The Cotton Mill is an upscale chain of women\'s clothing stores, located in the

Question

The Cotton Mill is an upscale chain of women's clothing stores, located in the southwestern United States. Do to recent success, The Cotton Mill's top management is planning to expand by locating new stores in other regions of the country. The director of planning has been asked to study the relationship between yearly sales and the store size. As part of the study, the director selects a sample of 25 stores and determines the size of the store in square feet and the sales for the last year. The sample data follows.

Using store size as the independent variable, run the data using excel and answer the following: 1. Write the regression equation. 2. Interpret the regression constant and regression coefficient. 3. Forecast a value for the dependent variable. 4. Test the significant of the regression coefficient using alpha = .05. 5. Test the overall significant of the regression model. 6. Interpret the coefficient of determination. 7. Are there any indications of a violation of the general linear model? Be specific.

Store size (1000s of square feet) Sales (millions of$) 3.7 2.0 5.0 0.7 2.6 2.9 5.2 5.9 3.0 2.4 2.4 0.5 5.0 0.4 4.2 3.1 2.6 5.2 3.3 3.2 4.9 5.5 2.9 2.2 2.3 9.18 4.58 8.22 1.45 6.51 2.82 10.45 9.94 4.43 4.75 7.30 3.33 6.67 0.55 7.56 2.23 4.49 9.90 8.93 7.60 3.71 5.47 8.22 7.17 4.35

Explanation / Answer

1. Write the regression equation.

The regression equation is given by:

sales =  2.1748 + 1.1768*store_size

2. Interpret the regression constant and regression coefficient.

The regression constant is the intercept which is equal to 2.1748. It can be interpreted as the value of the dependent variable (sales) when the value of independent variable (store_size) is zero.

The regression coefficient is the slope which is equal to 1.1768. It can be interpreted as if store_size increases by one unit, then the sales increases by 1.1768.

3. Forecast a value for the dependent variable.

The regression equation is given by: sales =  2.1748 + (1.1768*store_size).

Since a value for store_size is not given for forecasting, I am taking store_size to be say 4.1. Now if the store_size is 4.1, then according to our regression equation, our sales is:

sales =  2.1748 + (1.1768*store_size)

sales =  2.1748 + (1.1768*4.1)

sales =  6.99968.

4. Test the significant of the regression coefficient using alpha = .05.

In the results of the regression analysis, we find that the p-value (of t-test) of the regression coefficient is 0.000362. Since the p value is less than alpha 0.05, we can say that the  regression coefficient is highly significant. (Here we reject our null hypothesis that the regression coefficient is not significant and accept that the regression coefficient is significant).

5. Test the overall significant of the regression model.

The overall significance of the regression model is given by the p-value (of the f-test) which is equal to 0.0003621. Thus even at alpha = 0.01 i.e. 00% confidence, we see that the p value( 0.0003621 ) is less than the alpha and thus can say that the regression model is also highly significant. (here since p-value is less than the significance level 0.01, we reject our null hypothesis that the model is not significant and conclude that the model is highly significant).

6. Interpret the coefficient of determination.

The coefficient of determination is 0.4313665. Thus the independent variable store_size is able to explain 43.14% of the variation that happens in the sales of the store. The rest of the sales occur due to some other factors.

7. Are there any indications of a violation of the general linear model? Be specific.

The independent and the dependent variable looks linearly related on seeing the scatter plot. The residuals of the model is also normally distributed which is clear when using the qq plot. Thus the normality condition is also satisfied. Thus there appears no evident violation of the general linear model.

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