Regression and Prediction

Regression and Prediction

Regression analysis is among the most common statistical tools used in investigating relationships among variables and prediction. This paper considers Adam’s Pizza outlet, which operates in six cities, namely: Dallas, Houston, San Antonio, OKC, Tulsa, and Norman. The study conducted a multiple regression analysis using the data collected on quarterly intervals between the years 2013 and 2014. The study is aimed at determining the impacts of prices and advertising on the sales of pizza. It also seeks to assess the effect of other factors such as income and population on demand for pizza. This report, therefore, discusses the economic and statistical significance of the variables, interpretation of coefficient of determination, estimation of the 2015 first-quarter sales, and confidence intervals.

The Statistical Significance of the Variables

The multiple regression model uses six independent variables: price, Advertising expenditure, price of the competitors, income, population, and the trend. The price is the cost at which Adam’s Pizza sells in their outlets. The price affects the demand of the pizza, because a high price may scare away customers while low price attracts more customers into purchasing the product. The advertising expenditure affects the costs of production, which in turn affects the profit margin. The price of the competitors determines the maximum price at which Adam’s Pizza will set to avoid losing customers to the competitors. The income affects the willingness of the customers to spend on pizza while the population determines potential customers for the pizza. The last variable, trend, affects the pattern of the consumption of pizza.

Interpretation of the Coefficient of Determination

The coefficient of determination, R-squared, illustrates the effectiveness of the model to predict the dependent variable.  According to Uyanık & Güler (2013), it shows the amount of variation, effected by the independent variables. In this case, the coefficient of determination, R is 0.9332, which implies that the independent variables explain 93.32% of the response variable. That is, the independent variables predict 93.32% of the response variable.

Estimation of the 2015 Sales

The estimation of the 2015 sales involves the use of the average of each of the independent variables. We then insert the respective averages in the multiple regression model, as shown below:

Unit Sales= 582751 -122607*Price + 5.84* AdvertExp +29867* Comp Price + 2.04*Income + 0.03036*Pop + 309T

Unit Sales = 582751 -122607* 7.4105625+ 5.84*32124.0104 +29867* 6.70427083

+ 2.04*58425.0333 + 0.03036*7796206.963 + 3097*1

Unit Sales= 421042.6209

Confidence Intervals

95% Confidence Interval

The  95% Confidence Interval is obtained by using 2-standard deviations as the standard value to calculate both lower and upper bounds (Alhamide, Ibrahim & Alodat, 2016). The results are as shown in the section below.

Thus: The lower bound           = Predicted value – 2 (Standard error)

= 421042.6209 – (2*74342.1531)

=272358.3147

The Upper bound        = Predicted value +2 (Standard error)

=421042.6209 – (2*74342.1531)

=569726.9271

The results of the confidence show that we are 95% confident interval that the unit sales in 2015 first quarters were between  272358.3147 and 569726.9271.

99% Confidence Interval

We use 3-standard deviation to calculate to 99% Confidence Interval

Thus: The lower bound           = Predicted value – 3 (Standard error)

=421042.6209 -3 (74342.153)

=198016.1616

The Upper bound                    = Predicted value +3(Standard error)

=421042.6209 +3 (74342.153)

=644069.0802

The results of the confidence show that we are 99% confident interval that the unit sales in 2015 first quarters were between  198016.1616 and 644069.0802.

References

Uyanık, G. K., & Güler, N. (2013). A study on multiple linear regression analysis. Procedia-Social and Behavioral Sciences, 106(1), 234-240.

Alhamide, A. A., Ibrahim, K., & Alodat, M. T. (2016). Pak. J. Statist. 2016 Vol. 32 (2), 81-96 Inference for multiple linear regression model with extended skew normal errors. Pak. J. Statist, 32(2), 81-96.