Linear Regression Analysis: Hypothesis Testing, Slope Interpretation, R-Squared, ANOVA Essay.

Nature Of Probability Distribution

The descriptive statistics from part 1 along with the frequency distribution and related histogram from part 2 provide a host of information from which the following meaningful conclusions are obtained..

Nature Of Probability Distribution – One of the key conditions to be met for a distribution to be labeled as normal is the absence of any skew, A key reflection of this is a symmetrical bell shaped curve. Also, when skew is zero, the various measures of central tendency tend to converge on a single point. However, this is not true for either of the variables given as the histograms have a tail on either side and non-converging central tendency measures are observed from part 1.

Startup cost variation – Also, based on the information provided, there seems to be quite significant variation in the startup costs pattern. This is observable by comparison of mean and median startup costs for thee various businesses. Further, the histogram also confirms this trend which is especially observable in case of X5 or pet stores where the startup cost seems quite less in comparison with other businesses.

A series of the following step need to be implemented in order to conduct the hypothesis test for model significance.

Step 1: Hypothesis Formation (H_o and H₁)

H₀: µ_X1= µ_X2= µ_X3= µ_X4= µ_X5

H₁: At a bare minimum, the startup costs on an average of one particular business differes from the other startup costs on an average

Step 2: Defining the level of significance utilised for testing

In the given case, level of significance or α has been assumed as 5% or 0.05. 

Step 3: Output Interpretation

P value corresponding to the above F stat value = 0.02

Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁ acceptance.

Step 5: Conclusion

The above analysis clearly reflects that the linear relation between the variables taken is statistically significant as some slope does have non-zero value and hence proves the significance. 

The given data has been provided and the underlying relation between the given variables needs to be brought out using regression as the appropriate computational method. In order to facilitate the same, the key tool that has been used in MS-Excel which through the Data Analysis option enables the user to run linear regression.  

The regression coefficients are essentially the slopes which are critical to the derivation of exact relation between the given variables. Also, there are five predictor variables in the form of x2, x3, x4, x5 & x6 and one variable which id dependent on these and indicated by x1. The equation is listed below.

The fit of the regression model is a key characteristic of the underlying model which essentially presents the underlying strength of the linear relationship. In case of a regression model, it is advisable that the underlying fit ought to be high. In the absence of the same, more independent variables are included in the model and certain existing variables which are not significant may be removed. In order to opine on the underlying fit of regression model, the following indicators are considered relevant.  

Startup Cost Variation

R²– The given regression model has an immensely high value amounting to 0.9932 and is quite close to 1 which is the maximum possible value. The implication of the R² is that 99.32% of changes observed in the annual sales are offering an explanation through the usage of the predictor variables on a joint basis.

Significance – Through the ANOVA testing the regression model significance can be determined. A significant model typically implies a good fit. In the given case, the ANOVA indicates high significance of the model which again reiterates the conclusion drawn using R²,

A series of the following step need to be implemented in order to conduct the hypothesis test for model significance. 

Step 1: Hypothesis Formation (H_o and H₁)

H₀: β_X2= β_X3= β_X4= β_X5= β_X6=0

H₁: All the slopes cannot be considered zero and there is presnece of atleast one non-zero or significance slope.

Step 2: Defining the level of significance utilised for testing

In the given case, level of significance or α has been assumed as 5% or 0.05.  

Step 3: Output Interpretation

From the table above, F statistic = 611.59

P value corresponding to the above F stat value = 0.000

Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁ acceptance.

Step 4: Conclusion

The above analysis clearly reflects that the linear relation between the variables taken is statistically significant as some slope does have non-zero value and hence proves the significance.

The slope interpretation essentially involves analyzing the respective coefficients considering their sign as well as magnitude. The presence of a + sign denotes that the change in both dependent and independent variable would be in the same direction. However, the presence of a – sign denotes that the change in both dependent and independent variable would be in the opposite direction. Further, the quantum of change that is expected to be seen in the dependent variable would depend on the magnitude of the slope along with the respective change in the independent variable. Considering the general norms highlighted, the slope interpretation is offered.

X2 (Value is 16.20) – The given slope reflects that any change in the franchise store area by 1 unit (i.e. 1 sq. ft) would produce an alteration in annual sales by $ 16.20 and that too in the same direction.

X3 (Value is 0.17) - The given slope reflects that any change in the franchise store inventory by 1 unit (i.e.$1) would produce an alteration in annual sales by $ 0.17 and that too in the same direction.

X4 (Value is 11.53) - The given slope reflects that any change in the franchise store advertising spending by 1 unit (i.e.$1) would produce an alteration in annual sales by $ 11.53 and that too in the same direction.

X5 (Value is 13.58) - The given slope reflects that any change in the franchise store customer coverage by 1 unit (i.e. one family) would produce an alteration in annual sales by $ 13,58 and that too in the same direction..

Regression Model Fit

X6 (Value is -5.31) - The given slope reflects that any change in the franchise store competing stores by one would produce an alteration in annual sales by $ 5,310 and that too in the opposite direction.  

There is a probability of 0.95 that the given variable coefficient or slope would tend to fall in the interval which has been highlighted in red. 

A series of the following steps need to be implemented in order to conduct the hypothesis test for slope significance.

Step 1: Hypothesis Formation (H_o and H₁)

H₀:  The slope for the concerned independent variable can be considered to zero highlighting the inherent insignificance.

H_i: The slope for the concerned independent variable cannot be considered to zero highlighting the inherent significance.

Step 2: Defining the level of significance utilised for testing

In the given case, level of significance or α has been assumed as 5% or 0.05.

Step 3: Excel output

The test of significance which is applicable for the given case is based on the output obtained for regression analysis  particularly in context of the coefficient determination.

Step 4: Output Interpretation

The available approaches for hypothesis testing are in the form of p value and also critical value. For the purposes of this hypothesis testing.

Th value for slope of X2 variable as 0. Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁

The p value for slope of X3 variable as 0.01. Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁

The p value for slope of X4 variable as 0. Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁

The p value for slope of X5 variable as 0. Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁

The p value for slope of X6 variable as 0.01. Comparing the p value obtained above with the level of significance assumed in Step 2, it is evident that α is higher than p value. This provides existence of meaningful evidence warranting H_o rejection and H₁

Step 5: Conclusion

The above analysis clearly reflects that the linear relation between the variables taken is statistically significant as all the slopes that have been tested do have non-zero value and hence prove their significance at 95% confidence level. 

The implementation of hypothesis testing technique as underlined above has clearly reflected that the various slopes have managed to imply that they are indeed significant and hence cannot be ignored or considered as zero. Under such a scenario, there is no reason to bring any changes in the existing model and the same model is retained. 

While the regression model to be deployed has already been finalized, the critical inputs for consideration. 

Given franchise store area = 1,000 sq. ft

Given franchise level of Inventory = $150,000

Given franchise annual advertisement spending = $ 5,000

Given franchise targeted families = 5000

Given franchise competing stores in the vicinity = 2 

The inputs that are listed above are put in the regression model obtained to reflect. 

Therefore based on computation above, it can be concluded for the given franchise store, the annual sales would be $ 138, 448.