Certified Six Sigma Black Belt Exam Advanced Hypothesis Testing 3 — Questions and Answers
Question 1: A paired t-test is preferred over a two-sample t-test when:
- The sample sizes are unequal
- Measurements are taken on the same subject under two conditions (Correct answer)
- The variances of the two groups are unequal
- The data are not normally distributed
Correct answer: Measurements are taken on the same subject under two conditions
Pairing on the same subject removes between-subject variability, making the paired t-test more powerful when natural pairs exist.
Question 2: Bartlett's test is preferred over Levene's test when:
- Data are highly skewed or contain outliers
- Data are approximately normally distributed (Correct answer)
- Sample sizes are very small (n < 10)
- Categorical data are involved
Correct answer: Data are approximately normally distributed
Bartlett's test assumes normality and is more powerful than Levene's test when that assumption holds.
Question 3: In a chi-square test of independence, the degrees of freedom are calculated as:
- (number of rows) × (number of columns)
- (number of rows − 1) + (number of columns − 1)
- (number of rows − 1) × (number of columns − 1) (Correct answer)
- Total sample size − 1
Correct answer: (number of rows − 1) × (number of columns − 1)
Degrees of freedom for a chi-square test of independence = (r − 1)(c − 1), where r = rows and c = columns.
Question 4: A Black Belt conducts 20 simultaneous hypothesis tests each at α = 0.05. The probability of at least one false positive (family-wise error) is approximately:
- 0.05
- 0.36
- 0.64 (Correct answer)
- 1.00
Correct answer: 0.64
Family-wise error rate = 1 − (1 − 0.05)²⁰ ≈ 0.64, meaning there is about a 64% chance of at least one spurious rejection.
Question 5: Which correction is commonly applied to control the family-wise error rate when performing multiple comparisons?
- Box-Cox transformation
- Bonferroni correction (Correct answer)
- Anderson-Darling adjustment
- Yates' continuity correction
Correct answer: Bonferroni correction
The Bonferroni correction divides alpha by the number of tests (α/k), maintaining the family-wise error rate at the desired level.
Question 6: When sample size increases, holding alpha constant, what happens to the power of a hypothesis test?
- Power decreases because variance increases
- Power increases because the standard error decreases (Correct answer)
- Power stays the same because alpha is fixed
- Power becomes unpredictable
Correct answer: Power increases because the standard error decreases
Larger samples reduce the standard error, shrink the confidence interval, and improve the test's ability to detect true differences.
Question 7: A Mood's Median test is used to:
- Compare variances of two or more populations
- Test whether multiple population medians are equal (Correct answer)
- Assess normality of a single sample
- Compare proportions from two independent samples
Correct answer: Test whether multiple population medians are equal
Mood's Median test is a nonparametric test for comparing medians across two or more independent groups.
Certified Six Sigma Black Belt Exam Advanced Hypothesis Testing 3
A paired t-test is preferred over a two-sample t-test when: