7.4 Hypothesis Testing Terminology, Errors, and Single-Sample Tests

Hypothesis testing provides a rigorous, objective framework for making data-driven statistical decisions in the Analyze phase. It allows Six Sigma Black Belts to determine whether observed differences between process parameters (means, variances, proportions) and target benchmarks represent genuine physical effects or merely random sampling variation.

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Fundamentals of Hypothesis Testing

The 7-Step Hypothesis Testing Roadmap

1. State the Operational Problem: Define the business question in clear process terms.
2. Formulate Null ($H_0$) and Alternative ($H_a$) Hypotheses: Express the question mathematically.
3. Select Significance Level ($\alpha$): Define acceptable Type I risk (typically $\alpha = 0.05$).
4. Check Statistical Assumptions: Confirm data type, independence, stability, and normality.
5. Calculate Sample Size & Statistical Power: Ensure $n$ is sufficient to detect target effect $\delta$.
6. Compute Test Statistic & $p$-Value: Execute mathematical test formula on sample data.
7. State Statistical & Operational Conclusion: Compare $p$-value against $\alpha$ and translate into business action.

Null ($H_0$) vs. Alternative ($H_a$) Hypotheses

• Null Hypothesis ($H_0$): The default baseline statement of "no effect," "no difference," or "status quo." It always contains an equality sign ($=, \le, \ge$).
• Alternative Hypothesis ($H_a$ or $H_1$): The research hypothesis the Black Belt seeks to prove (e.g., "the process mean has changed" or "new tooling reduces cycle time"). It contains inequalities ($\neq, <, >$).

Decision Risks & Error Matrix

Statistical decision-making carries two distinct types of decision errors:

• Type I Error ($\alpha$): Rejecting a true null hypothesis. The significance level $\alpha$ (typically set to $0.05$ or $5%$) represents the maximum allowable probability of committing a Type I error.
• Type II Error ($\beta$): Failing to reject a false null hypothesis when a true process difference exists.
• Statistical Power ($1 - \beta$): The probability of correctly detecting a true treatment effect of size $\delta$. Power is typically targeted at $\ge 0.80$ ($80%$).

Factors Influencing Statistical Power

1. Sample Size ($n$): Increasing $n$ reduces standard error ($\text{SE} = \frac{\sigma}{\sqrt{n}}$), significantly increasing power.
2. Effect Size ($\delta = |\mu - \mu_0|$): Larger physical differences are easier to detect, yielding higher power.
3. Significance Level ($\alpha$): Increasing $\alpha$ (e.g., from $0.01$ to $0.05$) increases power but elevates Type I risk.
4. Process Variability ($\sigma$): Lower process noise/variance increases statistical power.

The $p$-Value Decision Rule

The $p$-value is the exact probability of obtaining a test statistic at least as extreme as the observed sample value, assuming the null hypothesis $H_0$ is true.

• If $p$-value $\le \alpha$: Reject $H_0$. The result is statistically significant.
• If $p$-value $> \alpha$: Fail to reject $H_0$. Insufficient evidence to claim a significant effect.

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Single-Sample Hypothesis Tests

Single-sample tests evaluate whether a sample parameter differs significantly from a known benchmark or target value ($\mu_0$, $\sigma_0^2$, or $p_0$).

1. One-Sample $z$-Test for Mean (Known $\sigma$)

Used when sample data are continuous, normally distributed, and population standard deviation ($\sigma$) is known.

• Test Statistic: $z_0 = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$
• Critical Region (Two-Tailed at $\alpha = 0.05$): Reject $H_0$ if $|z_0| > 1.96$.

2. One-Sample $t$-Test for Mean (Unknown $\sigma$)

Used when population standard deviation $\sigma$ is unknown and estimated using sample standard deviation ($s$).

• Test Statistic: $t_0 = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$
• Degrees of Freedom: $\text{df} = n - 1$.
• Distribution: Student's $t$-distribution, featuring heavier tails than $Z$ for small sample sizes, converging to $Z$ as $\text{df} \to \infty$.

Worked 1-Sample $t$-Test Example

A team tests if a new injection molding machine meets a target cycle time $\mu_0 = 45.0\text{ seconds}$. Sample size $n = 25$, sample mean $\bar{x} = 43.2\text{ seconds}$, sample standard deviation $s = 4.0\text{ seconds}$, $\alpha = 0.05$.

1. Hypotheses: $H_0: \mu = 45.0$ vs. $H_a: \mu \neq 45.0$.
2. Standard Error: $\text{SE} = \frac{4.0}{\sqrt{25}} = \frac{4.0}{5} = 0.80$.
3. Test Statistic: $t_0 = \frac{43.2 - 45.0}{0.80} = \frac{-1.80}{0.80} = -2.25$.
4. Critical Value ($t_{0.025, 24}$): $\pm 2.064$.
5. Conclusion: Since $|-2.25| > 2.064$ ($p = 0.0338 < 0.05$), Reject $H_0$. The new machine cycle time is significantly lower than 45.0 seconds.

3. One-Sample Proportion $z$-Test

Evaluates whether a sample binomial success proportion ($\hat{p} = \frac{x}{n}$) differs significantly from a target baseline proportion ($p_0$).

• Test Statistic: $z_0 = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
• Validity Assumptions: $n p_0 \ge 5$ and $n (1-p_0) \ge 5$.

4. One-Sample Chi-Square Test for Variance ($\sigma^2$)

Evaluates whether process variance ($s^2$) differs from a target baseline variance ($\sigma_0^2$):

$\chi^2_0 = \frac{(n - 1) s^2}{\sigma_0^2}$

• Degrees of Freedom: $\text{df} = n - 1$. Follows a Chi-Square distribution with $n-1$ degrees of freedom.

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Summary of Single-Sample Test Formulas