Statistics Analysis: Determine Whether The Median Gets Impacted by The Selected Variable

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Introduction

The following pages discuss analysis of a quantitative study designed for purpose of this project. For this purpose, two variables, Weekly median rent of new bonds and number of bedrooms has been considered. Two quarter data, namely March and June has been taken into consideration.

The data has been downloaded from Australian website.

Research Design

The data downloaded for June quarter provided a number of variables from which one particular variable, number of bedrooms has been selected. The data was for June and March quarters.

Further, the data had to be scrubbed to remove blanks, not applicable cases etc. Also, the number of bedrooms considered is limited to one, two, three and four. Other types of settings such as Bedsitter have been excluded for the purpose of this study. After scrubbing, the number of observations was 6,059 for June and 2503 for March. The data table is available in Microsoft Excel.

The research in the following pages intends to conduct various statistical tests, such as, ANOVA, correlation, t-tests and linear regression so as to determine the extent and strength of relationship between the two variables.

The purpose of this research is to determine whether the median weekly rent gets impacted by the selected variable, that is, number of bedrooms.

Hypothesis Development

Linear Regression

Linear Regression was conducted in Microsoft Excel with following output for June and March separately:

It can be seen that correlation coefficient is reasonable at 0.41 for June, indicating some degree of linear relationship between the median rent and number of bedrooms in Australian region. The coefficient of determination is 0.16, which indicates that only 16% of the change in variable (median weekly rent) can be attributed to the other variable (number of bedrooms). This shows that there are many other variables that should be considered before arriving at a conclusion regarding median rent. Additionally, the F is larger than Significance F, indicating that the null hypothesis can be rejected.

The regression equation can be explained as: y = 227.36 + 103.13x, where y is weekly median rent and x is the number of bedrooms. Clearly, there is a positive relationship between the two variables. That is, when number of bedroom increases by one unit, the median weekly rent increases by $103.13.

It can be seen that correlation coefficient is reasonable at 0.39 for March, indicating some degree of linear relationship between the median rent and number of bedrooms in Australian region. The coefficient of determination is 0.15, which indicates that only 15% of the change in variable (median weekly rent) can be attributed to the other variable (number of bedrooms). This shows that there are many other variables that should be considered before arriving at a conclusion regarding median rent. Additionally, the F is larger than Significance F, indicating that the null hypothesis can be rejected.

The regression equation can be explained as: y = 265.5 + 106.04x, where y is weekly median rent and x is the number of bedrooms. Clearly, there is a positive relationship between the two variables. That is, when number of bedroom increases by one unit, the median weekly rent increases by $106.04.

Hence, the above analysis indicates that there is positive relationship between the two variables, but that the number of bedrooms is not the only variable that should be considered when determining various factors that impact the median rent.

Scatter Plot

The above discussion regarding the relationship between median weekly rent and number of bedrooms is reflected in the scatter plot also:

From the above plot, it is clear that as the number of bedrooms increases, the median weekly rent also increases although the rent seems to be range-bound around $500.

ANOVA Test

Single factor ANOVA test was conducted in MS-Excel. The median weekly rent was segregated in four groups, basis the number of bedrooms for the June data. The test was run at alpha of 0.05. The hypothesis is:

H0: μ1bdr = μ2bdr = μ3bdr = μ4bdr

H1: μ1bdr ≠ μ2bdr ≠ μ3bdr ≠ μ4bdr

The result is as follows:

From above, we can see that the value of F (429.1) > F crit (2.6). Hence, we are able to reject the null hypothesis. Concluding, we can say that there is significant difference in the means of weekly median rents for 1 bedroom, 2 bedroom, 3 bedroom and 4 bedroom houses in Australia.

Similarly, same procedure was followed for March Data. The test was run at alpha of 0.05. The hypothesis is:

H0: μ1bdr = μ2bdr = μ3bdr = μ4bdr

H1: μ1bdr ≠ μ2bdr ≠ μ3bdr ≠ μ4bdr

The result is as follows:

From above, we can see that the value of F (154.8) > F crit (2.6). Hence, we are able to reject the null hypothesis. Concluding, we can say that there is significant difference in the means of weekly median rents for 1 bedroom, 2 bedroom, 3 bedroom and 4 bedroom houses in Australia.

Paired t-tests

A paired sample t-test compares same group mean across time periods. Hence, we will be conducting paired sample t-tests to compare mean of various size houses across two quarters: March and June. However, since number of data points needs to be same for the periods, we have taken a sample of 380 observations for each of the tests. The output is as follows for each of the bedroom size:

From the above output, we can identify whether there is significant difference in the two pairs that is from June quarter and March quarter:

  1. In case of one bedroom, two tail P value (0.50) is > 0.05 indicating that change in median rent from March to June is not significant.
  2. In case of two bedrooms, two tail P value (0.00) is < 0.05 indicating that change in median rent from March to June is significant. This is reflected in variance and mean for the two quarters which have changed drastically.
  3. In case of three bedrooms, two tail P value (0.00) is < 0.05 indicating that change in median rent from March to June is significant. This is reflected in mean for the two quarters which has changed drastically.
  4. In case of four bedrooms, two tail P value (0.20) is > 0.05 indicating that change in median rent from March to June is not significant

Hence, we can conclude that the median rent has changed significantly between the two quarters for two bedrooms and three bedrooms houses.

Independent Samples t-tests

Independent sample t-test compares two groups to indicate the difference in means. This has been done with whole set of different bedroom sizes across quarters. Hence, the entire sample could be considered unlike pared t-test where we had to reduce the sample set. The output is as follows:

From the above output, we can identify whether there is significant difference in the median rent for June quarter and March quarter:

  1. In case of one bedroom, two tail P value (0.00) is < 0.05 indicating that change in median rent from March to June is significant.
  2. In case of two bedrooms, two tail P value (0.00) is < 0.05 indicating that change in median rent from March to June is significant. This is reflected in variance and mean for the two quarters which have changed drastically.
  3. In case of three bedrooms, two tail P value (0.00) is < 0.05 indicating that change in median rent from March to June is significant. This is reflected in variance and mean for the two quarters which have changed drastically.
  4. In case of four bedrooms, two tail P value (0.053) is > 0.05 indicating that change in median rent from March to June is not significant. However, the difference in P value and significance level is very slight and even in this we can say that median rents are differing significantly.

Hence, we can conclude that the median rent has changed significantly between the two quarters, irrespective of bedroom size.

Conclusion and Implication

In the above analysis, we saw that the weekly median rent for the two quarters is different significantly in majority cases. We also saw linear relationship between the bedroom size and rent, that is, as the number of bedroom increased, the rent also increased. The correlation as there but it is not too strong. Further, the coefficient of determination was low, indicating that although the number of bedrooms impacts the rent, this is not the only factor doing so. There might be other more significant factors that must be looked into. Some of these cannot be modelled into regression, such as attitude; however, the other factors can be modelled. Such as, the location, utilities, construction age etc. These factors together will give a more holistic picture about factors that impact the median weekly rent in Australian region.

References

  1. Weisberg (1992) Central Tendency and Variability, Sage University Paper.
  2. Garver (1932) Concerning the limits of a measure of skewness. Ann Math Stats 3(4) 141–142
  3. Freedman, David (2005) Statistical Models: Theory and Practice, Cambridge University Press
  4. Australian Website:

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