Q: What does a Gauge Repeatability & Reproducibility (Gauge R&R) study measure?

Gauge R&R (Measurement System Analysis) quantifies how much of observed variation comes from the measurement system (gauge) vs. the actual process. Repeatability measures variation when one operator uses the same gauge multiple times; Reproducibility measures variation across different operators. A well-accepted benchmark: %GR&R 30% is unacceptable.

Q: A Six Sigma team compares the mean cycle time of two production lines. The team uses a two-sample t-test and gets a p-value of 0.03 with α = 0.05. What is the correct conclusion?

Since p = 0.03 < α = 0.05, the result is statistically significant. The team rejects the null hypothesis (H₀: μ₁ = μ₂) and concludes there is sufficient evidence that the two production lines have different mean cycle times.

Q: What is a Type II error in hypothesis testing?

A Type II error (β error) occurs when you fail to reject a false null hypothesis — you miss a real difference or effect. The probability of a Type II error is β; statistical power (1 – β) is the probability of correctly detecting a real effect.

Q: A team wants to test whether defect rates differ significantly across four machine centers. Which test is most appropriate?

When comparing proportions or counts across multiple categories (four machine centers × defective/non-defective), the chi-square test of independence is appropriate. A two-sample t-test handles means for two groups; ANOVA handles means for multiple groups.

Q: Statistical power in hypothesis testing is defined as the probability of:

Statistical power = 1 – β = P(reject H₀ | H₀ is false). It is the probability of detecting a real effect when one exists. Power increases with larger sample size, larger effect size, and higher significance level (α).

Q: A non-parametric alternative to the one-sample t-test, used when the data does not follow a normal distribution, is:

The Wilcoxon signed-rank test is the non-parametric equivalent of the one-sample (or paired) t-test. It tests whether the median of a sample differs from a hypothesized value without assuming a normal distribution — appropriate for small samples or non-normal data.

Question 1

In hypothesis testing, a Type I error (α error) is defined as:

Accepted Answer

A Type I error (false positive) occurs when you reject the null hypothesis (H₀) when it is actually true. The probability of a Type I error is α (significance level). A Type II error (β) is failing to reject H₀ when H₁ is actually true (false negative).

Question 2

Which test is used to compare the means of two independent groups when population variances are unknown?

Accepted Answer

The two-sample (independent) t-test compares the means of two independent groups when population standard deviations are unknown. ANOVA is used for three or more groups. The F-test compares variances. Chi-square tests are for categorical (attribute) data.

Question 3

A p-value of 0.03 with α = 0.05 means:

Accepted Answer

The p-value is the probability of observing a result as extreme or more extreme than the sample data, assuming H₀ is true. p = 0.03 < α = 0.05, so you reject H₀. A p-value does NOT equal the probability that H₀ is true — that is a common misinterpretation.

Question 4

What does a Gauge Repeatability & Reproducibility (Gauge R&R) study measure?

Accepted Answer

Gauge R&R (Measurement System Analysis) quantifies how much of observed variation comes from the measurement system (gauge) vs. the actual process. Repeatability measures variation when one operator uses the same gauge multiple times; Reproducibility measures variation across different operators. A well-accepted benchmark: %GR&R < 10% is excellent; 10–30% may be acceptable; >30% is unacceptable.

Question 5

When is a chi-square test of independence appropriate?

Accepted Answer

The chi-square test of independence tests whether two categorical (attribute) variables are associated. For example, testing whether defect type and shift are independent. It operates on count data in a contingency table. The chi-square goodness-of-fit test (different) tests whether observed frequencies match expected frequencies.

Question 6

A Six Sigma team compares the mean cycle time of two production lines. The team uses a two-sample t-test and gets a p-value of 0.03 with α = 0.05. What is the correct conclusion?

Accepted Answer

Since p = 0.03 < α = 0.05, the result is statistically significant. The team rejects the null hypothesis (H₀: μ₁ = μ₂) and concludes there is sufficient evidence that the two production lines have different mean cycle times.

Question 7

What is a Type II error in hypothesis testing?

Accepted Answer

A Type II error (β error) occurs when you fail to reject a false null hypothesis — you miss a real difference or effect. The probability of a Type II error is β; statistical power (1 – β) is the probability of correctly detecting a real effect.

Question 8

A team wants to test whether defect rates differ significantly across four machine centers. Which test is most appropriate?

Accepted Answer

When comparing proportions or counts across multiple categories (four machine centers × defective/non-defective), the chi-square test of independence is appropriate. A two-sample t-test handles means for two groups; ANOVA handles means for multiple groups.

Question 9

Statistical power in hypothesis testing is defined as the probability of:

Accepted Answer

Statistical power = 1 – β = P(reject H₀ | H₀ is false). It is the probability of detecting a real effect when one exists. Power increases with larger sample size, larger effect size, and higher significance level (α).

Question 10

A non-parametric alternative to the one-sample t-test, used when the data does not follow a normal distribution, is:

Accepted Answer

The Wilcoxon signed-rank test is the non-parametric equivalent of the one-sample (or paired) t-test. It tests whether the median of a sample differs from a hypothesized value without assuming a normal distribution — appropriate for small samples or non-normal data.

Hypothesis Testing & Statistics Practice Questions

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