and understand the relationship between various elements of
strata (Selvanathan, Selvanathan, and Keller, 2017).
c. The kind of issues the researcher would likely face in data collection include
misinterpretation of data, non-response errors and selector errors (Selvanathan,
Selvanathan, and Keller, 2017). Other issues would include financial problems in cases
where travel and human resources would be needed in the case of personal interview.
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Part 2
Problem Statement
The researcher has collected data from a sample of 150 individuals for the study where for each
individual, the weekly take-home pay and weekly food expenditure were recorded. We are
required to use the dataset and EXCEL to answer a series of questions.
Solutions
a. The researcher arrived at 8 class intervals using the formula:
class interval= Largest Value−Smallest Value
ApproximateClass width
For example, in the case of the weekly take-home pay the class interval 8 class intervals
are determined as follows:
class interval=1090−105
125 =7.88 ≅ 8 class intervals
b. The data for drawing the histogram for weekly home pay variable is prepared as below:
Table 1: Frequency Distribution for Weekly-Take-Home Pay
The histogram for weekly home pay variable is as shown below:
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Figure 1: Histogram for Weekly-Take-Home Pay
The data for drawing of the weekly food expenditure histogram is prepared as below:
Table 2: Frequency Distribution for Food Expenditure
The histogram for the weekly food expenditure is as shown below:
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Figure 2: Histogram for Weekly Food Expenditure
c. The numerical summary report for the weekly take-home pay variable is as shown in the
table below:
Table 3: Summary Statistic for Weekly-Take-home Pay
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The numerical summary report for the weekly food expenditure variable is as shown in
the table below:
Table 4: Summary statistic for food expenditure
d. Both the weekly take-home pay and the weekly food expenditure are positively skewed,
this can be seen by observing the histograms in both the cases. Additionally, when the
summary statistics are used, it is seen that the arithmetic mean in both cases is greater
than the median and the mode, followed by the median and the least of all is the mode
thereby indicating that the two distributions are positively skewed (Bruce, 2008).
However, weekly take-home pay is more positively skewed than the weekly food
expenditure.
Part 3
Problem
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We are required to assist the researcher to investigate the association between the weekly take
home pay and weekly food expenditure.
Solution
a. Since the researcher wants to determine how income impacts the spending patterns of
people, the independent variable (X) is the weekly take home pay while the dependent
variable (Y) is the food expenditure.
b. Appropriate plot to investigate the relationship between the two variables is a scatter plot
and is fitted with a linear trend line is shown below:
Figure 3: Scatter Plot
It shows there is a positive linear relationship between the two variables.
c. In excel, the numerical summary measure to measure the strength and the direction of the
linear relationship between the variables is computed using the “CORREL (A2:A151,
B2:B151)” function to show the correlation (Bissett, 2007). Where the A and B
represents the arrays of the two variables.
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Table 5: Correlation Coefficient
The correlation coefficient is 0.899683 meaning there is a strong positive relation
between weekly take-home pay and weekly food expenditure. It’s strong because the
value approaches 1 and its positive since the absence of a sign on the left of it is a default
indication of positive.
d. The regression summary table is as shown below:
Table 6: Regression Table
The least square regression equation can be estimated using the following equation:
Food expenditure ( Y ) =0.3133∗Weekly take home pay ( X ) + 40.8585
The intercept of the above regression equation is 40.8585 meaning that when the weekly
take home pay is zero, the value of food expenditure would be 40.8585. The slope of the
regression equation is 0.3133 and it indicates the degree with which the dependent
variable would be affected by the independent variable (Croucher, 2016). Example if the
income was 50 then the food expenditure would be:
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food expenditure= ( 0.3133∗50+ 40.8585 )=56.5235
e. Using the regression summary output table, we conduct a hypothesis testing to conclude
whether there is a linear relationship between weekly take-home pay and weekly food
expenditure. We begin by stating the null (There is no linear relationship) and the
alternative hypothesis (There is linear relationship).
The null hypothesis H0 : ρ=0
The alternate hypothesis H 1 : ρ ≠ 0
The test statistic is given by:
t= r √ n−2
√ 1−r 2
t= 0.899683 √150−2
√1−0.8996832 =10.945
0.4365 =25.07
The p-value for the test statistic obtained from the student t-distribution with n-2 (148) degrees
of freedom is P (T≤25.07) =1.0000. Since the p-value is greater than the significance level
(α=0.05) we accept the null hypothesis (Selvanathan, Selvanathan, and Keller, 2017). We
conclude that there is sufficient evidence that there is no linear relationship between the weekly
take-home pay and the food expenditure.
Method 2
Uses the slope and the regression line. The assumptions in this case are: There exists a linear
relationship between the dependent and the independent variable, for every value of X, the
probability distribution for the dependent variable have the equal standard deviation and for any
value of the independent variable, the values of the dependent variables are independent and
almost normally distributed. A significant relationship between the variables only exists when
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the slope is not equal to zero. The null hypothesis is therefore, the gradient is zero while the
alternative is that the slope of the regression equation is not zero.
The null hypothesis H0 : m=0
The alternate hypothesis H 1 :m≠ 0
The significance level is 0.05. Using a linear regression t-test. From the regression output table
the standard error of slope is 6.931, the slope is 40.8585, the degrees of freedom (n-2=148). The
test statistic will be
t= m
SE = 40.8585
6.931 =5.895
The p-value for test statistic from the t distribution table is P (T≤5.895) =1.000. Since the p-value
is greater than the significance level (α=0.05) we accept the null hypothesis (Selvanathan,
Selvanathan, and Keller, 2017). We conclude that there is sufficient evidence that there is no
linear relationship between the weekly take-home pay and the food expenditure.
Part 4
Problem Statement
a. We are to write a report about the findings of the study conducted by the researcher.
Solution
Report
The manner in which people spend in their daily life is greatly influenced by various factors such
as income, gender, age and the level of education. This research was aimed at determining the
relationship between food expenditure and take-home pay in Australia. A survey was conducted
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on a sample of 150 individuals with different incomes to determine how much they spend on
food using interviewer administered interview method.
The result of the survey indicated that the average weekly take-home pay was 501.5867. The
minimum take home pay for the sample was 105 while the maximum was 1090. The standard
deviation was 237.6588 and the median was 465. On the other hand, the average expenditure on
food was 197.991266. The minimum expenditure on food was 44.3137 while the maximum was
373.4779. The standard deviation was 82.7533 and the median was 190.87045. The above result
indicated that the distribution for food expenditure and the weekly take home pay was positively
skewed since in both cases, the mean is greater than the median and the median is greater than
the mode respectively. Additionally, a hypothesis was conducted to determine whether there was
a linear relationship between the weekly take-home pay and the expenditure on food for the
population from which the sample was collected. The result of the hypothesis indicated that there
was a strong positive relationship between the two variables meaning expenditure on food
increased with increase in the weekly take-home pay.
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References
Bruce, P. (2015). Introductory statistics and analytics. New Jersey: Wiley.
Bissett, B. (2007). Automated data analysis using Excel. Boca Raton: Chapman & Hall/CRC.
Croucher, J. S. (2016). Introductory mathematics & statistics. 6th ed. Australia: North Ryde,
N.S.W. McGraw-Hill Education.
Selvanathan, E. A., Selvanathan, S., and Keller, G. (2017). Business statistics abridged. 7th ed.
South Melbourne, Victoria: Cengage Learning.
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