Certified Six Sigma Black Belt Exam Certified Six Sigma Black Belt Hypothesis Testing Applications 1 — Questions and Answers
Question 1: A Black Belt suspects that cycle time has increased from a historical mean of 8 minutes, and believes any change would only be an increase. Which type of hypothesis test should be used?
- Two-tailed test
- Left-tailed (lower-tail) test
- Right-tailed (upper-tail) test (Correct answer)
- Chi-square goodness-of-fit test
Correct answer: Right-tailed (upper-tail) test
When the team has a directional suspicion (increase only), a right-tailed (upper-tail) test is appropriate. It places all of α in the upper tail, giving more power to detect a shift in the specified direction versus splitting α across both tails.
Question 2: A Black Belt sets the significance level α = 0.05 before running a hypothesis test. Which statement correctly interprets this value?
- There is a 5% chance the null hypothesis is true
- There is a 5% probability of rejecting a true null hypothesis (Correct answer)
- The test will correctly detect a real effect 95% of the time
- The alternative hypothesis has a 95% probability of being correct
Correct answer: There is a 5% probability of rejecting a true null hypothesis
α (alpha) defines the Type I error rate — the probability of incorrectly rejecting a true null hypothesis. It does not describe the probability that H₀ is true, nor is it the same as power (which is 1 − β).
Question 3: Cycle time data from two independent production lines fails a normality test. A Black Belt wants to compare central tendency between the two lines. Which test is most appropriate?
- Paired t-test
- Two-sample (independent) t-test
- Mann-Whitney U test (Correct answer)
- Levene's test
Correct answer: Mann-Whitney U test
The Mann-Whitney U test is the nonparametric equivalent of the two-sample t-test for independent groups. It compares medians and does not require the normality assumption, making it appropriate when data violates normality.
Question 4: A CSSBB wants to test whether observed defect counts across five product categories follow a historically expected distribution. Which test should be applied?
- One-sample t-test
- Chi-square goodness-of-fit test (Correct answer)
- Kruskal-Wallis test
- One-way ANOVA
Correct answer: Chi-square goodness-of-fit test
The chi-square goodness-of-fit test compares observed frequencies to expected frequencies derived from a theoretical or historical distribution. It is designed specifically for this use case, unlike the t-test or ANOVA which compare means.
Question 5: A Black Belt increases sample size from n = 25 to n = 100 while keeping α = 0.05 fixed. What is the most likely result?
- The Type I error rate increases
- The significance level automatically adjusts downward
- The power of the test increases (Correct answer)
- The null hypothesis becomes more likely to be accepted
Correct answer: The power of the test increases
Power (1 − β) increases with larger sample sizes because larger samples reduce standard error, making it easier to detect a true effect. The Type I error rate (α) remains fixed by the researcher's choice and is not affected by sample size.
Question 6: In a simple linear regression model, a Black Belt tests H₀: β₁ = 0 and obtains a p-value of 0.002 at α = 0.05. What is the correct conclusion?
- Fail to reject H₀; the predictor has no significant effect
- Reject H₀; the slope coefficient is statistically significant (Correct answer)
- The model explains 99.8% of the variation in the response
- The regression intercept is also significantly different from zero
Correct answer: Reject H₀; the slope coefficient is statistically significant
Since the p-value (0.002) is less than α (0.05), the null hypothesis H₀: β₁ = 0 is rejected. This means the predictor variable has a statistically significant linear relationship with the response variable. The p-value does not equal R², and the conclusion about β₁ says nothing about the intercept.
Certified Six Sigma Black Belt Exam Certified Six Sigma Black Belt Hypothesis Testing Applications 1
A Black Belt suspects that cycle time has increased from a historical mean of 8 minutes, and believes any change would only be an increase.
Which type of hypothesis test should be used?