Income can have significant effects on people’s spending patterns. Research studies have revealed that consumer expenditure is influenced by various factors such as their income,gender, age and level of education. In order to investigate the relationship between food expenditure and take home pay in Australia, a researcher plans to survey a sample of individuals across the country.

Briefly explain
(a) What type of survey method could the researcher use and why
(b) What sampling method could the researcher use to select his/her sample and why
(c) What kind of issues the researcher may face in this data collection

Suppose a researcher has collected data form a sample of 150 individuals for this study. For each individual, the weekly take-home pay and weekly food expenditure were recorded. The data are stored in file FOODEXP.XLS. Using this data set and EXCEL, answer the questions below.

First, the researcher wishes to use the graphical descriptive methods and numerical descriptive mesures to present the data for the two variables.
(a) The researcher decides to use 8 class intervals such as 100<X≤225, 225<X≤350,350<X≤475, ….., 975<X≤1100 for weekly take-home pay variable and class intervals 0<X≤50, 50<X≤100, 100<X≤150, …. , 350<X≤400 for weekly food expenditure variable.
Explain how the researcher would have decided on 8 class intervals.
(b) Draw a histogram for each variable using appropriate BIN values from part (a).
(c) Prepare a numerical summary report for the two variables; weekly take-home pay and food expenditure by including summary measures such as mean, median, range, variance,standard deviation, smallest and largest values and the three quartiles, for each variable.

Notes: Use QUARTILE.EXC command to generate the three quartiles
(d) Based on your histograms in part (b) and descriptive summary measures in part (c),comment on the skewness of the distribution of the two variables.

Second, the researcher wishes to investigate the association between the two variables.
(a) Explain what could be the independent variable (X) and the dependent variable (Y).
(b) Using an appropriate plot, investigate the relationship between weekly take-home pay and weekly food expenditure. On the same plot, fit a linear trend line.
(c) Using Excel, compute a numerical summary measure to measure the strength and the direction of the linear relationship between weekly take-home pay and weekly food expenditure. Interpret this value.

(d) Display the regression summary table. Using the summary results, estimate the Least Squares Regression equation and interpret the intercept and slope coefficient estimates of the estimated linear regression model.
(e) Using the regression summary output table, conduct a hypothesis testing to conclude whether there is a linear relationship between weekly take-home pay and weekly food expenditure. Use .

(a) Based on your answers above, write a summary report about the findings of the study conducted by this researcher (maximum 200 words). 

Answer:

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.
The researcher could use non-probability sampling or convenience sampling for the survey.
The researcher might face problems related to biasness in the data collection procedure.

Answer Part 2

• The length of the intervals of Take Home Pay (THP) and Weekly Food Expenditur
  e (WFE) were considered approximately equal to the value of Range (R_THP = 985, R_WFE = 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).Histograms for both the variables have been provided in Figure 1 and Figure 2.

Numerical summary for Take Home Pay (THP) and Weekly Food Expenditure (WFE) have been provided in Table 1.

Table 1: Descriptive Summary of THP and WFE

• 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.
• 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

• The most likely preference for independent variable was Take home pay, and Weekly food expenditure for the dependent variable.
• The relation between the variables was established by two-way scatter plot. The relation between the two variables was found to be highly positive.

• 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.
• 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.

• Let the regression slope be denoted by.

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 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).

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.