Unit 4 Practical Exercise: The Following Example Illustrates ✓ Solved

Unit 4 Practical Exercise The following example illustrates

Unit 4 Practical Exercise The following example illustrates how probability values using areas under the standard normal distribution can be used to help make business decisions.

Explain how probability can be used to help solve management-type questions/problems.

Think of something at work, past or present, where you could apply the techniques in the example to assist in making the best decision.

If you can’t draw on life experience, think of a product/issue where this process could be applied.

Please explain your answer. Remember to cite your resources and use your own words in your explanation.

Paper For Above Instructions

Probability and decision making in business frequently hinge on understanding how a variable of interest behaves under uncertainty. The Goodyear example described in the prompt leverages the normal distribution to translate a performance target (miles driven by a tire) into a probability statement that informs policy decisions such as guarantees, discounts, or replacements. In managerial contexts, this approach helps decision makers quantify risk and set thresholds that balance customer expectations, warranty costs, and brand reputation. Foundational texts in statistics and probability for business and engineering repeatedly emphasize that many real-world outcomes can be modeled, approximately, by the normal distribution when sample sizes are large or when central limit effects apply (Walpole et al., 2012; Montgomery & Runger, 2014). The core idea is to compute the likelihood of an event (e.g., mileage exceeding a threshold) using the standardized z-score and the standard normal table, then translate that probability into actionable policy (Phi or 1 − Phi values) (Moore, McCabe, & Craig, 2014; De Veaux, Velleman, & Bock, 2009).

Key steps in applying this approach to management problems include: (1) specify the objective or threshold that matters to the decision (e.g., guarantee mileage, minimum service life, or stockout risk); (2) estimate the distribution parameters (mean μ and standard deviation σ) from reliable data; (3) convert the threshold to a z-score using z = (x − μ)/σ; (4) read the corresponding probability from the standard normal distribution; and (5) use that probability to set policy, such as the cutoff point for a warranty or the service level target for inventory. This workflow is widely discussed in statistics for business contexts, where the translation from data to decision requires careful modeling and interpretation (Anderson et al., 2016; McClave, Benson, & Sincich, 2014).

One common managerial application is setting a product guarantee or warranty level. Suppose a product has a lifetime that is roughly Normally distributed with a mean μ and standard deviation σ. If management wants to guarantee a threshold x such that only a certain small fraction p of units fail before x (i.e., the left tail up to x equals p), the z-score z_p corresponding to p is determined from the standard normal CDF, and then x = μ + z_p σ. This logic underpins the Goodyear example where a 10% tail is used to determine a mileage guarantee: the z value corresponding to 0.10 is found in the standard normal table, and the target mileage x is computed accordingly (Walpole et al., 2012; Montgomery & Runger, 2014).

Beyond warranties, the same probabilistic framework can guide decisions such as service levels, inventory replenishment, quality control, and pricing strategies. For example, a firm with daily demand that is approximately Normal can set a reorder point by combining demand during lead time with safety stock. If mean demand during lead time is μ_LT and standard deviation is σ_LT, the reorder point is ROP = μ_LT + z_α σ_LT, where z_α corresponds to the desired service level α. This approach aligns with standard methods taught in statistics for business and quality control (Neter et al., 1996; Box, Jenkins, & Reinsel, 2015). It illustrates how probability can translate into concrete operational thresholds that affect costs and customer satisfaction (Moore et al., 2014; Walpole et al., 2012).

From a managerial perspective, several cautions are warranted. The normal model assumes symmetry and well-behaved tails; real-world data may exhibit skewness, heavy tails, or multimodality. In such cases, relying solely on a single z-score for decision thresholds may misstate risk. Practitioners should perform goodness-of-fit checks, consider alternative distributions, and use simulations or bootstrapping to assess the robustness of the chosen threshold (Montgomery & Runger, 2014; De Veaux et al., 2009). Additionally, sample size and data quality influence the reliability of μ and σ estimates; poor estimates can lead to misguided policies. Consequently, it is prudent to update thresholds as new data accumulates and to assess sensitivity to different tail probabilities (Anderson et al., 2016; McClave et al., 2014).

Applying these concepts in a concrete workplace scenario helps illustrate the value and limits of probability-based decision rules. Consider a company launching a new electronic device with a mean lifetime of μ = 1200 hours and a standard deviation σ = 180 hours. If management wants to guarantee at least 95% of units will surpass a certain lifetime, they would seek x such that P(X > x) = 0.95, equivalently P(X ≤ x) = 0.05. The corresponding z-score is z ≈ −1.645. Then x = μ + zσ = 1200 + (−1.645)(180) ≈ 1200 − 296.1 ≈ 903.9 hours. In practice, this means the company would set a warranty threshold around 900 hours to ensure only about 5% of units would fail before the threshold, given the normal assumption. This example demonstrates how probability informs risk-taking in product guarantees and after-sales policies, balancing customer value with warranty costs (Montgomery & Runger, 2014; Walpole et al., 2012).

In another scenario, a firm evaluating promotional claims might model daily sales as Normal with mean 500 units and standard deviation 60 units. If the company wants a 99% service level for stock availability, it could compute the safety stock using z_0.99 ≈ 2.33 and determine the appropriate reorder point by incorporating lead-time demand variability. The same mathematical structure—convert a service level into a z-score, then translate into an inventory threshold—applies across many management contexts (Moore et al., 2014; Neter et al., 1996). These examples reinforce the practical takeaway: probability theory provides a principled basis for setting operational targets that align with strategic goals, while also highlighting the need for good data and thoughtful model validation (McClave et al., 2014; De Veaux et al., 2009).

In sum, probability values derived from areas under the standard normal distribution offer a powerful toolkit for managerial decision making. By converting performance targets into probabilistic statements, managers can quantify risk, justify policies, and articulate expected outcomes to stakeholders. The Goodyear example serves as a concrete illustration of this process, and the broader approach can be adapted to a wide range of product, process, and service decisions. As with any model, care must be taken to validate assumptions, monitor results, and adjust thresholds in light of new data and changing conditions. The core lesson for managers is clear: use the normal distribution as a decision aid, but couple it with data quality checks, scenario analyses, and ongoing learning to make informed, responsible choices (Anderson et al., 2016; McClave et al., 2014; Walpole et al., 2012).

References

• Walpole, R. E., Myers, R. H., Myers, S. L., Ye, Y. (2012). Probability & Statistics for Engineers & Scientists (8th ed.). Pearson.
• Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
• Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., Cochran, S. (2016). Statistics for Business and Economics (11th ed.). Cengage.
• McClave, J. T., Benson, P. G., Sincich, T. (2014). Statistics for Business and Economics (13th ed.). Pearson.
• Moore, D. S., McCabe, G. P., Craig, B. A. (2014). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman.
• De Veaux, R. D., Velleman, P. F., Bock, D. (2009). Intro Stats (2nd ed.). Pearson.
• Neter, J., Kutner, M. H., Nachtsheim, C. J., Wasserman, W. (1996). Applied Linear Statistical Models (4th ed.). McGraw-Hill.
• Box, G. P., Jenkins, G. M., Reinsel, G. C. (2015). Time Series Analysis: Forecasting and Control (5th ed.). Wiley.
• Mendenhall, W., Beaver, B., Beaver, R. (2013). Introduction to Probability and Statistics for Engineers and Scientists. Brooks/Cole.
• Freund, J. E., & Wilson, W. J. (2010). Statistics: A Foundation for Analysis. Cengage Learning.