Source: Mid-Term Formula Sheet, STAT 601 (UCF)
Tags: hypothesis testing, null hypothesis, alternative hypothesis, Type I error, Type II error, p-value, t-test, z-test, two-sample test, paired difference, pooled variance, F-test, chi-square, STAT 601
Difficulty: Intermediate to Advanced Prerequisites: Chapters 7–8 (sampling distributions, confidence intervals, standard error). Comfort with z and t distributions.
Confidence intervals estimate a parameter. Hypothesis tests decide whether the data provide enough evidence to reject a specific claim about that parameter. Chapter 9 covers the one-sample case: is this population mean (or proportion) equal to some proposed value? Chapter 10 extends this to comparing two populations: are the two means (or proportions) equal, or is one larger? These chapters bring together everything from the course so far: distributions, sampling theory, and estimation. The logic, state a hypothesis, pick a decision rule, compute a test statistic, decide, is the same in every test; only the formulas change depending on what you are testing and what you know about the population.
Hypothesis testing formalises the question "is the observed difference real, or could it just be sampling noise?" You set up a null hypothesis (no effect / no difference), compute a test statistic, and compare it to a critical value or p-value to decide whether the evidence is strong enough to reject the null. The framework is identical for one-sample and two-sample problems; the test statistic formula changes based on what you know about the populations.
Null hypothesis (H₀)
The default claim, typically "no effect" or "no difference." It is assumed true until the data provide strong enough evidence against it.
Alternative hypothesis (H₁)
The claim you are testing for. It can be one-sided (μ < μ₀ or μ > μ₀) or two-sided (μ ≠ μ₀).
Type I error (α)
Rejecting H₀ when it is actually true. The probability of this error is α, which you choose before collecting data (commonly 0.05 or 0.01). Think of it as a false alarm.
Type II error (β)
Failing to reject H₀ when it is actually false. The probability of this error is β. Think of it as a missed detection.
Power
Power = 1 − β = P(reject H₀ | H₀ is false). The probability of correctly detecting a real effect. Higher power is better; you increase it by increasing sample size, increasing α, or when the true effect is larger.
Test statistic
A standardised value (z, t, χ², or F) computed from the sample data. It measures how far the sample result is from the null hypothesis value, in units of standard error.
Critical value
The threshold from the reference distribution (z, t, χ², or F table) that separates the rejection region from the non-rejection region. For a one-tailed test, determined by α; for two-tailed, by α/2.
p-value
The probability of observing a test statistic as extreme as, or more extreme than, the one computed, assuming H₀ is true. If p-value ≤ α, reject H₀. If p-value > α, fail to reject.
Paired differences
When two samples are naturally linked (before/after, matched subjects), compute dᵢ = x₁ᵢ − x₂ᵢ and run a one-sample t-test on the differences.
Pooled variance (sₚ²)
When two independent populations are assumed to have equal variances, sₚ² = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2). In simple terms, it is a weighted average of the two sample variances.
Pooled proportion (p̄)
For two-proportion tests: p̄ = (x₁ + x₂) / (n₁ + n₂), where xᵢ is the count of successes in each sample.
State H₀ and H₁
Choose significance level α and determine the decision rule (critical value or p-value approach)
Compute the test statistic from the sample data
Make a decision: reject or fail to reject H₀
State the conclusion in context
All three approaches (critical value in original units, standardised test statistic, p-value) yield the same conclusion for the same data and α.
Left-tailed: H₀: μ ≥ μ₀, H₁: μ < μ₀ Two-tailed: H₀: μ = μ₀, H₁: μ ≠ μ₀ Right-tailed: H₀: μ ≤ μ₀, H₁: μ > μ₀
σ known: z = (x̄ − μ₀) / (σ / √n)
σ unknown: t = (x̄ − μ₀) / (s / √n), d.f. = n − 1
For proportions, substitute π for μ: z = (p − π₀) / √[π₀(1 − π₀) / n].
The hypotheses compare μ₁ − μ₂ to a hypothesised difference D₀ (often D₀ = 0):
Left-tailed: H₀: μ₁ − μ₂ ≥ D₀, H₁: μ₁ − μ₂ < D₀ Two-tailed: H₀: μ₁ − μ₂ = D₀, H₁: μ₁ − μ₂ ≠ D₀ Right-tailed: H₀: μ₁ − μ₂ ≤ D₀, H₁: μ₁ − μ₂ > D₀
Three scenarios:
Both σ known: z = [(x̄₁ − x̄₂) − D₀] / √(σ₁²/n₁ + σ₂²/n₂)
σ unknown, assumed equal (σ₁ = σ₂): use pooled variance sₚ², t-test with d.f. = n₁ + n₂ − 2
sₚ = √[(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)]
t = [(x̄₁ − x̄₂) − D₀] / [sₚ · √(1/n₁ + 1/n₂)]
σ unknown, not assumed equal: use separate variances, t-test with Welch's approximation for degrees of freedom
t = [(x̄₁ − x̄₂) − D₀] / √(s₁²/n₁ + s₂²/n₂)
d.f. = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]
Compute dᵢ = x₁ᵢ − x₂ᵢ for each pair
Run a one-sample t-test on the d values: t = (d̄ − D₀) / (s_d / √n), d.f. = n − 1
z = (p₁ − p₂) / √[p̄(1 − p̄)(1/n₁ + 1/n₂)], where p̄ = (x₁ + x₂) / (n₁ + n₂) and D₀ = 0
If D₀ ≠ 0: z = [(p₁ − p₂) − D₀] / √(p₁q₁/n₁ + p₂q₂/n₂)
One-population variance: χ² = (n − 1)s² / σ₀². Sensitive to departures from normality.
Two-population variances (equal variance test): F = s₁² / s₂², with d.f.₁ = n₁ − 1 and d.f.₂ = n₂ − 1. Alternatively, use Levene's test, which is more robust.
You can always use a confidence interval to conduct a two-tailed test. If the hypothesised value falls outside the CI, reject H₀. If it falls inside, fail to reject. This equivalence is worth remembering: it gives you two ways to check your answer.
Decision errors:
H₀ True | H₀ False | |
|---|---|---|
Reject H₀ | Type I (α) | Correct |
Fail to reject H₀ | Correct | Type II (β) |
One-sample z (σ known): z = (x̄ − μ₀) / (σ / √n)
One-sample t (σ unknown): t = (x̄ − μ₀) / (s / √n), d.f. = n − 1
Pooled standard deviation: sₚ = √[(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)]
Welch's d.f.: d.f. = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]
Proportion z-test: z = (p − π₀) / √[π₀(1 − π₀) / n]
Chi-square for variance: χ² = (n − 1)s² / σ₀²
F-test for equal variances: F = s₁² / s₂²
Pharmaceutical companies use two-sample hypothesis tests to determine whether a new drug performs better than a placebo. The FDA requires statistically significant results (typically α = 0.05) before approving a treatment.
Paired-difference tests are common in education research: the same students are tested before and after an intervention, and the test determines whether the intervention produced a meaningful improvement.
"Fail to reject H₀" does not mean "accept H₀." You have not proven H₀ true; you simply lack sufficient evidence to reject it. The distinction matters both conceptually and on exams.
Students sometimes set up H₁ to match the claim they want to test, but place the equality in H₁. The equality (= or ≥ or ≤) always belongs in H₀.
A small p-value does not tell you the effect is large. It tells you the effect is unlikely to be zero. With a large enough sample, even a trivially small difference can produce a tiny p-value.
The pooled-variance t-test assumes equal population variances. If you are unsure whether variances are equal, either run an F-test or Levene's test first, or default to the Welch (separate-variance) t-test, which does not require the assumption.
⚠️ Know the five-step process cold. Many exam questions are structured around it.
⚠️ Be able to choose the correct test: z vs. t, pooled vs. separate variance, paired vs. independent.
⚠️ Type I vs. Type II error: be able to describe each in the context of a specific problem (e.g., "concluding the drug works when it does not" is Type I).
⚠️ The p-value definition is commonly tested: "the probability of observing a result as extreme or more extreme than the sample, assuming H₀ is true."
⚠️ The summary table on page 6 of the formula sheet is a decision tree for the entire course. Know which row to use for each scenario.
⚠️ χ² and F tests are very sensitive to non-normality. This is a frequently tested caveat.
True or False: The null hypothesis always contains the equality.
Fill in the blank: Power = 1 − ________.
True or False: A p-value of 0.03 means there is a 3% chance that H₀ is true.
Fill in the blank: In a paired-difference test, d.f. = ________.
True or False: You can use a confidence interval to conduct a two-tailed hypothesis test.
Answers: 1. True. 2. β. 3. False (it means there is a 3% chance of observing data this extreme if H₀ were true; it is not the probability that H₀ is true). 4. n − 1 (where n is the number of pairs). 5. True.
Q: A company claims its light bulbs last at least 1,000 hours on average. A sample of 36 bulbs has x̄ = 980 hours and σ = 60. At α = 0.05, is there enough evidence to reject the claim?
A: H₀: μ ≥ 1000, H₁: μ < 1000 (left-tailed). z = (980 − 1000) / (60 / √36) = −20 / 10 = −2.0. Critical value z₀.₀₅ = −1.645. Since −2.0 < −1.645, reject H₀. There is enough evidence at the 5% level to conclude the bulbs last fewer than 1,000 hours on average.
Q: Two teaching methods are compared. Method A: n₁ = 25, x̄₁ = 78, s₁ = 10. Method B: n₂ = 30, x̄₂ = 74, s₂ = 12. Assume equal variances. Test whether the means differ at α = 0.05.
A: H₀: μ₁ − μ₂ = 0, H₁: μ₁ − μ₂ ≠ 0 (two-tailed). sₚ = √[(24 × 100 + 29 × 144) / 53] = √[(2400 + 4176) / 53] = √124.08 ≈ 11.14. t = (78 − 74) / [11.14 × √(1/25 + 1/30)] = 4 / [11.14 × √0.0733] = 4 / [11.14 × 0.2707] = 4 / 3.016 ≈ 1.326. With d.f. = 53, the critical t₀.₀₂₅ ≈ 2.006. Since 1.326 < 2.006, fail to reject H₀. There is not enough evidence to conclude the means differ.
Q: In a sample of 500, 280 support a policy. Test whether the population proportion exceeds 0.50 at α = 0.01.
A: H₀: π ≤ 0.50, H₁: π > 0.50 (right-tailed). p = 280/500 = 0.56. z = (0.56 − 0.50) / √(0.50 × 0.50 / 500) = 0.06 / √0.0005 = 0.06 / 0.02236 ≈ 2.683. Critical z₀.₀₁ = 2.326. Since 2.683 > 2.326, reject H₀. There is sufficient evidence at the 1% level that more than half the population supports the policy.
Q: What is the relationship between a 95% confidence interval and a two-tailed test at α = 0.05?
A: They are equivalent. If the hypothesised value of the parameter falls outside the 95% CI, the two-tailed test at α = 0.05 would reject H₀. If it falls inside, the test would fail to reject.
Hypothesis testing is the direct counterpart of the confidence intervals from Chapter 8; every CI maps to an equivalent two-tailed test. The pooled-variance assumption in the two-sample t-test can be checked with the F-test for equal variances, connecting to the variance material from Chapters 1–3. The concept of power and Type II error extends into more advanced topics like sample-size planning for experiments (covered in later courses or the final portion of this one).
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