Green Belt · Analyze phase
Lesson 3 covered how a hypothesis test is set up in general: a null and alternative hypothesis, an alpha level, a p-value decision rule, and the risk of Type I and Type II error. This lesson puts that framework to work on the most common family of tests a Green Belt runs: tests for normally distributed continuous data. Which specific test applies depends on how many samples are being compared, whether they are independent or paired, and whether the question is about a mean or a variance. Getting that classification right, more than any formula, is what the exam and real projects actually test.
The 1-sample t test compares a single sample's mean to a fixed target value, such as a spec, a contract requirement, or a historical baseline. The test statistic is t = (xbar - μ0) / (s / √n), where xbar is the sample mean, μ0 is the target value under the null hypothesis, s is the sample standard deviation, and n is the sample size. That statistic is compared to a critical value from the t distribution with n-1 degrees of freedom, or its p-value is compared to alpha.
Worked example. A county permitting office has told GCHS that its typical residential rehab permit turns around in 45 days. GCHS pulls a sample of n = 9 recent permits and finds a mean turnaround of 52 days with a sample standard deviation of 6 days. Is the office's actual performance different from the stated 45 days, at alpha = 0.05?
Decision: reject the null hypothesis. The sample gives strong evidence that actual permit turnaround differs from the office's stated 45 days, in this case running longer, which is worth escalating with data in hand rather than anecdote.
The 2-sample t test compares the means of two independent groups, such as punch-list closeout time for two different subcontractor crews. Before running it, check whether the two groups' variances are reasonably equal. If they are, the standard pooled-variance t test is appropriate. If they are not, use the unequal-variance form, commonly called Welch's t test, which adjusts the degrees of freedom downward instead of pooling the variances. Most statistical software runs the unequal-variance version by default because it performs well whether or not variances are actually equal, so when in doubt, use it.
The equal-variance assumption itself is checked with a test for equal variances, most commonly an F test (the ratio of the two sample variances) or Levene's test, which is less sensitive to non-normality than the F test. A significant result on that test is the signal to switch to the unequal-variance 2-sample t test rather than the pooled one.
A paired t test applies when the two sets of observations are not independent but are linked, such as the same 20 rental units inspected before and after a preventive-maintenance program, or the same appraiser's estimate compared to the final contractor bid on the same 15 jobs. Instead of comparing two group means directly, the test takes the difference for each pair, then runs a 1-sample t test on those differences against a hypothesized mean difference of zero. Pairing removes the unit-to-unit variation that would otherwise mask a real before/after effect, so it is usually more powerful than treating the same data as two independent samples, provided the paired readings are positively correlated, as before/after readings on the same unit typically are.
Sometimes the question is not about the mean but about consistency itself, for example whether the variability in square-footage estimates from a new estimating tool matches a target variance from the old process. The 1-sample chi-square test for variance compares a sample variance to a hypothesized target variance. Because the chi-square distribution is not symmetric, its lower and upper critical values differ, and the test is sensitive to departures from normality, so it should only be used when the underlying data are reasonably normal.
One-way analysis of variance (ANOVA) extends the 2-sample t test to three or more independent groups, for example comparing average work-order completion time across four maintenance technicians. Rather than running several pairwise t tests, which inflates the overall false-positive rate, ANOVA partitions the total variation in the data into variation between the group means and variation within each group, then forms one test statistic:
F = MS between / MS within
where MS between is the mean square between groups (variation of the group means around the overall mean) and MS within is the mean square within groups (average variation of individual observations around their own group mean). A large F means the between-group variation is large relative to the noise within each group, evidence that at least one group mean differs from the others.
| What a significant F does tell you | What it does not tell you |
|---|---|
| At least one group mean differs from at least one other | Which specific group or pair of groups differs |
| The differences seen are unlikely to be due to random sampling alone, at the chosen alpha | Whether the difference is large enough to matter practically |
Finding out which groups differ requires a post-hoc comparison test run after a significant F, such as Tukey's method, which compares every pair of group means while controlling the overall false-positive rate across all of those comparisons. Running plain, unadjusted t tests on every pair instead of a post-hoc method is a common exam trap and a common real mistake, since it lets the true error rate climb well above the stated alpha.
One-way ANOVA carries three assumptions: the data in each group are approximately normally distributed, the observations are independent of each other, and the variance is approximately equal across groups (sometimes checked with Levene's test or Bartlett's test before trusting the F test). A violation of normality points toward the nonparametric alternatives covered in Lesson 5.
Why this matters
A significant ANOVA F is the start of the investigation, not the end of it. A team that stops at "F is significant" without a post-hoc test and without asking whether the practical size of the difference matters can end up chasing a technician's schedule difference that is statistically real but operationally irrelevant.
1. Which test compares a single sample mean to a fixed target value?
2. In the 1-sample t test formula t = (xbar - μ0) / (s / √n), what does μ0 represent?
3. A GCHS sample of n = 9 permits has a mean turnaround of 52 days and a sample standard deviation of 6 days. The county claims a 45-day mean. What is the test statistic?
4. In that same permit example, df = 8 and the two-tailed critical t at alpha = 0.05 is 2.306. What is the correct conclusion?
5. Before choosing between the pooled and unequal-variance forms of the 2-sample t test, what should be checked first?
6. Which test is less sensitive to non-normality when checking whether two groups have equal variances?
7. GCHS inspects the same 20 rental units before and after a maintenance program and wants to test whether average condition scores changed. Which test fits this design?
8. Why is pairing typically more powerful than treating linked before/after data as two independent samples?
9. A 1-sample chi-square test for variance is used to test which kind of hypothesis?
10. One-way ANOVA is the appropriate test in which of these scenarios?
11. In one-way ANOVA, the F statistic is calculated as which ratio?
12. A one-way ANOVA returns a significant F. What does that result tell you by itself?
13. After a significant one-way ANOVA result, what is the correct next step to find which specific groups differ?
14. Which of the following is NOT one of the three core assumptions of one-way ANOVA?