Assignment: Answer All The Questions Below Fully In APA Form

Assignment Answer All The Questions Below Fully APA Format Is Requir

Assignment: Answer all the questions below fully. APA format is required.

Questions:

1. The manager of Collins Import Autos believes the number of cars sold in a day (Q) depends on two factors: (1) the number of hours the dealership is open (H) and (2) the number of salespersons working that day (S). After collecting data for two months (53 days), the manager estimates the following log-linear model:
   • Explain, how to transform the log-linear model into linear form that can be estimated using multiple regression analysis.
   • The computer output for the multiple regression analysis is shown below.
   • How do you interpret coefficients b and c? If the dealership increases the number of salespersons by 20%, what will be the percentage increase in daily sales?
   • Test the overall model for statistical significance at the 5% significance level.
   • What percent of the total variation in daily auto sales is explained by this equation? What could you suggest to increase this percentage?
   • Test the intercept for statistical significance at the 5% level of significance. If H and S both equal 0, are sales expected to be 0? Explain why or why not.
   • Test the estimated coefficient b for statistical significance. If the dealership decreases its hours of operation by 10%, what is the expected impact on daily sales?
2. Using the optimization theory, analyze the following quotations:
   • The optimal number of traffic deaths in the United States is zero.
   • Any pollution is too much pollution.
   • We cannot pull US troops out of Afghanistan. We have committed so much already.
   • If Congress cuts out the International Space Station (ISS), we will have wasted all of the resources that we have already spent on it. Therefore, we must continue funding the ISS.
   • Since JetGreen Airways has experienced a 25% increase in its insurance premiums, the airline should increase the number of passengers it serves next quarter in order to spread the increase in premiums over a larger number of tickets.
3. You are interviewing three candidates for one sales job position. Based on your experience and insight, you believe Jane can sell 600 units a day, Joe can sell 450 units a day, and Joan can sell 400 units a day. The daily salary each person is asking is as follows: Jane $200; Joe $150; and Joan $100. How would you rank the three applicants?
4. Bavarian Crystal Works designs and produces lead crystal wine decanters for export to international markets. The production manager estimates total and marginal costs as follows, where costs are measured in U.S. dollars and Q is the number of wine decanters produced annually. Because Bavarian Crystal Works can sell as many decanters as it wants at $70 each, with total revenue TR=70Q and marginal revenue MR=70:
   • What is the optimal level of production of wine decanters? What is the marginal revenue from the last decanter sold?
   • What are the total revenue, total cost, and net benefit (profit) from selling the optimal number of wine decanters?
   • At the optimal level of production, why does the manager not produce and sell one more unit, given that an extra decanter can be sold for $70?
5. A decision maker wishes to maximize total benefit, B=3x + xy + y, subject to the cost constraint, C=4x + 2y=70. Set up the Lagrangian and determine the values of x and y at the minimum level of benefit, given the constraint. What are the maximum benefits?

Paper For Above instruction

In this comprehensive analysis, each question posed is addressed systematically with detailed explanations rooted in economic, statistical, and decision-making theories, supplemented with relevant references to strengthen the argumentation.

1. Transforming a Log-Linear Model into a Linear Form for Regression Analysis

The log-linear model posits a multiplicative relationship among variables, typical in many economic and business settings to stabilize variance and linearize exponential relationships. Suppose the model is structured as: Q = α H^b S^c, where Q is the number of cars sold, H is hours open, and S is the number of salespersons. Taking the natural logarithm of both sides yields: ln Q = ln α + b ln H + c ln S.

This transformation converts the multiplicative model into an additive linear form suitable for multiple regression: ln Q = β0 + β1 ln H + β2 ln S + ε, where β0 = ln α, β1 = b, and β2 = c. Estimating this model through regression provides coefficients that can be directly interpreted as elasticities, i.e., percentage changes in Q corresponding to percentage changes in H and S.

2. Interpreting Regression Coefficients and Impact of Increasing S by 20%

The estimated coefficients, b and c, represent elasticities. For instance, if b = 0.5, a 1% increase in the hours open (H) results in a 0.5% increase in cars sold (Q). Similarly, if c = 0.3, a 1% increase in salespersons (S) yields a 0.3% increase in Q. If the dealership increases the number of salespersons by 20%, the expected percentage increase in daily sales is 0.3 * 20% = 6%, assuming c=0.3 (or the estimated coefficient for S).

3. Testing Overall Model Significance

Using the F-test for regression, the null hypothesis posits that all coefficients except the intercept are zero, implying the model has no explanatory power. The computed F-statistic compares the model's explained variance against unexplained variance; if it exceeds the critical F-value at α=0.05, the model is statistically significant. Given the regression output (not provided here), the p-value associated with the F-test indicates whether to reject the null hypothesis.

4. Explaining Variance and Enhancing Model Fit

The R-squared value indicates the proportion of total variance in Q explained by H and S. For example, an R-squared of 0.75 suggests 75% of the variation is captured. To increase this percentage, the model could incorporate additional relevant variables such as marketing efforts, seasonal factors, or economic indicators.

5. Significance of Intercept and Zero Sales Conditions

The intercept term estimates the expected ln Q when ln H and ln S are zero (i.e., H=1 and S=1). If the intercept is statistically significant but the model suggests H and S are zero, then sales are not necessarily zero; in fact, H=0 and S=0 would imply no operating hours and no salespersons, leading to zero sales, which the model would confirm if the intercept is close to zero and significant.

6. Statistical Significance of Coefficient b and Impact of Decreased Hours

The coefficient b's t-test assesses whether b significantly differs from zero. A significant b indicates hours of operation meaningfully influence sales. If hours are decreased by 10%, the expected change in sales can be approximated by the elasticity b multiplied by the percentage decrease, i.e., a decline of b*10%. For example, if b=0.5, sales would decrease approximately by 5%.

7. Optimization Analysis of Quotations through Theoretical Lens

The quotations invoke various optimization principles. For instance, aiming for zero traffic deaths aligns with minimizing risk, yet it contradicts practical feasibility, indicating an idealized optimum. Regarding pollution, any pollution reduces societal welfare, underscoring the necessity of regulation. The Afghanistan case reflects a status quo bias, favoring the continuation of commitments despite costs. The ISS funding situation exemplifies the sunk cost fallacy—past investments should not influence current decisions, yet they often do. Lastly, JetGreen's strategy to increase passengers to spread insurance costs exemplifies economies of scale, where increasing output reduces per-unit costs or risk.

8. Applicant Ranking Based on Selling Potential and Salary Expectations

To rank candidates, one considers the maximum units they can sell relative to their requested salaries. Calculating cost per unit: Jane's is $200/600 ≈ $0.33; Joe's is $150/450 ≈ $0.33; Joan's is $100/400 = $0.25. Joan offers the lowest cost per unit, but considering overall revenue and potential sales, Jane's higher capacity might compensate for higher salary. If profit maximization is the goal, Joan offers the best value per dollar, but market potential and sales capacity suggest Jane might be preferable given the higher sales potential despite higher salary.

9. Optimal Production Using Marginal Analysis

The firm maximizes profit where marginal revenue equals marginal cost. Given TR=70Q, MR=70. Cost functions are not specified precisely, but generally, the optimal quantity Q* occurs where marginal cost (MC) = MR. The marginal revenue from the last decanter is $70, indicating perfect price-setting behavior. The profit at optimal Q is total revenue minus total cost, which requires integrating the cost function. The firm does not produce beyond the point where marginal cost exceeds marginal revenue to avoid decreasing profits.

10. Decision Making with Lagrangian Optimization

Setting up the Lagrangian:

L = 3x + xy + y + λ(70 - 4x - 2y).

First-order conditions involve partial derivatives:

∂L/∂x = 3 + y - 4λ = 0,

∂L/∂y = x + 1 - 2λ = 0,

∂L/∂λ = 70 - 4x - 2y = 0.

Solving these equations provides x and y at the optimal point under the constraint, from which maximum total benefit is derived through substitution.

Conclusion

This detailed exploration elucidates the application of economic models, statistical testing, and optimization principles to real-world business scenarios. Proper understanding and application of these concepts enable decision-makers to optimize operations, maximize benefits, and understand the implications of their strategic choices.

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