QUME507_M2

QUME 507 Module 2: Decision and Regression Analysis

2.1 Decision Analysis Introduction The six steps of decision making will be discussed in Topic 2.1 of the Module. Also, this module will differentiate concepts such as risk and uncertainty. Models for decision making under uncertainty such as Maximax, Maximin, Hurwicz, Laplace and Minimax will be discussed. Tools such as Bayesian analysis and decision trees will be integrated. Finally, the useful theory to explain the behavior of risk-averse and risk- taking consumers and companies Decision Analysis Decision analysis can be used to develop an optimal strategy when a decision maker is faced with several decision alternatives and an uncertain or risky pattern of future events (Anderson et al., 2016). Decision theory is an analytical and systematic approach to the study of decision making (Render et al., 2018). There are different mathematical models which are useful to help managers make the best possible decisions. However, first, the six steps in decision making must be evaluated. Steps in Decision Making In Module 1 we evaluate the seven steps in the statistical analysis approach. In this Module we present the six steps to be considered in decision making theory according to Render et al. (2018). 1. Clearly define the problem you are facing. 2. Make a list of possible alternatives. 3. Identify possible outcomes or states of nature. 4. Number the payments (typically the profits) for each combination of alternatives and outcomes. 5. Choose one of the mathematical models of decision theory. 6. Apply the model and make the decision.

Example: Thompson Lumber Company Step 1 - Define the problem. The company is considering expanding by manufacturing and marketing a new product: backyard storage sheds. Step 2 - List alternatives. Build a large new plant. Build a small new plant. Do not develop the new product line at all. Step 3 - Identify possible outcomes. The market could be favorable or unfavorable. Step 4 - List the returns. Identify the conditional values of profits for a large plant, a small plant and no development for both possible market conditions. Step 5 - Select the decision model. This depends on the environment and the amount of risk and uncertainty. Step 6 - Apply the model to the data. The solution and analysis are used to assist in decision making. Scenarios in Decision Making There are three possible scenarios in decision making in which the situation occurs with which to decide what to do. The scenarios are mentioned below. Type 1: Decision-making under certainty The decision-maker knows with certainty the consequences of each alternative or decision. Type 2: Decision-making under uncertainty The decision-maker does not know the probabilities of the various outcomes.

Type 3: Low risk decision making The decision maker knows the probabilities of the various outcomes. Decision-making in Uncertainty Making decisions under uncertainty involves modeling the initial uncertainty so that conclusions can be drawn from the evidence and from what is initially thought. In addition, we have to decide what to do, taking into account that future events and observations can change our conclusions. In the case of companies, this uncertainty occurs when there are several states of nature and a manager cannot assess the probability of the outcome with confidence. This is because probability data are virtually unavailable. According to Render et al. (2018) there are several criteria for making decisions under these conditions. 1. Maximax (optimistic) 2. Maximin (pessimist) 3. Realism criterion (Hurwicz) 4. Equally likely (Laplace) Minimax of regret Criterion: Maximum (optimistic) The optimistic criterion is known as Maximax. According to Anderson et al. (2016) the optimistic criterion is an approach to choosing a decision alternative without using probabilities. For a maximization problem, it leads to choosing the decision alternative corresponding to the highest reward; for a minimization problem, it leads to choosing the decision alternative corresponding to the smallest reward. In other words, the best (maximum) payment is considered for each alternative and the alternative with the best (maximum) payment is chosen. Let's look at the example of Thompsom Lumber:

Step 1: The problem John Thompson identifies is whether to expand his product line by manufacturing and marketing a new product: patio storage sheds. Thompson's second step is to generate the alternatives that are available. In decision theory, an alternative is defined as a course of action or strategy that the decision maker can choose. Step 2: John decides that his alternatives are to build 1. a large new plant to manufacture the booths, 2. a small plant, or, 3. no plant (that is, he has the option of not developing the new product line). In the following example shown in figure 1, we can see that after locating the maximum return for each alternative and selecting the alternative with the maximum number; the alternative to choose is to build a large plant. Figure 1. Maximax Use Adapted by Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson Criterion: Maximin: Pessimistic When using the pessimistic criterion, the minimum return of each alternative is considered and the one with the best (maximum) return is chosen. Therefore, the pessimistic criterion is sometimes called the Maximin criterion. In the following example shown in figure 2, we can see that after locating the minimum return for each alternative and selecting the alternative with the maximum number; the alternative to choose is to do nothing. Figure 2. Maximin Use

Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Criterion of Realism (Hurwicz) Often called a weighted average, the criterion of realism (Hurwicz's criterion) is a compromise between an optimistic and a pessimistic decision. To begin with, a coefficient of realism (); is selected; this measures the level of optimism of the decision maker. The value of this coefficient is between 0 and 1. When  is 0, the decision maker is 100% pessimistic about the future. The advantage of this approach is that it allows the decision maker to manage personal feelings about relative optimism and pessimism. It is calculated as follows: Weighted average =  (maximum of rows) + (1 - ) (minimum of rows) Example Thompson Lumber For the large plant alternative, where  = 0.8: (0.8) (200,000) + (1 - 0.8) (- 180,000) = 124,000 For the small plant alternative, where  = 0.8: (0.8) (100,000) + (1 - 0.8) (- 20,000) = 76,000 Figure 3 shows the example for the criterion of realism. Figure 3. Realism criterion (Hurwicz)

Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Equally likely (Laplace) The equally likely criterion is also known as Laplace. It uses all payments for each alternative. The average payment for each alternative must now be found and the alternative with the best or highest average will be chosen. The equal probability approach assumes that all probabilities of occurrence for the states of nature are the same and thus each state of nature has equal probabilities. Example Thompson Lumber The option of equal probabilities for Thompson Lumber is the second alternative to build a small plant whose strategy since it has the maximum average return. This is shown in Figure 4. Figure 4. Equally Probable (Laplace) Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson.

Minimax Regret The minimum criterion of regret is an approach to choosing a decision alternative without using probabilities. For each alternative, maximum regret is calculated, leading to the choice of the decision alternative that minimizes maximum regret (Anderson et al., 2016). The first step is to create the loss of opportunity table by determining the losses from not choosing the best alternative for each state of nature. The loss of opportunity for any state of nature, or any column, is calculated by subtracting each payment in the column from the best payment in the same column as shown in Figure 5 and Figure 6. Figure 5. Determination of loss of opportunity for Thompson Lumber Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Figure 6. Loss of opportunity table for Thompson Lumber Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson.

As shown in Figure 7, the minimax regret choice is the second alternative, to build a small plant. This minimizes the loss of maximum opportunity. Figure 7. Minimax Thompson Lumber Decision Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. To see an article related to the importance of choosing the optimal software solution in uncertain conditions you can access the following article in the Supplementary Material Folder at the end of the Modules. Aldea, C. & Olariu, C. C. (2014). Selecting the optimal software solution under conditions of uncertainty. Procedia - Social and Behavioral Sciences, 109, 333-337. Low Risk Decision Making Low-risk decision making is when there are several possible states of nature and the probabilities associated with each possible state are known. One of the most popular methods of making decisions with risk: selecting the alternative with the highest expected monetary value. Expected monetary value (EMV) is an approach for choosing a decision alternative based on the expected value of each decision alternative. The recommended decision alternative is the one that provides the best expected value (Anderson et al. 2016). The expected value or mean value is the long-term average value of that decision. The EMV

for an alternative is just the sum of the possible payments for the alternative, each weighted by the probability of that payment occurring. It is calculated as follows: EMV (alternative i) = (first state of nature reward) x (probability of first state of nature) + (reward of the second state of nature) x (probability of second state of nature) +... + (reward of the last state of nature) x (probability of the last state of nature) Example Thompson Lumber Suppose that each market outcome has a probability of occurrence of 0.50. What alternative would give the highest EMV? EMV (large plant) = ($ 200,000) (0.5) + (- $ 180,000) (0.5) = $ 10,000 EMV (small plant) = ($ 100,000) (0.5) + (- $ 20,000) (0.5) = $ 40,000 EMV (do nothing) = ($ 0) (0.5) + ($ 0) (0.5) =$0 As can be seen in Figure 8, the largest expected value ($40,000) is the result of the second alternative, \"build a small plant\". Figure 8. EMV Thompson Lumber

Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Expected value of perfect information Sometimes you have to make decisions where you need to know whether you will change your environment from one of risky decision-making to one of certainty. In this case, external information can be obtained to assess the situation. There are two terms related to this issue: perfect information expected value (EVPI) and perfect information expected value (EVwPI). The EVPI is the average or expected return, in the long term, if we have perfect information before making a decision. To calculate this value, we choose the best alternative for each state of nature and multiply its payment by the probability of occurrence of that state of nature. EVPI places an upper limit on what you must pay for additional information. EVPI= EVwPI- Maximum EMV On the other hand, the EVwPI is the long-term average return if we have the perfect information before making a decision. It is calculated as follows: EVwPI =(best reward for first state of nature) x (probability of first state of nature) + (best reward for second state of nature) x (probability of second state of nature) +... + (Best reward for the last state of nature) x (probability of the last state of nature) Thompson Lumber Example Let's assume that Scientific Marketing, Inc. offers an analysis that will provide certainty about the (favorable) market conditions. The additional information will cost $65,000. Should Thompson Lumber buy the information?

Thompson can calculate the maximum you would pay for information, that is, the expected value of perfect information or EVPI. The process consists of three stages. First, the best retribution is found in each state of nature. If the perfect information indicates that the market will be favorable, it will build the large plant and the profit will be $200,000. If the perfect information indicates that the market will be unfavorable, the \"do nothing\" alternative is chosen, and the profit will be $0. These values are shown in the \"with perfect information\" row in Figure 9. Figure 9. EVwPI Thompson Lumber Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Second, the expected value is calculated with perfect information. Then, using this result, the EVPI is calculated. The expected value with perfect information is: EVwPI. Therefore, if we had perfect information, the payment would average $100,000. The maximum EMV without additional information is $40,000: EVPI = EVwPI – EMV maximum = $ 100,000 - $ 40,000 = $ 60,000

Thus, the most Thompson would be willing to pay for perfect information is $60,000. Of course, this is again based on the assumption that the probability of each state of nature is 0.50. This EVPI also tells us that the most we would pay for any information (perfect or imperfect) is $60,000. Expected Opportunity Loss (EOL) The expected opportunity loss (EOL) is the amount of loss (lower profit or higher cost) for not making the best decision for each state of nature. EOL is the cost of not choosing the best solution. To calculate it, first a table of opportunity loss is constructed. For each alternative, multiply the loss of opportunity by the probability of that loss for each possible outcome and add them up. The minimum EOL will always result in the same decision as the maximum EMV. In addition, the minimum EOL will always equal EVPI. Example Thompson Lumber EOL (large plant) = (0.50) ($0) + (0.50) ($180,000) = $90,000 EOL (small plant) = (0.50) ($100,000) + (0.50) ($20,000) = $60,000 EOL (do nothing) = (0.50) ($200,000) + (0.50) ($0) = $100,000 Figure 10. EOL: Thompson Lumber

Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Sensitivity Analysis Sensitivity analysis is the study of how changes in the probability assessments for states of nature or changes in payments affect the recommended decision alternative (Anderson et al., 2016). It investigates how our decision would change given a change in the problem data. Let's look at its application with the example of Thompson Lumber. For the example of Thompson Lumber: P = probability of a favorable market (1 - P) = probability of an unfavorable market The EMVs can now be expressed in terms of P, as indicated in the following equations. EMV (Large Plant) = $200,000P - $180,000) (1 - P) = $200,000P – $180,000 + $180,000P = $380,000P – $180,000 EMV (Small plant) = $100,000P – $20,000) (1 – P) = $100,000P – $20,000 + $20,000P = $120,000P – $20,000 EMV (Do nothing) = $0P + 0 (1 – P) = $0 The best decision is to do nothing as long as P is between 0 and the probability associated with point 1, where the EMV for doing nothing is equal to the EMV of the small plant. When P is between the probabilities of points 1 and 2, the best decision is to build the small plant. Point 2 is where the EMV for the small plant is equal to the EMV for the large plant. When P is greater than the probability for point 2, the best decision is to build the large plant. Of course, this is what you would expect when P increases. The value of P in points 1 and 2 is calculated as follows: Point 1: EMV (do nothing) = EMV(small plant) 0 = $120,000P − $20,000 P = 20,000 = 0.167 120,000 Point 2:

EMV (large plant) =EMV (large plant) A graphical representation of the Thompson Lumber sensitivity analysis is shown in Figure 10. Figure 10. Sensitivity Analysis Graphical Representation Thompson Lumber EMV value Point 2 EMV (Large Plant) $300,000 EMV (Small Plant) $200,000 .615 $100,000 Value of P EMV (do nothing) 0 1 –$100,000 –$200,000 Point 1 .167 Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. To see the application of the use of the sensitivity analysis in a real situation you can access the following article in the Complementary Material folder which is linked at the end of the links to the Modules. Suresh. A. S. (2012). A Study on Application of Sensitivity Analysis Techniques in Capital Budgeting Decisions. Asia Pacific Journal of Management & Entrepreneurship Research, 1(2), 210-234.

Decision Tree Any problems that may arise in a decision table can also be illustrated with a graph called a decision tree. All decision trees are similar in that they contain decision points or decision nodes and nature state points or nature state nodes: • A decision node is one where you can choose one of several alternatives. • A nature state node indicates the states of nature that can occur. Problem analysis with decision trees includes five steps: 1. Define the problem. 2. Structure or draw the decision tree. 3. Assign probabilities to the states of nature. 4. Calculate the benefits for each possible combination of alternatives and states of nature. 5. Solve the problem by calculating the expected monetary values (EMV) for each state of nature node. When drawing a tree, we start from the left and move to the right. Thus, the tree presents decisions and results in sequential order. The lines or branches that come out of the boxes (decision nodes) represent alternatives; while the branches that come out of the circles represent states of nature. The example of Thompson Lumber is shown in Figure 12 and Figure 13. Figure 12. Thompson Lumber Decision Tree representation State of Nature Favorable Market Decision node 1 Unfavorable Market Build small plant 2 Favorable Market Unfavorable Market

Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Figure 13. Decision tree for Thompson Lumber complete and resolved Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Observe that in the example in Figure 13 the payments are placed to the right of each tree branch. The probabilities are shown in parentheses next to each state of nature. Starting with the payments to the right of the figure, the EMVs are calculated for each state of nature and then placed next to their respective nodes. The EMV of the first node is $10,000. This represents the branch from the decision node to build a large plant. The EMV for node 2, building a small plant, is $40,000. Not building or doing nothing, of course, has a $0 payment. In the Thompson case, a small plant would have to be built.

Bayes' Theorem The Bayes' theorem approach recognizes that a decision maker does not know with certainty what state of nature will occur. It allows the manager to revise his or her initial assessment of the probabilities based on new information. The revised probabilities are called subsequent probabilities. For the Thompson Lumber example, a market study must first be conducted. From discussion with market research specialists at his hometown university, John knows that special studies like his will either be positive (that is predict a favorable market) or negative (that is predict an unfavorable market). These results are presented in Figure 14. Figure 14. Reliability of market research to predict states of nature State of Nature Study result Favorable Market Unfavorable Market Positive (predict a P(study positive) favorable market for P(study negative) the product) Negative (predict a favorable market for the product) Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. The general form of Bayes' theorem presented in Module 1 is: P(A | B) = P(B | P(B | A)  P( A) A)  P( A) + P(B | A)  P( A) Where: A, B = either of two events A = complement of A

We can say that A represents a favorable market and B represents a positive market study. Then, by substituting the appropriate numbers in this equation, we get the conditional probabilities, since the market research is positive: After substituting in the formula and solving; the results probabilities revised given a positive study and a negative study. These are shown in Figure 15 and Figure 16. Figure 15. Positive Study Results Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. Figure 16. Negative Study Results Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson.

To see application of the use of Bayes' Theorem in the marketing area you can access the following article in the Supplementary Material folder. Dave, H. P. & Dave, K. H. (2015). Application of Bayesian Decision Theory in Management Research Problems. International Journal of Scientific Research Engineering & Technology, 191-195. -------------------------------------------------------------------------------- 2.2 Regression Introduction Topic 2.2 of the Module will also teach the techniques of linear and multiple regression, ANOVA and correlation. Discussion will be initiated by presenting tools such as Cartesian planes and scatter plots. Students will be introduced to concepts such as intercept, slope, and estimation errors. Regression Analysis In business, the key to decision making often lies in understanding the relationships between two or more variables. For example, financial experts, when studying bond market behavior, may find it useful to know whether bond interest rates are related to the prime rate set by the Federal Reserve. On the other hand, a marketing executive might want to know how strong the relationship is between advertising dollars and sales dollars of a product or company (Black, 2016). Regression analysis is a very valuable tool for today's manager. It is the process of building a mathematical model or function that can be used to predict or determine a variable by another variable or other variables (Black, 2016). Usually the first step in simple regression analysis is to construct a scatter plot. In the diagram, the independent variable is usually plotted on the X-axis. The dependent variable is usually plotted on the Y-axis. Example Triple A Construction Triple A Construction Company remodels old houses in Albany. Over time, the company found that its dollar volume of remodeling work depended on the Albany area payroll. The figures for Triple A's revenue and the amount

of money earned by Albany workers over the past six years are presented in Figure 1. Economists have anticipated that the payroll in the local area will be $600 million next year and Triple A wants to plan according to this. Figure 1. Triple A Construction Data Triple A sales Local payroll ($100,000,000s) Adapted from Render, B., Stair, R. M., Hanna, M. E. & Hale, T. S. (2018). Quantitative Analysis for Management. (13 ed). Pearson. The scatter plot in Figure 2 indicates that higher values for the local payroll seem to result in higher sales for the company. There is no perfect relationship because not all the points are in a straight line, but there is a relationship. Figure 2: Triple A Construction Scatter Plot sales Payroll

Simple Linear Regression In any regression model there is an implicit assumption (which can be tested) that there is a relationship between the variables. There is also a random error that cannot be predicted. The fundamental simple linear regression model is: Y = B0 + B1X + Ɛ where Y = dependent variable (response) X = independent variable (predictive or explanatory) B0 = interception (value of Y when X = 0) B1 = pending the regression line Ɛ = random error The actual values for the intersection are not known: they are estimated using the sample data. The regression equation based on the sample data is given by: ���̂��� = ������0 + ������1������ Where = forecast value of Y B0 = estimate of B0, according to the results of the sample B1 = estimate of B1, according to the results of the sample By calculating the slope and the intersection of the regression equation for the Triple A Construction company example we have: Estimated equation above means the following: The Sales = 2 + 1.2X

As we saw, if the payroll for next year is $600 million (X = 6), then the anticipated value would be: = 2 + 1.25 x 6 = 9.5 ó $950,000 To find out if the model is really useful for predicting X-based Y, you could simply take the average error. However, positive and negative errors would cancel each other out. Then there are three measures of variability to consider, which are listed below: Sum of squares total (SST) - Total variability over mean. Sum of squares error (SSE) - Variability on the regression line. ������������������ = ∑ ������2 = ∑(������ − ���̂���)2 Sum of squares due to regression (SSR) - The total variability that is explained by the model. SSR =  (Yˆ − Y )2 Coefficient of Determination A widely used measure of adjustment for regression models is the coefficient of determination, or r2. The coefficient of determination is the proportion of variability of the dependent variable (y) explained or accounted for by the independent variable (x) (Black, 2016). The following formula is used to determine this coefficient: ������2 = ������������������ = 1 − ������������������ ������������������ ������������������ Correlation Coefficient Another measure related to the coefficient of determination is the correlation coefficient. This measure also expresses the degree or strength of the linear relationship. In general, it is expressed as r and can be any number between +1 and -1, including both values. The following formula is used to determine this coefficient: r =  r2 Analysis of variance table (ANOVA) SST =  (Y − Y )2

Analysis of Variance, or ANOVA, is a group of statistical tools. Analysis of the concept of variance begins with the notion that the responses of dependent variables (measurements, data) are not all the same in a given study. In other words, measures of dependent variables such as employee performance, sales, and length of stay, customer satisfaction and product viscosity often vary from one element to another, from one observation to another. Using various types of experimental designs, we can explore some possible reasons for this variation by analyzing the variation techniques (Black, 2013). Triple-A example in Excel 2016 The example of Triple A Construction is shown in Figure 3, Figure 4 and Figure 5. Figure 3. Step 1 Linear Regression Triple A Construction

Figure 4. Step 2 Triple A Construction Linear Regression

Figure 5. Triple A Construction Linear Regression Results A high r2 is A low F value of less than 0.5 desirable indicates a significant rounded to 1 relationship between X and Y The SSR regression. SSE error or residue and total SST are shown in the sum of SS squares column of the Analysis of Variance table ANOVA The relationship coefficients are shown in this column To see application of the use of simple regression analysis in the finance area you can access the following article in the Supplementary Materials folder. Brenes-González, H. A. (2017). Application of simple linear regression analysis for the estimation of the prices of Facebook, Inc. Revista Electrónica de Investigación en Ciencias Económicas (REICE), 5(10), 133-155. Multiple Regression The first step in the multiple regression is to write a model that captures the assumed relationship between the dependent variable and the independent variables. In general, we can label the variables with subscripts of 1 to the number of independent variables included in the model. The model equation developed for the multiple regression is as follows:

���̂��� = ������0 + ������1������1 + ������2������2+. . . +������������������������ where Y = dependent variable Xi = intersection (value of Y when Xi 0, ordered to the origin) 0 = intersection (value of Y when Xi 0, ordered at the origin) i = coefficient of the i-th independent variable k = Number of independent variables  = random error It should be highlighted that a multiple regression model is evaluated in a similar way as the simple linear regression model was evaluated. In multiple regression models, the p- value for the F test and r2 is interpreted in the same way as in simple linear regression models. To determine which of the independent variables in a multiple regression model is significant, a significance test is performed on the coefficients of each variable. While the statistics books provide details of these tests, the results are automatically displayed in the Excel output, as we will be viewing later on. Example Jenny Wilson Realty in Excel 2016 Below are the results of the Jenny Wilson Realty example. Figure 6. Jenny Wilson Realty example The determination A low significance level for F proves coefficient r2 is that there is a relationship between Y 0.67 and at least one of the independent variables x The p-values are used to test the significance of the individual variables The regression coefficients can be found here In the Jenny Wilson Realty example shown in figure e, the full model is statistically significant and useful for predicting the sales price of the house, since the p-value of

test F is 0.02. The r2-value is 0.6719, so 67% of the variability in the sales price of these houses could be explained by the regression model. However, there are two independent variables in the model: square footage of construction and age. It is possible that one of them is significant and the other is not. The F test only indicates that the model as a whole is significant.