MTH302 GDB Solution Spring 2012

 MTH302 GDB Solution Spring 2012

CONSTRUCT A BUSINESS PROBLEM WHICH SHOWS A POSITIVE RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT VARIABLES USING REGRESSION ANALYSIS

Solution:
Marketing research professionals often use inferential or descriptive statistics to guide major marketing decisions. There are a number of statistical tests that explore the relationship between the independent variable(s) and the dependent variable. The key is to translate the business problem into a statistical problem, solve the problem statistically, then translate the statistical solution into an actionable business solution.

Dependent Variable

The dependent variable — also called the response variable — is the output of a process or statistical analysis. Its name comes from the fact that it depends on or responds to other variables. Typically, the dependent variable is the result you want to achieve. In marketing, the results desired are tied to sales revenue. Sales as a dependent variable can be looked at in many ways, such as sales of a specific doll, sales of a category like toy cars, overall sales at a particular store, or even sales for the entire company.

Independent Variable

An independent variable is an input to a process or analysis that influences the dependent variable. While there can only be one dependent variable in a study, there may be multiple independent variables. When the dependent variable is sales revenue, the elements of the marketing mix — product, price, promotion and place — will definitely influence the dependent variable and can therefore be identified as independent variables.

Regression Analysis in Marketing

Marketing research employs a statistical tool called regression analysis to measure the strength of the relationship between the dependent variable and the independent variables. For example, a frozen yogurt shop could set loyalty card discounts, base price, and time of day as the independent variables to test not only the direct effect each factor has on parfait sales, but whether there is interaction between the variables. If, when the base price is low, loyalty card discounts influence sales less than when the base price is high, there is an interaction between the two factors.

Choosing the Right Variables

Asking the right question will lead you to the right answer. The more specific you can make your dependent variable — for instance, sales of a single MP3 player model as opposed to sales of all electronics — the better chance you have of isolating the independent variables that truly influence it. Also, even when you know your goal, you look at it a variety of different ways. For instance, “At what price can we make $100,000 per quarter in sales of product A?” is a subtly different question than, “At a price of $10, how many people will buy product A per quarter?” Look in the Resources section for further reading on how to start with the right question and use the right methodology to answer it.

A variable is an event, idea, value or some other object or category that a researcher or business can measure. Variables can be dependent or independent. Dependent variables vary by the factors that influence them, but independent variables stand on their own — changes in other variables have no effect on them. An independent variable in one context may be a dependent variable in another. An independent variable in business may affect sales, expenses and overall profitabilityIndependent variables that affect sales include customer demographics, store location and weather. Customer demographics include age, occupation, family status, income level and gender. These factors affect what a customer needs, which affects sales and ultimately profits. A store located in a densely populated metropolitan area may have higher sales than a store in a sparsely populated rural area. Similarly, customers may go shopping when the weather is pleasant, but few would venture outside in stormy or snowy weather. Some variables have a circular relationship with sales. For example, sales depend on advertising, but the level of advertising expenses also depends on sales.

Expenses

The prices of raw materials, labor wage rates and facility rental rates are independent expense variables. The prices of raw materials, such as food commodities, metals and minerals, do not change, regardless of how much a small business spends on them. Labor wage rates and facility rental rates are other examples of independent expense variables. They affect the cost structure of a small business, but the owner cannot change market wage rates or rental rates by himself.

Economy

Economic variables affect business profitability. The income of individual customers and profits of business customers are independent economic variables that affect overall business performance. During a recession, customers earn and spend less, which leads to declining business sales. Conversely, during a period of economic growth, customers earn and spend more, which increases business sales and profits. The interest rate on a bank loan or line of credit is an independent variable because it affects expenses and profits. However, the borrowing needs of a small business do not change interest rates.

Considerations: Dependent Variables

In the business context, profit is a dependent variable because it depends on the economy, sales and expenses. Product quality depends on the manufacturing and design processes. The number of employees laid off during a recession depends partly on declining business revenues. Government tax revenue depends on customer income, business profits, capital gains and other variables.

Solution # 2:- 

Regression Basics For Business Analysis If you've ever wondered how two or more things relate to each other, or if you've ever had your boss ask you to create a forecast or analyze relationships between variables, then learning regression would be worth your time. In this article, you'll learn the basics of simple linear regression - a tool commonly used in forecasting and financial analysis. We will begin by learning the core principles of regression, first learning about covariance and correlation, and then move on to building and interpreting a regression output. A lot of software such as Microsoft Excel can do all the regression calculations and outputs for you, but it is still important to learn the underlying mechanics. Variables At the center of regression is the relationship between two variables, called the dependent and independent variables. For instance, suppose you want to forecast sales for your company and you've concluded that your company's sales go up and down depending on changes in GDP. The sales you are forecasting would be the dependent variable because their value "depends" on the value of GDP, and the GDP would be the independent variable. You would then need to determine the strength of the relationship between these two variables in order to forecast sales. If GDP increases/decreases by 1%, how much will your sales increase or decrease? Covariance The formula to calculate the relationship between two variables is called covariance. This calculation shows you the direction of the relationship as well as its relative strength. If one variable increases and the other variable tends to also increase, the covariance would be positive. If one variable goes up and the other tends to go down, then the covariance would be negative. The actual number you get from calculating this can be hard to interpret because it isn't standardized. A covariance of five, for instance, can be interpreted as a positive relationship, but the strength of the relationship can only be said to be stronger than if the number was four or weaker than if the number was six. Correlation Coefficient We need to standardize the covariance in order to allow us to better interpret and use it in forecasting, and the result is the correlation calculation. The correlation calculation simply takes the covariance and divides it by the product of the standard deviation of the two variables. This will bound the correlation between a value of -1 and +1. A correlation of +1 can be interpreted to suggest that both variables move perfectly positively with each other, and a -1 implies they are perfectly negatively correlated. In our previous example, if the correlation is +1 and the GDP increases by 1%, then sales would increase by 1%. If the correlation is -1, a 1% increase in GDP would result in a 1% decrease in sales - the exact opposite. (Correlation is also a well-known metric to diversify an investor's portfolio; see Diversification: Protecting Portfolios From Mass Destruction to learn more.) Regression Equation Now that we know how the relative relationship between the two variables is calculated, we can develop a regression equation to forecast or predict the variable we desire. Below is the formula for a simple linear regression. The "y" is the value we are trying to forecast, the "b" is the slope of the regression, the "x" is the value of our independent value, and the "a" represents the y-intercept. The regression equation simply describes the relationship between the dependent variable (y) and the independent variable (x). Free Trading Guide - GFT y=bx+a The intercept, or "a", is the value of y (dependent variable) if the value of x (independent variable) is zero. So if there was no change in GDP, your company would still make some sales - this value, when the change in GDP is zero, is the intercept. Take a look at the graph below to see a graphical depiction of a regression equation. In this graph, there are only five data points represented by the five dots on the graph. Linear regression attempts to estimate a line that best fits the data, and the equation of that line results in the regression equation. Figure 1: Line of best fit Source: Investopedia, 2009. Excel Now that you understand some of the background that goes into regression analysis, let's do a simple example using Excel's regression tools. We'll build on the previous example of trying to forecast next years sales based on changes in GDP. The next table lists some artificial data points, but these numbers can be easily accessible in real life. Year Sales GDP 2005 100 1.00% 2006 250 1.90% 2007 275 2.40% 2008 200 2.60% 2009 300 2.90% Just eyeballing the table, you can see that there is going to be a positive correlation between sales and GDP. Both tend to go up together. Using Excel, all you have to do is click the Tools drop-down menu, select Data Analysis, and from there choose Regression. The popup box is easy to fill in from there; your Input Y Range is your "Sales" column and your Input X Range is the change in GDP column; choose the output range for where you want the data to show up on your spreadsheet and press OK. You should see something similar to what is given in the table below (I've left out parts of the output that isn't relevant for this article). (See Microsoft Excel Features For The Financially Literate for some tips on how to efficiently use Excel) Regression Statistics Coefficients Multiple R 0.8292243 Intercept 34.58409 R Square 0.687613 GDP 88.15552 Adjusted R Square 0.583484 - - Standard Error 51.021807 - - Observations 5 - - Interpretation The major outputs you need to be concerned about for simple linear regression are the R-squared, the intercept and the GDP coefficient. The R-squared number in this example is 68.7% - this shows how well our model predicts or forecasts the future sales. Next we have an intercept of 34.58, which tells us that if the change in GDP was forecasted to be zero, our sales would be about 35 units. And lastly, the GDP correlation coefficient of 88.15 tells us that if GDP increases by 1%, sales will likely go up by about 88 units. So how would you use this simple model in your business? Well if your research leads you to believe that the next GDP change will be a certain percentage, you can plug that percentage into the model and generate a sales forecast. This can help you develop a more objective plan and budget for the upcoming year. Of course this is just a simple regression and there are models that you can build that use several independent variables called multiple linear regressions. But multiple linear regressions are more complicated and have several issues that would need another article to discuss.Regression Basics For Business Analysis If you've ever wondered how two or more things relate to each other, or if you've ever had your boss ask you to create a forecast or analyze relationships between variables, then learning regression would be worth your time. In this article, you'll learn the basics of simple linear regression - a tool commonly used in forecasting and financial analysis. We will begin by learning the core principles of regression, first learning about covariance and correlation, and then move on to building and interpreting a regression output. A lot of software such as Microsoft Excel can do all the regression calculations and outputs for you, but it is still important to learn the underlying mechanics. Variables At the center of regression is the relationship between two variables, called the dependent and independent variables. For instance, suppose you want to forecast sales for your company and you've concluded that your company's sales go up and down depending on changes in GDP. The sales you are forecasting would be the dependent variable because their value "depends" on the value of GDP, and the GDP would be the independent variable. You would then need to determine the strength of the relationship between these two variables in order to forecast sales. If GDP increases/decreases by 1%, how much will your sales increase or decrease? Covariance The formula to calculate the relationship between two variables is called covariance. This calculation shows you the direction of the relationship as well as its relative strength. If one variable increases and the other variable tends to also increase, the covariance would be positive. If one variable goes up and the other tends to go down, then the covariance would be negative. The actual number you get from calculating this can be hard to interpret because it isn't standardized. A covariance of five, for instance, can be interpreted as a positive relationship, but the strength of the relationship can only be said to be stronger than if the number was four or weaker than if the number was six. Correlation Coefficient We need to standardize the covariance in order to allow us to better interpret and use it in forecasting, and the result is the correlation calculation. The correlation calculation simply takes the covariance and divides it by the product of the standard deviation of the two variables. This will bound the correlation between a value of -1 and +1. A correlation of +1 can be interpreted to suggest that both variables move perfectly positively with each other, and a -1 implies they are perfectly negatively correlated. In our previous example, if the correlation is +1 and the GDP increases by 1%, then sales would increase by 1%. If the correlation is -1, a 1% increase in GDP would result in a 1% decrease in sales - the exact opposite. (Correlation is also a well-known metric to diversify an investor's portfolio; see Diversification: Protecting Portfolios From Mass Destruction to learn more.) Regression Equation Now that we know how the relative relationship between the two variables is calculated, we can develop a regression equation to forecast or predict the variable we desire. Below is the formula for a simple linear regression. The "y" is the value we are trying to forecast, the "b" is the slope of the regression, the "x" is the value of our independent value, and the "a" represents the y-intercept. The regression equation simply describes the relationship between the dependent variable (y) and the independent variable (x). Free Trading Guide - GFT y=bx+a The intercept, or "a", is the value of y (dependent variable) if the value of x (independent variable) is zero. So if there was no change in GDP, your company would still make some sales - this value, when the change in GDP is zero, is the intercept. Take a look at the graph below to see a graphical depiction of a regression equation. In this graph, there are only five data points represented by the five dots on the graph. Linear regression attempts to estimate a line that best fits the data, and the equation of that line results in the regression equation. Figure 1: Line of best fit Source: Investopedia, 2009. Excel Now that you understand some of the background that goes into regression analysis, let's do a simple example using Excel's regression tools. We'll build on the previous example of trying to forecast next years sales based on changes in GDP. The next table lists some artificial data points, but these numbers can be easily accessible in real life. Year Sales GDP 2005 100 1.00% 2006 250 1.90% 2007 275 2.40% 2008 200 2.60% 2009 300 2.90% Just eyeballing the table, you can see that there is going to be a positive correlation between sales and GDP. Both tend to go up together. Using Excel, all you have to do is click the Tools drop-down menu, select Data Analysis, and from there choose Regression. The popup box is easy to fill in from there; your Input Y Range is your "Sales" column and your Input X Range is the change in GDP column; choose the output range for where you want the data to show up on your spreadsheet and press OK. You should see something similar to what is given in the table below (I've left out parts of the output that isn't relevant for this article). (See Microsoft Excel Features For The Financially Literate for some tips on how to efficiently use Excel) Regression Statistics Coefficients Multiple R 0.8292243 Intercept 34.58409 R Square 0.687613 GDP 88.15552 Adjusted R Square 0.583484 - - Standard Error 51.021807 - - Observations 5 - - Interpretation The major outputs you need to be concerned about for simple linear regression are the R-squared, the intercept and the GDP coefficient. The R-squared number in this example is 68.7% - this shows how well our model predicts or forecasts the future sales. Next we have an intercept of 34.58, which tells us that if the change in GDP was forecasted to be zero, our sales would be about 35 units. And lastly, the GDP correlation coefficient of 88.15 tells us that if GDP increases by 1%, sales will likely go up by about 88 units. So how would you use this simple model in your business? Well if your research leads you to believe that the next GDP change will be a certain percentage, you can plug that percentage into the model and generate a sales forecast. This can help you develop a more objective plan and budget for the upcoming year. Of course this is just a simple regression and there are models that you can build that use several independent variables called multiple linear regressions. But multiple linear regressions are more complicated and have several issues that would need another article to discuss.