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AP Statistics

AP Statistics Help: Potential Errors When Performing Tests

Review real example questions for Potential Errors When Performing Tests in AP Statistics.

Question 1

A manufacturer tests whether the mean lifetime of a battery is greater than 8 hours. They test H0:μ=8H_0: \mu=8H0​:μ=8 vs. Ha:μ>8H_a: \mu>8Ha​:μ>8 at α=0.05\alpha=0.05α=0.05. The test result is to reject H0H_0H0​ and claim the mean lifetime exceeds 8 hours. In reality, the true mean lifetime is μ=8.6\mu=8.6μ=8.6 hours. Which type of error was made?

  1. Type I error: concluding μ>8\mu>8μ>8 when in fact μ=8\mu=8μ=8
  2. Type II error: failing to reject H0H_0H0​ when μ>8\mu>8μ>8
  3. Type II error: rejecting H0H_0H0​ when μ=8\mu=8μ=8
  4. No error: the conclusion matches the true state of the population
  5. Type I error: failing to reject H0H_0H0​ when μ>8\mu>8μ>8
Explanation: This question tests understanding of correct decisions in hypothesis testing. The manufacturer rejected H₀ (concluded μ > 8) when the alternative hypothesis is actually true (μ = 8.6 > 8). No error was made because the decision matches reality: the test correctly detected that the mean lifetime exceeds 8 hours. When we reject H₀ and H₀ is false (Ha is true), we've made a correct decision. Type I errors occur when we reject a true H₀, and Type II errors occur when we fail to reject a false H₀ - neither applies here.

Question 2

A hospital tests whether the mean waiting time in the emergency room is less than 30 minutes after adding staff. They test H0:μ=30H_0: \mu=30H0​:μ=30 vs. Ha:μ<30H_a: \mu<30Ha​:μ<30 at α=0.05\alpha=0.05α=0.05. They fail to reject H0H_0H0​ and state there is not convincing evidence the mean waiting time is below 30 minutes. In reality, the true mean waiting time is μ=30\mu=30μ=30 minutes. Which type of error was made?

  1. Type II error: failing to detect that the mean is below 30 when it is
  2. Type I error: concluding the mean is below 30 when it is not
  3. No error: the decision is consistent with the true population mean
  4. Type I error: failing to reject H0H_0H0​ when μ<30\mu<30μ<30
  5. Type II error: concluding the mean is below 30 when it is not
Explanation: This question tests understanding of correct decisions in hypothesis testing. The hospital failed to reject H₀ (concluded no evidence μ < 30) when the null hypothesis is actually true (μ = 30). No error was made because the decision matches reality: the test correctly concluded there's no evidence the mean is below 30 minutes, and indeed it equals 30. When we fail to reject H₀ and H₀ is true, we've made a correct decision. Type I errors occur when we reject a true H₀, and Type II errors occur when we fail to reject a false H₀ - neither applies here.

Question 3

A hospital tests whether a new sterilization procedure reduces the mean number of bacteria colonies on instruments below 12 colonies. They use H0:μ=12H_0: \mu=12H0​:μ=12 versus Ha:μ<12H_a: \mu<12Ha​:μ<12 at α=0.01\alpha=0.01α=0.01. The test is not statistically significant, so they do not switch procedures. In reality, the true population mean under the new procedure is μ=9\mu=9μ=9 colonies. Which type of error was made?

  1. Type II error (failing to reject a false null hypothesis)
  2. No error was made because α=0.01\alpha=0.01α=0.01 is very strict
  3. Type I error (rejecting a true null hypothesis)
  4. Type II error (rejecting a true null hypothesis)
  5. Type I error (failing to reject a false null hypothesis)
Explanation: This question demonstrates a Type II error in hypothesis testing. The hospital failed to reject the null hypothesis (μ = 12 colonies) when in reality the alternative hypothesis was true (μ = 9 colonies < 12 colonies). This is a Type II error - failing to reject a false null hypothesis. The new procedure actually was better (fewer bacteria colonies), but the test failed to detect this improvement. The very strict significance level (α = 0.01) makes Type II errors more likely because we require overwhelming evidence to reject H₀. This example shows how being too conservative (small α) can lead to missing real improvements, which is particularly concerning in medical contexts where better procedures could improve patient safety.

Question 4

A streaming company tests whether a new recommendation algorithm increases the mean number of minutes watched per user above the current mean of 50 minutes. They test H0:μ=50H_0: \mu=50H0​:μ=50 versus Ha:μ>50H_a: \mu>50Ha​:μ>50 at α=0.05\alpha=0.05α=0.05. The result is statistically significant, so they conclude the algorithm increases watch time. In reality, the true population mean with the new algorithm is μ=50\mu=50μ=50 minutes (no change). Which type of error was made?

  1. Type II error (failing to reject a false null hypothesis)
  2. Type I error (rejecting a true null hypothesis)
  3. Type I error (failing to reject a false null hypothesis)
  4. No error was made because the company used α=0.05\alpha=0.05α=0.05
  5. Type II error (rejecting a true null hypothesis)
Explanation: This question illustrates a Type I error in hypothesis testing. The streaming company rejected the null hypothesis (concluded the algorithm increases watch time) when the null hypothesis was actually true (μ = 50 minutes, no change). This is a Type I error - rejecting a true null hypothesis. The statistically significant result was a false positive, leading to an incorrect business decision. Type I errors occur with probability α = 0.05, meaning this outcome will happen about 5% of the time even when following proper procedures. Understanding that statistical significance doesn't guarantee practical truth helps students appreciate why we control but cannot eliminate Type I error risk.

Question 5

A school district tests whether a new tutoring program changes the mean math score compared with the current mean of 75. They perform a two-sided test with H0:μ=75H_0: \mu=75H0​:μ=75 versus Ha:μ≠75H_a: \mu\neq 75Ha​:μ=75 at α=0.10\alpha=0.10α=0.10. The test is not statistically significant, so they do not adopt the program, concluding there is not enough evidence of a change in mean score. In reality, the true population mean score with the tutoring program is μ=75\mu=75μ=75 exactly. Which type of error was made?

  1. Type I error (rejecting a true null hypothesis)
  2. Type II error (failing to reject a false null hypothesis)
  3. No error was made
  4. Type I error (failing to reject a false null hypothesis)
  5. Type II error (rejecting a true null hypothesis)
Explanation: This question demonstrates a correct decision in hypothesis testing. The school district failed to reject the null hypothesis (μ = 75), and in reality, the null hypothesis was true (μ = 75). This represents a correct decision - no error was made. When we fail to reject a true null hypothesis, we've made the right choice. This scenario helps students understand that not all hypothesis test outcomes involve errors. The two-sided alternative (μ ≠ 75) and higher significance level (α = 0.10) don't change the fact that the correct decision was made. Recognizing correct decisions is as important as identifying errors in understanding the complete framework of hypothesis testing outcomes.

Question 6

A quality-control engineer tests whether the proportion of defective parts produced by a machine is greater than 0.02. They test H0:p=0.02H_0: p=0.02H0​:p=0.02 versus Ha:p>0.02H_a: p>0.02Ha​:p>0.02 at α=0.05\alpha=0.05α=0.05. The test is statistically significant, so they conclude the defect rate is greater than 0.02 and shut the machine down for maintenance. In reality, the true population defect proportion is p=0.02p=0.02p=0.02. Which type of error was made?

  1. Type II error (failing to reject a false null hypothesis)
  2. Type II error (rejecting a true null hypothesis)
  3. No error was made because maintenance is a safe choice
  4. Type I error (rejecting a true null hypothesis)
  5. Type I error (failing to reject a false null hypothesis)
Explanation: This question illustrates a Type I error in quality control. The engineer rejected the null hypothesis (concluded p > 0.02) when the null hypothesis was actually true (p = 0.02). This is a Type I error - rejecting a true null hypothesis. The statistically significant result led to unnecessary machine maintenance when the defect rate was actually at the acceptable level. In quality control, Type I errors can lead to unnecessary downtime and costs. The irony is that while trying to maintain quality, the false alarm disrupted production unnecessarily. This example helps students understand that Type I errors have real-world consequences and why controlling the significance level α is important in balancing error risks.

Question 7

A city tests whether the proportion of commuters who use public transit is greater than 0.25 after a fare reduction. They test H0:p=0.25H_0: p=0.25H0​:p=0.25 vs. Ha:p>0.25H_a: p>0.25Ha​:p>0.25 at α=0.05\alpha=0.05α=0.05. The city fails to reject H0H_0H0​ and states there is not convincing evidence the fare reduction increased transit use. In reality, the true proportion is p=0.30p=0.30p=0.30. Which type of error was made?

  1. Type I error: concluding transit use increased when it did not
  2. Type II error: failing to detect an increase that is real
  3. No error: failing to reject H0H_0H0​ means p=0.25p=0.25p=0.25
  4. Type I error: failing to reject H0H_0H0​ when p>0.25p>0.25p>0.25
  5. Type II error: concluding transit use increased when it did not
Explanation: This question tests understanding of Type II errors in one-sided tests. The city failed to reject H₀ (concluded no evidence of increased transit use) when the alternative hypothesis is actually true (p = 0.30 > 0.25, transit use DID increase). A Type II error occurs when we fail to reject a false null hypothesis - we miss detecting a real effect. This matches the scenario: the fare reduction actually increased transit use to 30%, but the test failed to detect this increase. Type I errors involve rejecting a true H₀, which cannot occur when we fail to reject.

Question 8

A pharmaceutical company tests whether a generic drug is less effective than the brand-name drug by comparing mean symptom-reduction scores. They set up H0:μG−μB=0H_0: \mu_G-\mu_B=0H0​:μG​−μB​=0 vs. Ha:μG−μB<0H_a: \mu_G-\mu_B<0Ha​:μG​−μB​<0 at α=0.05\alpha=0.05α=0.05. The analysis leads them to reject H0H_0H0​ and claim the generic is less effective. In reality, the generic and brand are equally effective (the true difference is μG−μB=0\mu_G-\mu_B=0μG​−μB​=0). Which type of error was made?

  1. Type II error: failing to detect that the generic is less effective
  2. Type I error: concluding the generic is less effective when it is not
  3. No error: rejecting H0H_0H0​ guarantees the generic is less effective
  4. Type II error: concluding the generic is less effective when it is not
  5. Type I error: failing to reject H0H_0H0​ when the generic is less effective
Explanation: This question tests understanding of Type I errors with difference tests. The company rejected H₀ (concluded the generic is less effective) when the null hypothesis is actually true (μG - μB = 0, drugs are equally effective). A Type I error occurs when we reject a true null hypothesis - we falsely detect a difference that doesn't exist. This is exactly the situation: the drugs are equally effective, but the test incorrectly concluded the generic is inferior. Type II errors involve failing to reject a false H₀, which doesn't apply when H₀ is rejected.

Question 9

A tech company tests whether a new interface reduces average customer support time. They use H0:μ=12H_0: \mu=12H0​:μ=12 minutes vs. Ha:μ<12H_a: \mu<12Ha​:μ<12 minutes at α=0.05\alpha=0.05α=0.05. They reject H0H_0H0​ and announce the interface reduced mean support time. In reality, the true mean support time with the new interface is still μ=12\mu=12μ=12 minutes. Which type of error was made?

  1. Type II error: failing to detect a real reduction in support time
  2. Type I error: concluding there is a reduction when there is none
  3. Type II error: concluding there is a reduction when there is none
  4. No error: rejecting H0H_0H0​ proves the mean is less than 12
  5. Type I error: failing to reject H0H_0H0​ when the mean is less than 12
Explanation: This question tests understanding of Type I errors in one-sided tests. The company rejected H₀ (concluded support time was reduced) when the null hypothesis is actually true (μ = 12, no reduction). A Type I error occurs when we reject a true null hypothesis - we falsely claim an improvement that doesn't exist. This is precisely what happened: the new interface doesn't actually reduce support time, but the test incorrectly concluded it does. Type II errors involve failing to reject a false H₀, which doesn't apply when H₀ is rejected.

Question 10

A quality-control engineer tests whether the mean fill amount of a cereal box is less than the labeled 500 g. The hypotheses are H0:μ=500H_0: \mu=500H0​:μ=500 vs. Ha:μ<500H_a: \mu<500Ha​:μ<500 at α=0.01\alpha=0.01α=0.01. The test result is to fail to reject H0H_0H0​, so the engineer reports there is not convincing evidence of underfilling. In reality, the true mean fill is μ=495\mu=495μ=495 g (the boxes are underfilled). Which type of error was made?

  1. Type I error: concluding the boxes are underfilled when they are not
  2. No error was made because failing to reject H0H_0H0​ proves μ=500\mu=500μ=500
  3. Type II error: failing to detect underfilling when it is real
  4. Type I error: failing to reject H0H_0H0​ when μ<500\mu<500μ<500
  5. Type II error: concluding the boxes are underfilled when they are not
Explanation: This question tests understanding of Type II errors in hypothesis testing. The engineer failed to reject H₀ (concluded no evidence of underfilling) when the alternative hypothesis is actually true (μ = 495 < 500, boxes ARE underfilled). A Type II error occurs when we fail to reject a false null hypothesis - we miss detecting a real effect. This matches the scenario perfectly: the boxes are actually underfilled but the test failed to detect this problem. Type I errors involve rejecting a true null hypothesis, which cannot happen when we fail to reject H₀.