Finite Mathematics Quiz: Sensitivity Analysis
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Sensitivity AnalysisQuestion 1 of 14

In a sensitivity analysis, the range of a parameter over which the optimal decision remains unchanged is called the 'range of optimality.' If Alternative A has an EMV of $45,000 and Alternative B has an EMV of $38,000 when the probability of success is 0.60, and the payoff for Alternative A under success is $90,000 while under failure it is $15,000, what is the lower bound of the range of optimality for the probability of success?

The probability at which Alternative A's EMV equals Alternative B's EMV, requiring knowledge of Alternative B's payoffs
0.45, calculated by setting the EMVs equal and solving for the probability that makes Alternative A marginally preferred
0.55, found by determining when Alternative A's EMV drops below its current advantage over Alternative B
0.60, since this is the current probability and represents the minimum for maintaining current EMV relationships
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Finite Mathematics Quiz

Finite Mathematics Quiz: Sensitivity Analysis

Practice Sensitivity Analysis in Finite Mathematics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Sensitivity Analysis, giving you a quick way to practice the rules, question types, and explanations that matter most for Finite Mathematics.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

In a sensitivity analysis, the range of a parameter over which the optimal decision remains unchanged is called the 'range of optimality.' If Alternative A has an EMV of $45,000 and Alternative B has an EMV of $38,000 when the probability of success is 0.60, and the payoff for Alternative A under success is $90,000 while under failure it is $15,000, what is the lower bound of the range of optimality for the probability of success?

  1. The probability at which Alternative A's EMV equals Alternative B's EMV, requiring knowledge of Alternative B's payoffs (correct answer)
  2. 0.45, calculated by setting the EMVs equal and solving for the probability that makes Alternative A marginally preferred
  3. 0.55, found by determining when Alternative A's EMV drops below its current advantage over Alternative B
  4. 0.60, since this is the current probability and represents the minimum for maintaining current EMV relationships
Explanation: To find the range of optimality, we need to determine when Alternative A's EMV equals Alternative B's EMV. We know A's payoffs ($90,000 success, $15,000 failure) but need B's payoffs to solve for the crossover probability. The calculation requires: p(90,000) + (1-p)(15,000) = EMV of B, but B's individual payoffs are unknown. Choices B and C provide specific values without sufficient information, while D incorrectly assumes the current probability is a boundary.

Question 2

A manufacturing company is evaluating three production strategies using decision analysis. The initial analysis assumed that the probability of 'High Demand' is 0.45, 'Medium Demand' is 0.35, and 'Low Demand' is 0.20. The current optimal strategy is Strategy 2 with an EMV of $340,000, followed by Strategy 1 at $325,000, and Strategy 3 at $310,000.

The company's market research team indicates that the probability of 'High Demand' could be anywhere from 0.35 to 0.55, with corresponding adjustments to keep total probability equal to 1.0 (assuming 'Medium Demand' and 'Low Demand' probabilities change proportionally). Under this uncertainty, what additional information would be most critical for completing a meaningful sensitivity analysis?

  1. The individual payoffs for each strategy under each demand scenario, since EMV changes depend on specific payoff-probability combinations (correct answer)
  2. The correlation between demand scenarios and external economic factors, to better estimate the realistic probability ranges
  3. The costs associated with switching strategies after the demand level becomes known, to factor in implementation flexibility
  4. The confidence intervals around each current EMV estimate, since these affect the significance of strategy ranking changes
Explanation: To conduct sensitivity analysis on probability changes, we need the individual payoffs for each strategy under each scenario. EMV calculations require multiplying probabilities by payoffs, so we cannot determine how probability changes affect strategy rankings without knowing the specific payoff values. Choice B addresses probability estimation but not sensitivity analysis mechanics, C introduces implementation costs not relevant to basic sensitivity analysis, and D confuses confidence intervals with the deterministic sensitivity analysis process.

Question 3

A sensitivity analysis on a three-alternative decision problem reveals that Alternative X has a 'range of optimality' for a key probability parameter from 0.15 to 0.75. Alternative Y is optimal when this probability is below 0.15, and Alternative Z is optimal when it exceeds 0.75. If the current probability estimate is 0.45 with a standard error of 0.12, what is the most appropriate interpretation for decision-making purposes?

  1. Alternative X should be chosen with high confidence since 0.45 is near the center of its optimality range
  2. The decision is highly sensitive since small probability changes could shift optimality to Alternative Y or Z
  3. Additional data collection is essential because the standard error spans multiple optimality ranges for different alternatives
  4. Alternative X is robust since the entire confidence interval (approximately 0.21 to 0.69) falls within its optimality range (correct answer)
Explanation: When analyzing decisions under uncertainty, sensitivity analysis helps you understand how changes in key parameters affect which alternative is optimal. The critical insight here is comparing your confidence interval for the probability estimate against the ranges where each alternative is best. With a probability estimate of 0.45 and standard error of 0.12, you can construct an approximate 95% confidence interval using ±2 standard errors: 0.45 ± 2(0.12) = 0.21 to 0.69. Since Alternative X is optimal when the probability falls between 0.15 and 0.75, and your entire confidence interval (0.21 to 0.69) sits comfortably within this range, Alternative X remains the best choice regardless of the uncertainty in your probability estimate. Option A focuses on the point estimate being "near the center" but misses the key issue of uncertainty around that estimate. Option B incorrectly suggests high sensitivity—while the decision would be sensitive if the confidence interval crossed boundary points, it doesn't here. Option C overstates the need for additional data collection; since your confidence interval falls entirely within Alternative X's optimality range, the current uncertainty level is manageable for decision-making purposes. The correct answer is D because Alternative X proves robust against the parameter uncertainty you face. Even accounting for estimation error, you won't accidentally choose a suboptimal alternative. Study tip: In sensitivity analysis problems, always construct confidence intervals around uncertain parameters and check whether they cross the boundary points where optimal alternatives change. If they don't cross boundaries, your decision is robust to the uncertainty.

Question 4

A decision maker wants to determine how much the payoff for Alternative C can decrease before it is no longer optimal. Currently, Alternative C has payoffs of $120,000 (probability 0.3), $80,000 (probability 0.5), and $40,000 (probability 0.2), giving an EMV of $84,000. The second-best alternative has an EMV of 78,000.Ifonlythemiddlepayoff(78,000. If only the middle payoff (80,000) can change, by how much can it decrease while Alternative C remains optimal?

  1. $8,000, since the EMV advantage is $6,000 and the middle outcome has probability 0.5
  2. $15,000, calculated by dividing the EMV difference by the probability of the changing outcome
  3. $12,000, found by solving for the payoff reduction that eliminates Alternative C's EMV advantage (correct answer)
  4. $6,000, which equals the current EMV advantage and represents the maximum allowable total EMV decrease
Explanation: Let x be the decrease in the middle payoff. The new EMV would be: 0.3(120,000) + 0.5(80,000 - x) + 0.2(40,000) = 84,000 - 0.5x. For C to remain optimal: 84,000 - 0.5x ≥ 78,000, so 0.5x ≤ 6,000, giving x ≤ 12,000. Choice A incorrectly calculates 6,000/0.5, B uses 78,000-84,000 incorrectly, and D confuses the EMV difference with the payoff change.

Question 5

A company's optimal inventory management strategy is designated as 'Strategy A'. A sensitivity analysis is conducted on the estimated cost of holding unsold inventory, ChC_h. The analysis reveals that the range of optimality for Strategy A is [$5, $20]. The company's initial estimate for $C_h$ was $12. What is the correct interpretation of this result?

  1. The expected profit under Strategy A is maximized as long as the holding cost ChC_h is between $5 and $20.
  2. If the actual holding cost ChC_h is determined to be $4, Strategy A is still the optimal choice for the company.
  3. Strategy A is the optimal choice only if the actual holding cost ChC_h is exactly equal to the initial estimate of $12.
  4. As long as the actual holding cost ChC_h falls within the interval [$5, $20], Strategy A remains the company's optimal choice. (correct answer)
Explanation: The range of optimality defines the set of values for an input parameter over which the current optimal decision does not change. Here, the analysis shows that Strategy A is the best choice for any holding cost ChC_h from $5 to $20, inclusive. The initial estimate of $12 falls within this range, confirming the current decision, but the key insight is the decision's stability over the entire interval.

Question 6

A manufacturer is deciding between two production processes, Process X and Process Y. The choice depends on the uncertain cost of a raw material, cc. The optimal decision is Process X, based on an estimate of c = \50perunit.AsensitivityanalysisrevealsthattherangeofoptimalityforProcessXisverynarrow,specificallyper unit. A sensitivity analysis reveals that the range of optimality for Process X is very narrow, specifically[$49, $51]$. Which of the following is the most logical conclusion for the manufacturer?

  1. The expected profit from Process X is likely to be very low, regardless of the cost cc.
  2. Process X is definitively superior to Process Y because it was identified as the optimal choice.
  3. The decision to use Process X is highly sensitive to the cost estimate, so obtaining a more precise forecast of cc is critical. (correct answer)
  4. The cost estimate of c=\50$ must be incorrect because the resulting range of optimality is too small to be practical.
Explanation: A narrow range of optimality indicates that the decision is not robust. A small change in the input parameter, in this case the material cost cc, will cause the optimal decision to switch from Process X to Process Y. If the true cost is just slightly outside the [\49, $51]$ range, the current decision will be wrong. Therefore, the high sensitivity means that the accuracy of the cost estimate is critical to making the correct choice.

Question 7

A city is planning a new public transportation system and must choose between a light rail and a bus rapid transit (BRT) system. The decision depends on projected long-term ridership, RR. Based on a ridership forecast of R=25,000R=25,000 passengers/day, the BRT system is found to be optimal. Sensitivity analysis shows that the BRT system remains the optimal choice for any ridership level from R=10,000R=10,000 to R=40,000R=40,000. What does this wide range of optimality imply?

  1. The decision to build the BRT system is robust and not highly dependent on the precision of the initial ridership forecast. (correct answer)
  2. The BRT system will be more profitable than the light rail system for any possible ridership level.
  3. The light rail system is only a better option if ridership is less than 10,000 passengers/day.
  4. The initial ridership forecast of R=25,000R=25,000 is confirmed to be accurate and can be fully trusted for financial planning.
Explanation: A wide range of optimality signifies a robust decision. It means that even if the initial estimate for the parameter (R=25,000R=25,000) is inaccurate, the decision to choose the BRT system will still be correct as long as the true value of RR falls within the wide range of [10,000,40,000][10,000, 40,000]. This makes the decision less dependent on the precision of the forecast.

Question 8

A company is analyzing three potential projects: Alpha, Beta, and Gamma. The expected net present value (ENPV) of each project depends on a single variable, the market growth rate, gg. Sensitivity analysis produces the following findings:

  • Project Alpha is optimal if g<3%g < 3\%.
  • Project Beta is optimal if 3%<g<7%3\% < g < 7\%.
  • Project Gamma is optimal if g>7%g > 7\%. The company's current forecast for the market growth rate is g=5%g = 5\%. Which statement is necessarily true?
  1. At a growth rate of g=3%g=3\%, the ENPV of Project Alpha is equal to the ENPV of Project Gamma.
  2. The ENPV of Project Beta is higher than the ENPV of both Alpha and Gamma at the current forecast of g=5%g=5\%. (correct answer)
  3. Project Beta is the least risky of the three projects because its range of optimality is a finite interval.
  4. If the growth rate forecast is revised to g=8%g=8\%, the company should continue with Project Beta since it was the original optimal choice.
Explanation: The current forecast is g=5%g=5\%. According to the sensitivity analysis results, Project Beta is the optimal choice for any growth rate between 3% and 7%. Since 5% falls within this range (3%<5%<7%3\% < 5\% < 7\%), Project Beta must have the highest ENPV at that growth rate. The other statements are incorrect: at g=3%g=3\%, Alpha's ENPV equals Beta's, not Gamma's. We cannot make conclusions about risk from this information. If the forecast changes to 8%, the optimal decision would change to Gamma.

Question 9

A company has used decision analysis to choose an investment strategy from several alternatives. The analysis relied on estimated probabilities for future market conditions and projected cash flows for each alternative. A comprehensive sensitivity analysis has been performed. Which of the following questions is sensitivity analysis LEAST likely to be able to answer?

  1. Are the initial probability estimates for future market conditions, provided by the economics department, accurate? (correct answer)
  2. What is the range of probabilities for a market downturn over which the currently chosen strategy remains optimal?
  3. How much would the projected cash flow from the second-best alternative need to increase for it to become the optimal choice?
  4. Which of the input variables, market condition probabilities or projected cash flows, has a greater impact on the choice of strategy?
Explanation: When you encounter questions about sensitivity analysis in decision analysis, focus on understanding what sensitivity analysis can and cannot determine. Sensitivity analysis examines how changes in input parameters affect the optimal decision, but it doesn't validate the accuracy of the original estimates themselves. The correct answer is A because sensitivity analysis cannot determine whether the initial probability estimates are accurate. Sensitivity analysis takes the given probabilities as starting points and tests how changes to these values would affect the decision outcome. It's a "what-if" tool that shows the impact of parameter variations, but it has no mechanism to verify whether the original estimates reflect reality. Determining accuracy would require external validation, historical data analysis, or expert verification—not sensitivity analysis. Let's examine why the other options are wrong. Option B is incorrect because sensitivity analysis specifically determines threshold values and ranges where decisions change—this is a core capability. Option C is wrong because sensitivity analysis routinely calculates how much input parameters need to change to alter the optimal choice; this involves straightforward threshold analysis. Option D is incorrect because sensitivity analysis directly measures and compares the relative impact of different variables on decision outcomes through techniques like tornado diagrams. Remember this key distinction: sensitivity analysis is powerful for exploring "what happens if our estimates change" but cannot answer "are our estimates right in the first place." When you see sensitivity analysis questions, separate parameter validation (which it cannot do) from parameter impact assessment (which is its primary purpose).

Question 10

A company's decision analysis includes three alternatives with EMVs of $92,000, $88,000, and $85,000 respectively. A sensitivity analysis on the probability of the 'Economic Growth' scenario (currently 0.35) shows that Alternative 1 remains optimal until this probability drops below 0.28. At exactly p = 0.28, which of the following must be true?

  1. Alternative 1's EMV equals Alternative 2's EMV, and Alternative 3 becomes non-competitive in future probability ranges
  2. Alternative 1's EMV equals the EMV of whichever alternative becomes optimal next, which may be Alternative 2 or 3 (correct answer)
  3. All three alternatives have equal EMVs, creating a three-way tie that defines the boundary of the optimality range
  4. Alternative 1's EMV drops below both other alternatives simultaneously, requiring immediate decision revision at this threshold
Explanation: At the boundary of optimality (p = 0.28), Alternative 1's EMV equals the EMV of whichever alternative becomes optimal next. This could be Alternative 2 or Alternative 3, depending on how the probability change affects each alternative's EMV. We cannot assume it's Alternative 2 just because it currently has the second-highest EMV. Choice A assumes Alternative 2 is next optimal, C incorrectly suggests a three-way tie, and D misunderstands that optimality changes when EMVs become equal, not when one drops below multiple others.

Question 11

A decision problem involves two alternatives, Strategy 1 and Strategy 2, and two states of nature, S1 (with probability pp) and S2 (with probability 1−p1-p). The payoff for Strategy 1 under S1 is V11V_{11} and under S2 is V12V_{12}. The payoff for Strategy 2 under S1 is V21V_{21} and under S2 is V22V_{22}. A sensitivity analysis on pp reveals that Strategy 1 is optimal for any value of pp such that 0≤p≤10 \le p \le 1. What does this imply about the payoffs?

  1. The sum of payoffs for Strategy 1 is greater than the sum of payoffs for Strategy 2.
  2. V11>V12V_{11} > V_{12} and V21>V22V_{21} > V_{22}.
  3. V11≥V21V_{11} \ge V_{21} and V12≥V22V_{12} \ge V_{22}. (correct answer)
  4. The payoffs for Strategy 1 must be equal to the payoffs for Strategy 2.
Explanation: If Strategy 1 is optimal for all possible values of the probability pp, it means its expected monetary value is always greater than or equal to that of Strategy 2. For this to be true for the entire interval [0,1][0, 1], Strategy 1 must be at least as good as Strategy 2 at both endpoints. At p=1p=1, we must have V11≥V21V_{11} \ge V_{21}. At p=0p=0, we must have V12≥V22V_{12} \ge V_{22}. This is the definition of Strategy 1 dominating Strategy 2.

Question 12

A manager is deciding on an investment. The optimal choice depends on the probability of economic growth, pp, and the projected revenue, RR. A sensitivity analysis reveals that the range of optimality for pp is narrow, while the range of optimality for RR is very wide. Which of the following is the most important strategic priority for the manager based on this analysis?

  1. Focus on marketing efforts to increase the projected revenue RR as much as possible.
  2. Re-evaluate the decision using a different model, as the current one is too sensitive.
  3. Find ways to reduce the fixed costs associated with the investment to improve overall profitability.
  4. Devote more resources to obtaining a more reliable and accurate forecast for economic growth. (correct answer)
Explanation: The analysis shows the decision is highly sensitive to the probability of economic growth (pp) but insensitive to the projected revenue (RR). A narrow range for pp means a small error in its estimation could lead to the wrong decision. A wide range for RR means the decision holds even if the revenue forecast is off by a large amount. Therefore, the highest priority is to reduce uncertainty in the most sensitive parameter, which is pp. This is best achieved by getting a more accurate forecast.

Question 13

A financial analyst recommends purchasing a specific stock based on an expected monetary value (EMV) calculation, which assumes a 70% probability (p=0.7p=0.7) of a bull market. The analyst then performs a sensitivity analysis on the variable pp. The primary purpose of this analysis is to:

  1. find the exact future value of the stock by modeling all possible market conditions.
  2. identify the breakeven probability at which the EMV of purchasing the stock is exactly zero.
  3. determine how much the recommendation to purchase the stock depends on the accuracy of the 70% probability estimate. (correct answer)
  4. prove that purchasing the stock is guaranteed to be profitable regardless of the actual market condition.
Explanation: Sensitivity analysis is used to determine how the optimal decision is affected by changes in input variables. Its primary purpose is to assess the robustness of the decision. In this case, it tests how sensitive the 'purchase' recommendation is to the initial probability estimate of p=0.7p=0.7. A wide range of optimality would mean the decision is robust, while a narrow range would mean the decision is highly dependent on the accuracy of the estimate.

Question 14

An agricultural firm uses a model to decide whether to plant Crop A or Crop B. The model's recommendation depends on the probability, pp, of a 'wet' growing season. The current estimate is p=0.4p=0.4, and at this value, planting Crop A is optimal. A sensitivity analysis report contains the following statement: 'The optimal decision is to plant Crop A for all p<0.55p < 0.55 and to plant Crop B for all p>0.55p > 0.55.' Which of the following is the most accurate conclusion?

  1. If the firm's managers believe the probability of a wet season is actually 60%, they should stick with the current plan of planting Crop A.
  2. The expected profit from Crop A is greater than or equal to the expected profit from Crop B for any probability p≤0.55p \le 0.55. (correct answer)
  3. The model is flawed because it does not provide a recommendation for the specific case where the probability p=0.55p = 0.55.
  4. A more accurate weather forecast is unnecessary because the current plan (Crop A) is confirmed to be optimal at the current estimate of p=0.4p=0.4.
Explanation: The statement from the sensitivity analysis defines the ranges of optimality. Saying 'Crop A is optimal for all p<0.55p < 0.55' means that for any probability in that range, the expected monetary value (or profit) of choosing A is higher than that of choosing B. At p=0.55p=0.55, the two options have equal expected profit (it's the indifference point). So, for any p≤0.55p \le 0.55, EMV(A)≥EMV(B)EMV(A) \ge EMV(B).