Business Statistics Quiz: Type I Ii Errors And Tradeoffs
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Type I Ii Errors And TradeoffsQuestion 1 of 20

To decrease the probabilities of both Type I and Type II errors simultaneously in a hypothesis test, what single action must a business take?

Increase the significance level (α) of the test.
Decrease the significance level (α) of the test.
Increase the sample size (n) for the test.
This goal is statistically impossible to achieve.
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Business Statistics Quiz

Business Statistics Quiz: Type I Ii Errors And Tradeoffs

Practice Type I Ii Errors And Tradeoffs in Business Statistics 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 Type I Ii Errors And Tradeoffs, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Statistics.

How to use this quiz

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.

All questions

Question 1

To decrease the probabilities of both Type I and Type II errors simultaneously in a hypothesis test, what single action must a business take?

  1. Increase the significance level (α) of the test.
  2. Decrease the significance level (α) of the test.
  3. Increase the sample size (n) for the test. (correct answer)
  4. This goal is statistically impossible to achieve.
Explanation: There is an inherent tradeoff between Type I (α) and Type II (β) errors for a fixed sample size; decreasing one increases the other. The only way to decrease both error probabilities simultaneously is to collect more information, which means increasing the sample size. A larger sample provides a more accurate estimate of the population parameter, making the test more precise and powerful.

Question 2

A quality control test is performed on a manufacturing line. H₀: The product meets specifications. H₁: The product does not meet specifications. The test is conducted with α = 0.05, and the calculated p-value is 0.04. Which statement accurately describes the conclusion and the associated risk?

  1. The null hypothesis is rejected. There is a 4% chance that this conclusion is wrong.
  2. The null hypothesis is rejected. There is a 5% risk of having committed a Type I error in making this decision. (correct answer)
  3. The null hypothesis is not rejected. There is a 95% chance that the product meets specifications.
  4. The null hypothesis is not rejected, meaning there is a high probability of a Type II error.
Explanation: Since the p-value (0.04) is less than the significance level (α = 0.05), the correct decision is to reject the null hypothesis. The significance level, α, represents the pre-determined probability of committing a Type I error (rejecting a true null hypothesis) that the researcher is willing to accept. Therefore, the decision carries a 5% risk of being a false positive. The p-value is not the probability that the conclusion is wrong.

Question 3

A bank is developing a fraud detection algorithm. The system flags transactions as either 'fraudulent' (reject H₀: transaction is legitimate) or 'legitimate' (fail to reject H₀). Historical data shows that 2% of all transactions are actually fraudulent. The current algorithm has a 5% false positive rate (Type I error) and catches 85% of actual fraud cases.

If the bank processes 100,000 transactions monthly, what is the approximate ratio of false positives to true positives generated by this algorithm?

  1. Approximately 1.2:1 (more false positives than true positives)
  2. Approximately 2.9:1 (nearly three times more false positives) (correct answer)
  3. Approximately 0.35:1 (more true positives than false positives)
  4. Approximately 5.8:1 (nearly six times more false positives)
Explanation: With 100,000 transactions: 2% are fraudulent (2,000), 98% legitimate (98,000). True positives = 85% of 2,000 = 1,700. False positives = 5% of 98,000 = 4,900. Ratio = 4,900:1,700 ≈ 2.9:1. Choice A miscalculates the base rates or percentages. Choice C inverts the ratio. Choice D appears to double-count or misapply the error rates.

Question 4

A hospital is testing a new diagnostic procedure (H₀: new procedure is no better than current vs. H₁: new procedure is superior). The current procedure correctly identifies 70% of cases, while missing 30% (costing $15,000 per missed case in delayed treatment). The new procedure costs $500 more per test. If Type I error leads to adopting an ineffective expensive procedure and Type II error means staying with a less accurate procedure, which analysis correctly evaluates the error tradeoffs for 1,000 annual cases?

  1. Type I error costs $500,000 annually; Type II error costs depend on how much the new procedure actually improves accuracy beyond 70% (correct answer)
  2. Type I error costs $500,000 annually; Type II error costs $4,500,000 annually in current missed diagnoses, so minimize Type II error
  3. Type II error costs $500,000 annually in excess testing fees; Type I error costs $4,500,000 in continued missed diagnoses
  4. Both errors cost $500,000 annually since that's the difference in testing costs, so use α = 0.05 to balance them equally
Explanation: Type I error = adopting ineffective new procedure = $500 extra × 1,000 cases = $500,000 annually. Type II error = staying with current procedure when new is actually better = depends on the actual improvement the new procedure provides beyond the current 70% accuracy. We can't calculate Type II error cost without knowing the new procedure's true accuracy. Choice B incorrectly uses current missed diagnoses as Type II cost. Choice C reverses the error definitions. Choice D incorrectly equates both error costs.

Question 5

A quality control manager is testing whether the mean weight of cereal boxes is 500g (H₀: μ = 500g vs. H₁: μ ≠ 500g). She's considering two testing strategies: Strategy A uses α = 0.01 with n = 50, while Strategy B uses α = 0.05 with n = 50. If the true population mean is actually 495g with σ = 15g, which statement correctly compares the power of these two strategies?

  1. Strategy A has higher power because the smaller α value increases the probability of correctly rejecting a false null hypothesis
  2. Strategy B has higher power because the larger α value decreases the probability of Type II error when the null is actually false (correct answer)
  3. Both strategies have equal power because they use the same sample size and the power depends only on n and the effect size
  4. Strategy A has higher power because the smaller α value creates a wider acceptance region, making rejection more likely when H₀ is false
Explanation: Power = 1 - β, where β is the probability of Type II error. When α increases (0.01 to 0.05), the critical region becomes larger, making it easier to reject H₀ when it's actually false, thus increasing power and decreasing β. Choice A incorrectly states that smaller α increases power. Choice C is wrong because power depends on α as well as n and effect size. Choice D incorrectly describes the relationship between α and the acceptance region.

Question 6

An airline is testing a new boarding procedure. H₀: The new procedure has no effect on average boarding time. H₁: The new procedure reduces average boarding time. The airline decides to use a one-sided (left-tailed) test. What is the primary business tradeoff associated with this choice compared to using a two-sided test?

  1. It doubles the probability of a Type I error while keeping the probability of a Type II error constant.
  2. It eliminates the risk of a Type II error because the test is focused only on improvement.
  3. It reduces the required sample size to achieve a given power, but it is more susceptible to violations of statistical assumptions.
  4. It has greater power to detect a reduction in boarding time, but it cannot detect if the new procedure significantly increases boarding time. (correct answer)
Explanation: When you encounter questions about one-sided versus two-sided hypothesis tests, focus on the fundamental tradeoff: what you gain in power versus what you lose in detection capability. The correct answer is D because this captures the core business tradeoff. A one-sided (left-tailed) test concentrates all the alpha level (say, 5%) in one tail of the distribution, giving you more power to detect decreases in boarding time compared to a two-sided test that splits alpha between both tails (2.5% each). However, this focus comes at a cost—if the new procedure actually increases boarding time significantly, the one-sided test cannot detect this as statistically significant, even if the increase is substantial. Let's examine why the other options miss the mark: A is incorrect because a one-sided test doesn't change the Type I error rate—it's still set at your chosen alpha level (typically 5%). The test simply allocates this probability differently. B is wrong because Type II errors (failing to detect a true effect) can still occur in one-sided tests. The test being "focused" doesn't eliminate the risk of missing a real reduction in boarding time. C incorrectly links sample size and assumption violations. While one-sided tests do have higher power (potentially reducing required sample size), they aren't inherently more susceptible to assumption violations. Study tip: Remember that one-sided tests are a "focused bet"—you gain power in the direction you're testing but become blind to significant effects in the opposite direction. Always consider whether missing effects in the untested direction poses business risks.

Question 7

A company is considering two testing protocols to evaluate a cost-saving measure. Both protocols use the same sample size. Protocol A uses α = 0.05. Protocol B uses α = 0.01. Which of the following statements is necessarily true?

  1. Protocol B is less likely to result in a Type I error and more likely to result in a Type II error than Protocol A. (correct answer)
  2. Protocol B is more powerful than Protocol A.
  3. Protocol A is less likely to result in a Type I error and more likely to result in a Type II error than Protocol B.
  4. Protocol A has a lower combined probability of error (α + β) than Protocol B.
Explanation: The significance level α is the probability of a Type I error. Since 0.01 < 0.05, Protocol B has a lower chance of a Type I error. For a fixed sample size, there is an inverse relationship between α and β (the probability of a Type II error). Decreasing α increases β. Therefore, Protocol B is more likely to result in a Type II error than Protocol A.

Question 8

A pharmaceutical company is testing a new drug for reducing cholesterol levels. The null hypothesis states that the drug has no effect (mean reduction = 0 mg/dL), while the alternative hypothesis states that the drug reduces cholesterol by more than 0 mg/dL. The company's regulatory team has determined that falsely concluding the drug is effective when it isn't would cost the company $50 million in future lawsuits and recalls. However, failing to detect a truly effective drug would result in $200 million in lost market opportunity to competitors.

Based on the cost structure described, what should be the company's primary concern regarding statistical errors, and what does this suggest about their choice of significance level?

  1. Type I error is more costly; they should use a smaller significance level like α = 0.01 to reduce the probability of false approval
  2. Type II error is more costly; they should use a larger significance level like α = 0.10 to reduce the probability of missing an effective drug (correct answer)
  3. Type I error is more costly; they should use a larger significance level like α = 0.10 to reduce the probability of false rejection
  4. Type II error is more costly; they should use a smaller significance level like α = 0.01 to reduce the probability of missing an effective drug
Explanation: Type II error (failing to detect a truly effective drug) costs $200 million vs. Type I error (false approval) costing $50 million. Since Type II error is more costly, they should reduce β by increasing α (using α = 0.10), as there's an inverse relationship between Type I and Type II error probabilities. Choice A incorrectly identifies Type I as more costly. Choice C confuses the direction of the significance level adjustment. Choice D incorrectly suggests decreasing α would reduce Type II error probability.

Question 9

A marketing director is testing whether a new advertising campaign increases sales (H₀: μ ≤ $10,000 vs. H₁: μ > $10,000). She's concerned about two scenarios: launching an ineffective campaign (costing $500,000 in wasted spending) versus missing the opportunity to launch an effective campaign (costing $2,000,000 in lost revenue). If she sets α = 0.05, what is the most accurate characterization of her error priorities?

  1. She's prioritizing avoiding Type I error since the 5% significance level minimizes the cost of launching ineffective campaigns
  2. She's balancing both errors equally since α = 0.05 is a standard compromise between Type I and Type II error rates
  3. She's inadequately protecting against Type II error since missing effective campaigns is four times more costly than launching ineffective ones (correct answer)
  4. She's appropriately minimizing total expected costs since the 5% level optimally weights the relative probabilities of both error types
Explanation: Type II error (2Mcost)is4timesmoreexpensivethanTypeIerror(2M cost) is 4 times more expensive than Type I error (0.5M cost), suggesting she should use a higher α to reduce β. Using α = 0.05 doesn't reflect this cost imbalance. Choice A misidentifies which error α = 0.05 prioritizes. Choice B incorrectly suggests α = 0.05 equally balances errors regardless of costs. Choice D incorrectly implies α = 0.05 is optimal without considering the specific cost structure.

Question 10

A startup with limited funds tests a new feature. H₀: The feature has no effect on user retention. The p-value is 0.15. The standard α is 0.05. The CEO decides to roll out the feature anyway, arguing, "As a small company, we can't afford to miss any opportunities." This decision implies the CEO is implicitly acting as if...

  1. ...the cost of a Type I error (rolling out a useless feature) is far greater than the cost of a Type II error.
  2. ...the statistical test was flawed and its results should be completely disregarded.
  3. ...the p-value of 0.15 provides strong evidence in favor of the feature's effectiveness.
  4. ...the cost of a Type II error (missing a useful feature) is far greater than the cost of a Type I error. (correct answer)
Explanation: When you encounter hypothesis testing questions involving business decisions, focus on how the decision-maker is weighing Type I versus Type II errors based on their actions, not their words. In this scenario, the statistical evidence doesn't support rejecting H₀ (p = 0.15 > α = 0.05), meaning there's insufficient evidence the feature improves retention. However, the CEO chooses to implement it anyway. This action reveals their implicit cost-benefit calculation: they'd rather risk implementing a useless feature (Type I error) than miss out on a potentially beneficial one (Type II error). Answer D correctly identifies this logic. The CEO's statement about "can't afford to miss opportunities" directly indicates they view the cost of a Type II error as greater than a Type I error. Answer A reverses this relationship—it suggests the CEO fears implementing useless features more than missing good ones, which contradicts their decision to proceed despite weak evidence. Answer B is incorrect because the CEO isn't questioning the test's validity; they're making a business decision despite the statistical results. Answer C misinterprets the p-value—0.15 provides weak, not strong, evidence for effectiveness. Remember: when analyzing hypothesis testing decisions in business contexts, ignore what decision-makers say and focus on what they do. Their actions reveal their true assessment of error costs. If someone acts against statistical evidence, they're implicitly stating that the cost of the error they're risking (Type I) is less than the cost of the error they're avoiding (Type II).

Question 11

A chemical plant must ensure the concentration of a certain compound in its wastewater is below a regulatory limit. H₀: The mean concentration is at or below the limit. H₁: The mean concentration is above the limit. A Type I error would lead to an expensive and unnecessary shutdown for recalibration. A Type II error would lead to violating regulations and incurring massive fines. If the fine for a violation is 100 times more costly than an unnecessary shutdown, what is the most logical approach to setting the significance level, α?

  1. Set α to be extremely small (e.g., 0.001) to avoid the cost of an unnecessary shutdown.
  2. Increase the sample size so that α can be set to 0, eliminating Type I errors completely.
  3. Set α = 0.05 because it correctly balances the probabilities of the two types of errors.
  4. Set α to be relatively large (e.g., 0.10) to increase the test's power to detect a violation. (correct answer)
Explanation: When faced with hypothesis testing in high-stakes business decisions, you need to consider the relative costs of Type I and Type II errors, not just their statistical probabilities. A Type I error occurs when you reject a true null hypothesis (concluding concentration exceeds the limit when it doesn't), while a Type II error occurs when you fail to reject a false null hypothesis (missing an actual violation). The key insight is that these errors have dramatically different costs. The Type II error (missing a violation) costs 100 times more than the Type I error (unnecessary shutdown). When consequences are this imbalanced, you should adjust your significance level α\alpha to minimize the more expensive error. Setting α\alpha relatively large (like 0.10) makes your test more sensitive to detecting violations, reducing the probability of costly Type II errors. This increases the test's power - its ability to correctly identify when the concentration actually exceeds limits. Choice A is backwards - making α\alpha extremely small would reduce Type I errors but dramatically increase the risk of the much more expensive Type II errors. Choice B is impossible; you cannot eliminate Type I errors completely while maintaining any ability to detect violations. Choice C misses the point entirely - the standard 0.05 level ignores the severe cost imbalance and doesn't optimize decision-making. Study tip: In business statistics, always consider the economic consequences of different error types. The "optimal" significance level depends on relative costs, not statistical convention. When one error type is vastly more expensive, adjust α\alpha to minimize that risk.

Question 12

A corporation is testing a new cybersecurity system. H₀: The system does not reduce the number of security breaches. H₁: The system does reduce the number of breaches. An initial, small-scale study fails to find a significant result. A larger, more expensive follow-up study is proposed. A manager argues against the new study, stating, "If the effect was real, we would have seen it the first time." Why might this manager's argument be incorrect?

  1. The first study might have made a Type I error, incorrectly suggesting there was no effect.
  2. The manager is correct; a failure to reject the null hypothesis is definitive proof of no effect.
  3. Statistical significance is irrelevant for cybersecurity; all proposed improvements should be implemented.
  4. The first study may have had low power due to its small sample size, making it likely to commit a Type II error. (correct answer)
Explanation: When you encounter questions about failed hypothesis tests and proposed follow-up studies, you need to understand the difference between statistical power and Type I/II errors, and why "failing to find significance" doesn't mean "no effect exists." The manager's argument reflects a common misconception about statistical power. A small-scale study has limited ability to detect real effects because it lacks sufficient sample size to achieve adequate statistical power. Power is the probability of correctly rejecting a false null hypothesis - in other words, finding an effect when one truly exists. With low power, even a genuinely effective cybersecurity system might not show statistically significant results, leading to a Type II error (failing to reject H0H_0 when H1H_1 is actually true). A larger study would increase power by providing more data points, making it easier to detect the system's true effectiveness if it exists. Looking at the wrong answers: A incorrectly describes Type I error, which involves rejecting a true null hypothesis - but the first study failed to reject H0H_0, so Type I error isn't relevant here. B represents the exact misconception the question is testing - failing to reject H0H_0 is not proof of no effect, just insufficient evidence. C makes an illogical claim that statistical significance doesn't matter for cybersecurity decisions, which would be poor business practice. Study tip: Remember that "no significant result" ≠ "no effect exists." When you see questions about failed studies followed by larger studies, think about statistical power and sample size. Small studies often lack power to detect real but modest effects.

Question 13

A marketing team A/B tests a new website layout. H₀: The conversion rate of the new layout is the same as the old one. H₁: The conversion rate of the new layout is higher. An initial test with 1,000 users fails to reject H₀ at α = 0.05, yielding a p-value of 0.08. The team suspects a small positive effect exists. To have the best chance of correctly detecting a true, small improvement in a subsequent test, what is the most appropriate action?

  1. Decrease the significance level to α = 0.01 to make the test more rigorous.
  2. Increase the sample size substantially while keeping α = 0.05. (correct answer)
  3. Conclude the new layout is not better and discard the design to save resources.
  4. Keep the sample size the same but increase the significance level to α = 0.10.
Explanation: The power of a test (the probability of correctly rejecting a false null hypothesis, 1-β) is the ability to detect a true effect. To detect a small effect, more power is needed. The most effective way to increase power while maintaining the same Type I error rate (α) is to increase the sample size. Decreasing α would decrease power. Increasing α would increase power but also increase the risk of a false positive. Concluding failure might be a Type II error.

Question 14

A bank's fraud detection system analyzes transactions to identify potential fraud. The null hypothesis for any given transaction is H₀: The transaction is legitimate. The alternative hypothesis is H₁: The transaction is fraudulent. The bank is extremely concerned with missing actual cases of fraud, as this leads to direct financial losses and erodes customer trust.

Given the bank's priority as described in the passage, how should the system's hypothesis testing parameters be configured?

  1. The system should be set to a very low α to minimize the number of legitimate transactions that are flagged for review.
  2. The system should be configured to minimize β, the probability of a Type II error, even if it leads to a high number of false alarms. (correct answer)
  3. The system should prioritize a low p-value threshold for action, ensuring that only transactions with strong evidence of fraud are flagged.
  4. The system should aim to make α and β as close to equal as possible to balance the two types of errors.
Explanation: A Type II error (β) is failing to reject a false null hypothesis. In this context, it means failing to identify a fraudulent transaction when it is, in fact, fraudulent. Since the bank's priority is to avoid missing fraud, it must minimize β. This means maximizing the test's power. The consequence is that α, the probability of a Type I error (flagging a legitimate transaction), will likely increase, leading to more false alarms.

Question 15

A company tests a new employee wellness program to see if it reduces the average number of sick days taken. After a six-month trial, a hypothesis test (H₀: no reduction in sick days) results in a failure to reject the null hypothesis. A manager declares, "The test proves the wellness program has no effect on sick days." What is the primary flaw in the manager's statement?

  1. The manager is failing to consider the possibility of a Type I error, where a true effect was missed.
  2. The manager is committing the fallacy of 'accepting the null'; the test only failed to provide sufficient evidence of an effect, which is not the same as proving no effect exists. (correct answer)
  3. The manager should have used a lower significance level to increase the likelihood of finding a significant result.
  4. The manager's conclusion is correct, as failing to reject the null hypothesis is statistically equivalent to confirming its truth.
Explanation: A core principle of hypothesis testing is that failing to reject the null hypothesis (H₀) does not prove H₀ is true. It simply means the sample data did not provide strong enough evidence to conclude H₀ is false. There might be a real effect that the test was not powerful enough to detect (a Type II error). The manager's statement of proof is an overstatement of the statistical conclusion.

Question 16

A research team is designing an experiment. They find that to achieve their desired power of 90% (β = 0.10) with a significance level of α = 0.05 to detect a small effect size, they need a sample size of n=1000. If they are willing to accept a lower power of 80% (β = 0.20) but want to keep α = 0.05 and detect the same small effect size, how would the required sample size change?

  1. The required sample size will decrease. (correct answer)
  2. The required sample size will increase.
  3. The required sample size will remain the same because α is constant.
  4. The change cannot be determined without knowing the population standard deviation.
Explanation: When you encounter statistical power questions, remember that power analysis involves four interconnected elements: sample size (n), effect size, significance level (α), and statistical power (1-β). Changing any one of these affects the others. Statistical power represents your ability to correctly detect a true effect when it exists. Higher power means you're less likely to miss a real effect (lower Type II error rate). However, achieving higher power comes at a cost - you need more data to be more confident in detecting effects. In this scenario, you're reducing the power requirement from 90% to 80% while keeping α = 0.05 and the same effect size constant. With lower power, you're accepting a higher risk of missing the true effect, but this trade-off allows you to use fewer participants. The mathematical relationship shows that as power decreases, the required sample size decreases proportionally. Looking at the wrong answers: Choice B) suggests sample size increases, but this reverses the relationship - you'd need more participants for higher power, not lower. Choice C) incorrectly assumes that keeping α constant means sample size stays the same, ignoring power's independent effect on sample size calculations. Choice D) claims we need the population standard deviation, but power analysis formulas incorporate effect size, which already accounts for the relationship between mean differences and variability. Study tip: Remember the power analysis trade-offs: higher power requires larger samples, while accepting lower power allows smaller samples. This inverse relationship is fundamental to experimental design decisions.

Question 17

In hypothesis testing, how does an increase in the true difference between the null hypothesis parameter and the true population parameter (the effect size) affect the probabilities of Type I and Type II errors, assuming sample size and α are held constant?

  1. It decreases the probability of a Type II error (β) but has no effect on the probability of a Type I error (α). (correct answer)
  2. It increases the probability of a Type I error (α) but has no effect on the probability of a Type II error (β).
  3. It decreases the probabilities of both Type I and Type II errors.
  4. It has no effect on the probabilities of either Type I or Type II errors.
Explanation: When you encounter questions about Type I and Type II errors in hypothesis testing, focus on what each error represents and what factors influence them. Type I error (α) is rejecting a true null hypothesis, while Type II error (β) is failing to reject a false null hypothesis. The key insight here is that α is set by the researcher before conducting the test and remains fixed regardless of the true population parameter. Type I error depends only on your chosen significance level, not on reality. However, Type II error depends heavily on how far the true parameter is from the null hypothesis value—this distance is called the effect size. When the effect size increases (the true parameter moves further from the null hypothesis value), it becomes much easier for your test to detect this difference and correctly reject the false null hypothesis. This dramatically reduces the probability of Type II error (β), while α remains unchanged at your predetermined level. Answer A correctly identifies this relationship: larger effect size decreases β but doesn't affect α. Answer B reverses this relationship incorrectly—α is fixed by your significance level choice, not influenced by the true parameter. Answer C wrongly suggests both error probabilities decrease, but α is unaffected by effect size. Answer D incorrectly claims no effects occur, missing the crucial relationship between effect size and statistical power. Remember this pattern: α is what you control as the researcher, while β depends on the reality of how different the truth is from your null hypothesis. Larger effects are easier to detect.

Question 18

An analyst conducts 20 separate A/B tests on 20 different proposed changes to a company website, using α = 0.10 for each test. In reality, none of the 20 changes have any actual effect on user engagement (i.e., all 20 null hypotheses are true). What is the probability that the analyst will report at least one statistically significant result (i.e., commit at least one Type I error)?

  1. Exactly 0.10
  2. Approximately 0.88 (correct answer)
  3. Approximately 0.12
  4. Exactly 2.00, which is impossible.
Explanation: For a single test where H₀ is true, the probability of a Type I error is α = 0.10. The probability of NOT committing a Type I error is 1 - α = 0.90. For 20 independent tests, the probability of committing NO Type I errors is (0.90)^20 ≈ 0.1216. The probability of committing at least one Type I error is 1 minus the probability of committing none: 1 - 0.1216 ≈ 0.8784, or about 88%.

Question 19

A pharmaceutical company is testing a new drug against a placebo. The null hypothesis (H₀) is that the drug has no effect. The financial cost of launching an ineffective drug (a Type I error) is extremely high, involving manufacturing scale-up and marketing campaigns. The opportunity cost of failing to launch a truly effective drug (a Type II error) is considered significant but substantially lower than the cost of a Type I error. How should the researchers set their significance level (α) and what is the primary tradeoff?

  1. Set a very low α (e.g., 0.01) to minimize the chance of a false positive, which accepts a higher probability of failing to detect a truly effective drug. (correct answer)
  2. Set a high α (e.g., 0.10) to increase the test's power, which minimizes the chance of missing a truly effective drug at the cost of a higher false positive rate.
  3. Set α = 0.05 as is standard, and increase the sample size to reduce the Type II error rate without altering the Type I error rate.
  4. Set a very low α (e.g., 0.01) because this simultaneously decreases the probabilities of both Type I and Type II errors.
Explanation: The significance level, α, is the probability of a Type I error. Since the cost of a Type I error (launching an ineffective drug) is extremely high, the company should minimize this risk by setting a very low α. The tradeoff is that decreasing α for a fixed sample size will increase β, the probability of a Type II error (failing to detect a real effect).

Question 20

Consider a hypothesis test for the mean of a population where H₀: μ = 100. The test uses a fixed sample size and α = 0.05. Which of the following true values of the population mean (μ) would give the test the highest power?

  1. μ = 101
  2. μ = 99
  3. μ = 105 (correct answer)
  4. The power is the same for all alternatives because α and n are fixed.
Explanation: Power is the probability of correctly rejecting a false null hypothesis. This probability increases as the true parameter value moves further away from the value specified in the null hypothesis (i.e., as the effect size increases). The value μ = 105 is furthest from the null value of μ = 100. This larger difference is easier to detect, resulting in higher power compared to the smaller differences represented by μ=101 and μ=99.