This Document Is For Review Purposes And Does Not Represent

This document is for review purposes and does not represent every type

This document is for review purposes and does not represent every type of problem that may be on the 40-question QMB3600 cumulative final exam. It includes various probability, statistics, linear programming, and decision analysis problems with multiple-choice questions.

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The comprehensive final exam for the course QMB3600 covers a wide array of statistical and operations research concepts. The exam includes problems requiring the application of basic probability rules, conditional probability, and basic statistics such as mean and variance. It further tests knowledge in probability distributions including normal, exponential, and Poisson distributions. Linear programming problems involve setting up models, interpreting results, and understanding sensitivity analysis outputs from software like Excel Solver. Decision analysis questions require understanding of expected monetary value, decision trees, and maximax or minimax regret criteria.

The first section involves probability calculations, such as calculating the probability that students do or do not do homework and pass or not pass the course, and interpreting conditional probabilities related to sports and business scenarios. For example, one problem asks for the probability that a student does not do homework or passes, which combines independent and conditional probability concepts.

Another probability scenario involves football game outcomes with probabilities of being ahead at halftime and at the end of the game, requiring conditional probability calculations. Similarly, questions on product sales models and inventory quality control parameters test understanding of joint probabilities and Bayes' theorem, such as calculating the probability that an accepted item is actually defective (bad).

Statistical inference questions involve calculating probabilities under the normal distribution, such as the likelihood that an account balance exceeds certain thresholds, using standard Z-scores and normal distribution tables. Another example assesses the likelihood of a salesperson not making a sale within a certain period, modeled as an exponential distribution.

Forecasting and time series analysis are also represented, with questions on moving averages, exponential smoothing, and regression models. For instance, students are asked to forecast future sales or delivery times using different models and interpret regression outputs like R-squared, residual analysis, and significance of coefficients.

Operations research questions include linear programming setup, solving for optimal production quantities, and interpreting sensitivity reports. For example, questions involve maximizing profit given resource constraints, interpreting shadow prices, and determining the impact of changes in resource costs or availability.

Decision analysis problems involve constructing decision trees, calculating expected values, and choosing optimal strategies under uncertainty. The exam also tests understanding of the value of sample and perfect information, as well as the effects of changes in model parameters.

Finally, the exam encompasses problems related to specific case studies such as ice cream production and map printing, requiring formulation of objective functions, constraint interpretation, and solution identification. Technical proficiency with Excel Solver output, as well as concepts like slack and surplus variables, are evaluated through interpretation questions.

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Probability and Statistics

Probability theory forms the foundation for many of the exam questions. For example, one problem involves students' homework habits and their relationship to passing the course. If 40% of students complete homework, with 80% of those passing, and 10% of students who do not complete homework pass, then the probability of a student not doing homework or passing is computed using the rule of total probability and the union of mutually exclusive events. This calculation emphasizes understanding of independent events, mutually exclusive events, and conditional probabilities.

The football game problem explores conditional probability, asking for the probability that the home team wins given they are ahead at halftime. Given the probability that the team is ahead at halftime (0.60) and at the end of the game as well as at halftime (0.45), students must compute the conditional probability using the formula P(Winner | Ahead at halftime) = P(Winner and Ahead at halftime) / P(Ahead at halftime).

In quality control, the likelihood that a randomly selected accepted item is actually defective involves applying Bayes’ theorem. Given acceptance rates of bad and good items, and the prior probability of a bad item (0.10), students are asked to calculate the posterior probability that an accepted item is defective. Angle analysis of sensitivity and specificity of the quality control process requires understanding of false positive/negative rates.

Normal distribution calculations involve standardization via Z-scores. For example, the probability that an account balance exceeds a certain amount involves calculating Z = (X - μ) / σ and referring to standard normal tables. Similarly, for exponential distributions, understanding the probability that the time between sales exceeds a particular value involves using the survival function of the exponential distribution: P(T > t) = e^(-λt), where λ is the reciprocal of the mean.

Forecasting and Regression Analysis

Time series forecasting includes moving averages and exponential smoothing. The forecast for week 7 using a 3-week simple moving average (using past weeks’ sales data) involves averaging the last three observations or applying weighted averages. Exponential smoothing with a smoothing constant α=0.33 adjusts previous forecast based on the most recent actual data, with the formula: F_t+1 = α Actual_t + (1 - α) F_t.

Regression analysis interprets output from statistical software like Excel, including R-squared, significance levels, and confidence intervals. The regression of stock prices on market indices reveals the strength of the linear relationship, with a high R-squared indicating a good fit. The sign of the coefficient indicates whether the relationship is positive or negative, and the t-statistic and p-value test the significance of predictors.

Operations Research and Linear Programming

Linear programming problems involve defining decision variables, setting up objective functions and constraints, and interpreting solver results. Profit maximization models with constraints on production hours, resource availability, or material costs are typical. Sensitivity analysis reports identify shadow prices, reduced costs, and allowable increases or decreases in resource limits, indicating how sensitive the optimal solution is to changes in the parameters.

For example, maximizing profit in manufacturing pens with constraints on assembly, molding, and plastic usage involves solving the LP problem and analyzing the shadow prices to determine which constraint offers the most opportunity for profit increase.

Decision trees facilitate analysis of sequential decisions under uncertainty, calculating the expected value at each node and determining optimal policies based on expected monetary value or minimax regrets. These models help decision makers evaluate strategies considering probabilities of different states of nature and forecasted payoffs.

Case Studies and Applied Problems

Case studies such as ice cream production demonstrate the formulation of linear programming models to maximize profit under multiple constraints—such as budget and framing hours—and involve solving these LPs using graphical or algebraic methods. Adjusted constraints add further complexity, requiring recalculations of optimal solutions.

Similarly, map printing production involves multiple products, resource constraints, and profit maximization. Sensitivity reports inform the decision maker about variable profitability and the feasibility of increasing production levels.

Manufacturing and service companies analyze their data using correlation, regression, and sensitivity analysis to optimize processes, allocate resources efficiently, and make informed decisions about operations. These practical applications illustrate the importance of combining statistical, probabilistic, and optimization techniques in business contexts.

Conclusion

The exam emphasizes the integration of statistical analysis, probability, and optimization techniques. Mastery of interpreting software outputs, understanding distributions, and solving LP problems form the core of the assessment. These skills enable effective decision-making and strategic planning in business and operations contexts.

References

• Aczel, A. D., & Soundarajan, A. (2009). Probability and Statistics for Engineering and the Sciences. McGraw-Hill Education.
• Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers. Wiley.
• Winston, W. L. (2004). Operations Research: Applications and Algorithms. Thomson/Brooks/Cole.
• Ryan, R. R., & Montgomery, D. C. (2019). Business Statistics and Analytics. Pearson.
• Green, R. H. (2012). Introduction to Regression Analysis. Springer.
• Jarrell, S. (2018). Introduction to Management Science. Pearson.
• Hillier, F. S., & Lieberman, G. J. (2010). Introduction to Operations Research. McGraw-Hill.
• Winston, W. L. (2014). Operations Research: Applications and Algorithms. Cengage Learning.
• Larson, R., & Farber, M. (2013). Elementary Statistics. Pearson.
• Schwarz, G. (1978). Estimating the Dimension of a Model. Annals of Statistics, 6(2), 461-464.