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LDR 5301, Methods of Analysis for Business Operations 1

Course Learning Outcomes for Unit VII Upon completion of this unit, students should be able to:

1. Differentiate the steps of the quantitative analysis approach. 1.1 Explain how quantitative analysis can be used to make a business more efficient.

2. Distinguish between the approaches to determining probability.

2.1 Use probability approaches to analyze a business.

3. Contrast the major differences between the normal distribution and the exponential and Poisson distributions. 3.1 Determine the type of distribution exhibited by a business.

4. Explain the major steps in decision-making.

4.1 Apply the steps of decision-making to a business analysis.

7. Assess the differences between correlation and causation. 7.1 Explain the four values of the correlation coefficient. 7.2 Examine the coefficient of determination and the coefficient of correlation, and deduce their

meanings.

Course/Unit Learning Outcomes

Learning Activity

1.1 Unit VII Project 2.1 Unit VII Project 3.1 Unit VII Project 4.1 Unit VII Project

7.1 Unit Lesson Chapter 4 Unit VII Project

7.2 Unit Lesson Chapter 4 Unit VII Project

Required Unit Resources Chapter 4: Regression Models Unit Lesson

Introduction to Regression Analysis What is regression analysis? According to Render et al. (2018), regression analysis serves two purposes: it helps to explain the relationship between variables, and it can help predict the value of one variable in relation to the other. Therefore, what we have here is a comparison between variables to see if one reacts positively to the other, has no impact on the other, or has a negative impact on the other. Here is an example that might occur in the real world.

UNIT VII STUDY GUIDE Regression Models

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a. A company, Black Swan, has created a new weatherproof coat that is lightweight and adapts to the ambient temperature to either cool you or warm you.

b. The company must take on a massive advertising campaign to now grow sales worldwide. c. A regression model (see Figure 1) can be used to compare sales to the costs of advertising.

Notice that Figure 1 displays regression lines comparing marketing to sales. As you can see, there are many forms the regression can take (linear, concave, S-shaped, and convex). The data drive the curve.

Regression Analysis Definitions Let’s take a minute to review some definitions that will assist you as you read this lesson, the chapter readings, and watch the videos for this unit. The textbook offers five different definitions: coefficient of correlation, coefficient of determination, dependent variable, independent variable, and scatter plot diagrams. Each of the regression analysis terms being defined measures data in a different way that will help the user determine if there is a relationship between the data variables. The coefficient of correlation measures the strength of the relationship between the variables. The relationship can be positive, negative, or neutral as long as the variables are in line with the linear relationship. The results are determined from your calculations where r=1, r=-1, r=0, and 0<r<1 (Render et al., 2018). You will note all of the relationships in Figure 4.3 on page 117 and in the chart below. Next, the coefficient of determination is the calculated number received from the r2-r2 formula which is seen 0=1. The computed number ranges from 0 to 1. The closer to 1, the stronger the correlation in the event. The number indicates the percentage of variability with the dependent variable. The independent variable is the X variable in the regression equation. This variable helps predict the independent variable to the user in the equation (Render et al., 2018). Finally, there are scatter diagrams or scatter plots. These display the relationship between two variables and show how closely they are related. Scatter is used in the name because when a shotgun is fired, the pellets or buckshot that leave the barrel have a scattering effect on the target area, much like the data on the graph. They are compared to each other to see if there is correlation along the regression line. See Figure 2 below. The four charts display perfect positive, no correlation, positive correlation, and perfect negative correlation.

Figure 1: Regression Models

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Value Relationship Interpretation of the Value— What does it all mean?

0 No linear relationship

+1 Perfect positive linear relationship As one variable increases, so does the other in an exact linear rule

-1 Perfect negative linear relationship

As one variable increases, the other decreases in an exact linear rule

Between 0 and 0.3 Weak positive (or negative) linear relationship

Shaky linear rule

Between 0.3 and 0.7 (0.3 and -0.3)

Moderate positive (or negative) linear relationship

Fuzzy-firm linear rule

Between 0.7 and 1.0 (-0.7 and -1.0)

Strong positive (or negative) linear relationship

Firm linear rule

(Ratner, 2009)

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Regression analysis begins with the relationship. This begins by gathering and plotting data along two axis. The plotting of the data is used in a scatter diagram. This is where the independent variable is plotted on the horizontal access, and the dependent variable on the vertical axis (Render et al., 2018).

Figure 2: Scatter Plot Diagrams (Render et al., 2018)

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Figure 3 provides an excellent example of this. Note the independent variable payroll on the horizontal axis and the dependent variable on the vertical axis. So what does this tell you? At first, it communicates that as the payroll increases, sales should also increase. However, there is not a perfect relationship, because not all the data lie on the trend line. There is a relationship because both are increasing from the lower left to the upper right, but it is not 100% correlated (matching). What this indicates to the chief executive officer (CEO) of the company is that there is a percentage of error involved in trying to predict sales.

Scatter Plots and Correlation Take a few minutes to watch the following segments from the full video cited below: Understanding Correlation (Segment 9 of 17), Correlations and Scatter Plots-Weather (Segment 10 of 17), and Correlation Coefficients-Beach Attendance (Segment 11 of 17). Discovery Education (Producer). (2007). Discovering math: advanced—statistics and data analysis: part 1

[Video]. Films on Demand. https://libraryresources.columbiasouthern.edu/login?auth=CAS&url=https://fod.infobase.com/PortalPl aylists.aspx?wID=273866&xtid=117914

The transcript for this video can be found by clicking on “Transcript” in the gray bar to the right of the video in the Films on Demand database. The video segments provide a good review of correlation with super examples of beach attendance and weather. What to determine from this is whether there is correlation—a degree of association between variables. In the first segment, there is the comparison between shark attacks in Florida and the number of tourists, measured in millions. The second segment discusses the number of beachgoers to the average temperature X. What one must consider is coincidence. Is it by chance? Does an increase in tourists mean that more sharks arrive? Maybe not. Correlation does not indicate causation. You are encouraged to continue watching the video if this interests you. As the segments continue, you will see how two other variables are compared (math scores on exams to average temperature). As you would expect, there was no correlation displayed. When comparing two variables we also use the correlation coefficient, which is the following formula: -1< r > 1. This compares the data and determines which way the line is sloped. If the r=45, then the line is positively sloped upward from left to right. If the r = -13, then it is negatively sloped from left to right, and the same for r = 0: no slope. The closer r = 1, then the stronger the correlation and points along the line.

Simple Linear Regression On page 113, Render et al. (2018) provide the simple linear regression formula, as seen below: Y = β0+β1X+ϵY=β0+β1X+ϵ(4-1) where Y = dependent variable (response variable)

Figure 3: Scatter diagram (Render et al, 2018)

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X = independent variable (predictor variable or explanatory variable) β0 = intercept (value of Y when X = 0) β1 = slope of regression lineϵ=random error Render et al. (2018) provide a very good example of simple linear regression in their Triple A Construction example on page 114 of the textbook. In this scenario, Triple A Construction is trying to predict sales in the future. What, therefore, is the dependent variable? Sales is the dependent variable. The independent variable is the Albany payroll. Take your time, walk through the problem given the data chart in Table 4.2, and apply the formulas. The end result of 9.5 or $950K displays that if the payroll is $600 million the following year, predicted sales would be $950K.

Assumptions of the Regression Model As we have seen in decision-making, the decision-maker might have some assumptions on what will occur, therefore, leading to an assignment of probability. With regard to assumptions in the regression model, Render et al. (2018) determined that four are important: the errors

• are independent, • are distributed normally (recall from previous unit what a normal distribution looks like), • have a mean of zero, and • have a constant variance no matter the value of X.

Now, let’s look at it in the following figures that reinforce these assumptions. Look at Figure 4. Based on what you have read regarding correlation and regression in this lesson, what do you see? What you see is a pattern of randomness, right? There are random plots of data and there is a large area of error to the center regression line.

Look at Figure 5. What do you see? Focus your sight line moving from the error axis on the left to the end of the regression line on the right. What you should notice is that the error rate and breadth of the scattered points gets farther apart as X (horizontal axis) increases. An example of this was demonstrated in a previous lesson. The closer you walk toward a sound (radio, jackhammer, honking horn alarm), the louder the sound gets. The farther you walk away from the sound, the lower the sound is.

Figure 4: Pattern of Errors Indicating Randomness (Render et al., 2018)

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Finally, examine Figure 6. What do you see? In simple terms and observation, it looks like the dots present a hill. The hill begins lower on the left, rises as it moves with higher error rate and higher X value, but then it reverses course and comes back down on the right side of the graph to where X increases in value but the opposite error rate is decreasing. This data indicates that the data is not linear (straight line).

Conclusion In this unit, we examined regression. As a review, there are two purposes of regression analysis— to explain the relationship between variables and to help predict the value of one variable in relation to the other (Render et al., 2018).

Figure 5: Nonconstant Error Variance (Render et al., 2018)

Figure 6: Errors Indicate Relationship is Not Linear (Render et al., 2018)

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Why is regression important, and what can you do with regression analysis? The takeaway from this unit is being able to determine if there is a relationship between two variables. There are many examples in the real world. For example, an inverse relationship exists between interest rates and investments. When interest rates rise, stock and bond prices fall (and their yields increase). When interest rates are reduced, then stock and bond prices rise (and their yield falls). Think about the biotech industry and pharmacy industry with drug development to fight diseases. There is a lot of trial and error when finding a cure for a virus or disease. Researchers want to see if Vaccine X reduces Infection A. Hence, they are looking for a correlation of reduction in the disease spreading. If Vaccine X has no impact on Infection A, then researches know there is no correlation, and therefore must go back to the laboratory to modify Vaccine X and retry it. Can you think of any regression models that you have seen in society, your place of employment, or at home? Think about this as we move through the unit.

References Ratner, B. (2009, May 18). The correlation coefficient: Its values range between +1/-1, or do they? Journal of

Targeting, Measurement and Analysis for Marketing, 17, 139–142 https://link.springer.com/article/10.1057/jt.2009.5

Render, B., Stair, R. M., Jr., Hanna, M. E., & Hale, T. S. (2018). Quantitative analysis for management (13th

ed.). Pearson. https://online.vitalsource.com/#/books/9780134518558 Suggested Unit Resources In order to access the following resources, click the links below. The Chapter 4 PowerPoint Presentation will summarize and reinforce the information from this chapter in your textbook. You can also view a PDF of the Chapter 4 presentation. The following video segments were referenced in the unit lesson and provide some great real-world examples of correlation. Take a few minutes to watch these segments from the full video cited below: Understanding Correlation (Segment 9 of 17), Correlations and Scatter Plots-Weather (Segment 10 of 17), and Correlation Coefficients-Beach Attendance (Segment 11 of 17). Discovery Education (Producer). (2007). Discovering math: advanced—statistics and data analysis: part 1

[Video]. Films on Demand. https://libraryresources.columbiasouthern.edu/login?auth=CAS&url=https://fod.infobase.com/PortalPl aylists.aspx?wID=273866&xtid=117914

The transcript for this video can be found by clicking on “Transcript” in the gray bar to the right of the video in the Films on Demand database. Learning Activities (Nongraded) Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit them. If you have questions, contact your instructor for further guidance and information. For an overview of the chapter equations, review the Key Equations on page 136 of the textbook. Then, complete questions 1–12 on the Self-Test on page 139. You can use the key in the back of the book in Appendix H to check your answers for Self-Tests. Finally, complete Problem: 4–10 on page 140. You can use the answer key in Appendix G in the back of the textbook in order to check your answers