Certified Six Sigma Black Belt Exam Hypothesis Testing Applications 2 — Questions and Answers
Question 1: A Six Sigma team runs a two-sample t-test and obtains a p-value of 0.03 with α = 0.05. What is the correct conclusion?
- Fail to reject H₀ because p > α/2
- Reject H₀ because p < α (Correct answer)
- Accept H₀ because the difference is small
- Reject H₁ because the sample size is insufficient
Correct answer: Reject H₀ because p < α
When p < α, there is sufficient evidence to reject the null hypothesis.
Question 2: Which assumption must be verified before performing a paired t-test?
- Equal population variances
- The differences between pairs are approximately normally distributed (Correct answer)
- Both populations have the same mean
- Sample sizes must exceed 30
Correct answer: The differences between pairs are approximately normally distributed
A paired t-test requires that the within-pair differences are approximately normally distributed.
Question 3: An engineer wants to test whether a new process reduces defect rates compared to the old process. Which hypothesis setup is correct?
- H₀: μnew ≥ μold, H₁: μnew < μold (Correct answer)
- H₀: μnew = μold, H₁: μnew ≠ μold
- H₀: μnew ≤ μold, H₁: μnew > μold
- H₀: μnew < μold, H₁: μnew ≥ μold
Correct answer: H₀: μnew ≥ μold, H₁: μnew < μold
A one-tailed (lower-tail) test is appropriate when the goal is to show the new process has fewer defects.
Question 4: What is the consequence of increasing sample size on hypothesis test power, assuming all else remains constant?
- Power decreases as the test becomes more conservative
- Power increases because the standard error decreases (Correct answer)
- Power is unaffected by sample size
- Power increases only if α is also increased
Correct answer: Power increases because the standard error decreases
Larger samples reduce standard error, making it easier to detect true effects and increasing statistical power.
Question 5: When applying a chi-square goodness-of-fit test, which condition must be satisfied for valid results?
- All expected cell frequencies must be at least 5 (Correct answer)
- All observed frequencies must be equal
- The data must follow a normal distribution
- Degrees of freedom must equal the number of categories
Correct answer: All expected cell frequencies must be at least 5
The chi-square approximation is unreliable when expected cell counts fall below 5; cells may need to be combined.
Question 6: A Black Belt uses ANOVA to compare the means of four process lines. If the F-statistic is significant, what does this tell you?
- All four means are different from each other
- At least one pair of means is significantly different (Correct answer)
- The variances across lines are unequal
- The grand mean equals the overall target
Correct answer: At least one pair of means is significantly different
A significant ANOVA F-test indicates at least one group mean differs, but post-hoc tests are needed to identify which pairs.
Question 7: In hypothesis testing, a Type II error occurs when:
- H₀ is rejected when it is actually true
- H₀ is not rejected when it is actually false (Correct answer)
- The significance level is set too low
- The sample mean equals the population mean
Correct answer: H₀ is not rejected when it is actually false
A Type II error (β) is failing to reject a false null hypothesis, meaning a real effect goes undetected.
