Certified Six Sigma Black Belt Exam Certified Six Sigma Black Belt Advanced Hypothesis Testing 1 — Questions and Answers
Question 1: A Six Sigma Black Belt is comparing the variance of two independent processes to determine if they differ significantly before running a two-sample t-test. Which hypothesis test should be used?
- Chi-square goodness-of-fit test
- Levene's test or Bartlett's test for equal variances (Correct answer)
- Paired t-test
- One-proportion z-test
Correct answer: Levene's test or Bartlett's test for equal variances
Levene's test (robust to non-normality) or Bartlett's test is used to assess equality of variances between two or more groups. This is a prerequisite check before deciding whether to use a pooled or Welch's t-test.
Question 2: A Black Belt sets alpha = 0.05 and calculates a power of 0.72 for a hypothesis test. What does this mean?
- There is a 72% chance the null hypothesis is true
- There is a 28% probability of committing a Type I error
- The test has a 28% probability of failing to detect a true effect (Type II error rate) (Correct answer)
- The test will reject the null hypothesis 72% of the time regardless of the true effect
Correct answer: The test has a 28% probability of failing to detect a true effect (Type II error rate)
Power = 1 − β, where β is the probability of a Type II error (failing to reject a false null hypothesis). A power of 0.72 means β = 0.28, so there is a 28% chance the test misses a real effect.
Question 3: Which non-parametric test is the most appropriate alternative to a one-way ANOVA when the assumption of normality is severely violated?
- Mood's Median test
- Kruskal-Wallis test (Correct answer)
- Mann-Whitney U test
- Friedman test
Correct answer: Kruskal-Wallis test
The Kruskal-Wallis test is the non-parametric equivalent of one-way ANOVA and compares medians across three or more independent groups when normality cannot be assumed. The Mann-Whitney is limited to two groups.
Question 4: A Black Belt conducts a hypothesis test and correctly fails to reject a true null hypothesis. Which outcome does this represent?
- Type I error
- Type II error
- Correct decision — true negative (Correct answer)
- Correct decision — true positive
Correct answer: Correct decision — true negative
When the null hypothesis is actually true and the test fails to reject it, the result is a correct decision called a true negative. Type I error occurs when a true null is rejected; Type II error occurs when a false null is not rejected.
Question 5: A Six Sigma team wants to test whether defect counts across four production shifts follow the expected distribution based on historical proportions. Which test is most appropriate?
- Two-sample t-test
- One-way ANOVA
- Chi-square goodness-of-fit test (Correct answer)
- F-test for equal variances
Correct answer: Chi-square goodness-of-fit test
The chi-square goodness-of-fit test compares observed frequencies against expected frequencies derived from a known or hypothesized distribution. It is the correct choice when testing whether categorical count data matches a specified proportion pattern.
Question 6: When increasing sample size in a hypothesis test while holding alpha constant, what is the direct effect on the power of the test?
- Power decreases because there is more sampling variability
- Power remains unchanged because alpha is fixed
- Power increases because the standard error decreases and effect detection improves (Correct answer)
- Power decreases because the critical value increases
Correct answer: Power increases because the standard error decreases and effect detection improves
Larger sample sizes reduce the standard error of the test statistic, making it easier to detect a given effect size. This narrows the sampling distribution under H₁, increasing overlap with the rejection region and raising power.
Certified Six Sigma Black Belt Exam Certified Six Sigma Black Belt Advanced Hypothesis Testing 1
A Six Sigma Black Belt is comparing the variance of two independent processes to determine if they differ significantly before running a two-sample t-test.
Which hypothesis test should be used?