Simple Linear Regression and Application

 Simple Linear Regression and Application

Introduction

Throughout this analysis, the relationship between revenue collected from the stores and the size in the square foot for each store will be studied. The assumption that will be made in this analysis is that the square foot of the store locations is the independent variable while the revenue collected from the stores is the dependent variable. The assumption that is made above will make sense since the size of a store determines the amount that will be collected thus, the revenue collected from any store location depends on the size of that store. This will be made possible since considering a price per foot of the store, the larger the store, the higher the amount of revenue that will be collected from that store and vice versa. Therefore, there is a high possibility that the relationship that will be identified in the two variables will be a positive one.

Data

The data that will be used for the analysis of the relationship identified is shown below

The data above was collected by the earlier intern and this is what will be used in the linear regression analysis. The above data shows that revenue was collected from all the stores and the stores’ sizes were also measured in terms of the square foot as shown in the above table. It is possible that the data was collected by measuring the stores in each location and going through the records to determine the amount of revenue that each store generates while operational. The revenue collected is measured in terms of dollars while the sizes of the stores are measured in terms of the square foot as shown in the data results table above.

For the analysis, revenue will be used to represent the revenue that was collected from the stores and square foot to represent the sizes of the store’s location.

Regression results

The postulated results for the above relationship is shown like in the formula below

Revenue = β0 + β1(Square Foot) + u

The figure below shows the scatter plot with a fitted linear trend line. The trend line was generated through the use of simple linear regression of the data provided.

From the regression data shown in the excel sheet,

Interpreting the data literally, when the square foot of the stores in any location is zero, the amount of revenue that should be collected from the store is $3618.44. The amount of revenue collected is very reasonable following its positivity but since it is very difficult to have zero square foot of the store sizes in any of the locations, it shows that with increasing 1 square foot of the store size, the revenue collected should increase by $0.002591 since the coefficient of the independent variable is 0.002591.

Following the fact that this value is positive, this is exactly what we assumed when forming the relationship between the given two variables. It is also clear that with an increase in the square foot of a store, an individual is expected to collect more revenues.

The sufficient conditions for good estimates

Linear in parameters

From the sample data given, taking a closer look at the scatter-plot there is a clear correlation between square foot sizes of the stores and revenue collected from the stores in every location. It is clear that the linear approximation relationship between the dependent and the independent variables is a better way to show the relationship and linear approximation is very reasonable as shown in the case.

Random sampling

The data set that is provided is very limited and cannot be used to make a serious conclusion on the analysis but from the data set provided, there are very potential issues that are very clear. The first issue is that the data points provided are only 12 concerning the 12 stores in 12 different locations. For better conclusions, it could have been better to have more stores within different locations to have a larger data set. Thus, the conclusion made from this analyzed data cannot be enough to be used elsewhere. Secondly, data collection information is very limited due to lack of more information but there is a high possibility that 50% of stores have lower sizes in terms of square foot and the rest have higher sizes. A better data set for analysis could have covered all sizes from lower to higher with the same number of stores having a middle size thus the conclusion made here cannot be relied on that much.

Variation in the independent variable

As has been highlighted earlier, the sample size data used in the analysis is very small but still, there are situations where there is a variation in the variables provided. The only problem that the data set had was very few data between the 20,000 square foot and 60,000 square foot sizes of the stores as recorded from the different locations. The variation in the independent variable could have been analyzed well when more data could have been gathered between the sizes outlined above.

Zero mean of the error term conditional on the independent variable

There are a series of issues that can affect the revenue collected from a store which is not related to the sizes of the stores. For example, issues like the sizes of the store used and the number of stores occupied are all issues that might affect the amount of revenue collected from the stores but are all included as errors. It is true that the size of the store used is correlated with the store sizes and should be put into consideration when analyzing such data.