When do you use a paired t-test?
Answer: When observations come in natural pairs (before/after, matched subjects); test the mean of the differences.
Explanation
Reduces variability from subject differences.
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Statistics: Hypothesis Testing Practice (University)
Two samplesDesignInterpretationRegressionANOVAChi-squareSetupProportionsOne-sampleErrorsDecision
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Two samples
Two-proportion z-test: what is the pooled proportion?
Answer: p̂ = (x1 + x2) / (n1 + n2), used in the standard error under H0 that the proportions are equal.
Explanation
Confidence intervals use the separate proportions.
When do you use a two-sample t-test?
Answer: Comparing means of two independent groups.
Explanation
Welch's version does not assume equal variances.
Design
What sample size is needed for a margin of error m in a proportion CI?
Answer: n = (z*/m)² · p*(1 − p*), using p* = 0.5 if unknown.
Explanation
For m = 0.03 at 95%: n ≈ 1,068.
What is a one-sided test and when is it appropriate?
Answer: Ha specifies a direction (μ > μ0); appropriate only when the direction was decided in advance for a substantive reason.
Explanation
Doubles the power in that direction but ignores the other.
Why does randomization matter for inference?
Answer: It makes groups comparable so differences can be attributed to the treatment, and it justifies the probability model.
Explanation
Observational data risk confounding.
Interpretation
What is the Bonferroni correction?
Answer: Divide α by the number of tests to control the family-wise error rate.
Explanation
Conservative; reduces power.
Why is 'p-hacking' a problem?
Answer: Testing many hypotheses or tweaking analyses until something reaches p < 0.05 inflates false positives.
Explanation
Preregistration and corrections (Bonferroni) help.
What is the relationship between a confidence interval and a two-sided test?
Answer: If the 95% CI excludes the null value, the test rejects H0 at α = 0.05.
Explanation
They convey the same evidence, but the CI shows the magnitude.
Statistical vs practical significance?
Answer: A tiny effect can be statistically significant with a huge sample; practical significance asks whether the effect matters.
Explanation
Report effect sizes and confidence intervals.
Regression
How do you test whether a regression slope is significant?
Answer: t = b / SE(b) with n − 2 df; H0: β = 0.
Explanation
A confidence interval for the slope not containing 0 means significance.
ANOVA
What does a one-way ANOVA test?
Answer: Whether the means of three or more groups are all equal.
Explanation
F = between-group variance / within-group variance; follow up with post-hoc tests.
Chi-square
How are expected counts computed in a test of independence?
Answer: (row total × column total) / grand total for each cell.
Explanation
df = (rows − 1)(columns − 1); expected counts should be ≥ 5.
Chi-square test statistic formula
Answer: χ² = Σ (observed − expected)² / expected
Explanation
Large values indicate poor fit.
When do you use a chi-square goodness-of-fit test?
Answer: To test whether observed counts in one categorical variable match expected proportions.
Explanation
df = categories − 1.
Setup
Write the hypotheses for testing whether a coin is fair.
Answer: H0: p = 0.5; Ha: p ≠ 0.5 (two-sided).
Explanation
One-sided tests use < or > in Ha.
What is a test statistic?
Answer: A standardized value (z, t, χ², F) measuring how far the sample result is from what H0 predicts.
Explanation
Compare it with a critical value or convert it to a p-value.
What is a significance level (α)?
Answer: The threshold probability for rejecting H0; the maximum acceptable Type I error rate, commonly 0.05.
Explanation
Choose α before collecting data.
Proportions
Sample: 60 of 100 prefer A; test H0: p = 0.5. Compute z.
Answer: z = (0.60 − 0.50) / √(0.25/100) = 0.10 / 0.05 = 2.0
Explanation
Two-sided p ≈ 0.046; reject at α = 0.05.
Conditions for a one-proportion z-test?
Answer: Random sample, independence, and np0 ≥ 10 and n(1 − p0) ≥ 10.
Explanation
Expected successes and failures both at least 10.
One-proportion z statistic
Answer: z = (p̂ − p0) / √(p0(1 − p0)/n)
Explanation
Use p0 (not p̂) in the standard error for the test.
One-sample
Conditions for a t-test on a mean?
Answer: Random sample, independence (n < 10% of population), and approximately normal population or n ≥ 30.
Explanation
Check for outliers in small samples.
Sample: n = 25, x̄ = 52, s = 10, H0: μ = 50. Compute t.
Answer: t = (52 − 50) / (10/√25) = 2 / 2 = 1.0
Explanation
With 24 df, p ≈ 0.33 two-sided; fail to reject.
One-sample t statistic formula
Answer: t = (x̄ − μ0) / (s / √n)
Explanation
x̄ sample mean, μ0 hypothesized mean, s sample SD.
When do you use a one-sample z-test vs t-test for a mean?
Answer: z when σ is known (rare); t when σ is estimated from the sample.
Explanation
t has n − 1 degrees of freedom.
Errors
How can you increase the power of a test?
Answer: Larger sample size, larger true effect, higher α, or less variability.
Explanation
Power analysis is done before the study.
Define Type I and Type II errors in a drug trial.
Answer: Type I: concluding the drug works when it does not. Type II: concluding it does not work when it does.
Explanation
α controls Type I; power = 1 − β controls Type II.
Decision
What does 'fail to reject H0' mean?
Answer: The data do not provide enough evidence against H0; it does not prove H0 true.
Explanation
Absence of evidence is not evidence of absence.
A p-value is 0.03 with α = 0.05. What do you conclude?
Answer: Reject H0: there is statistically significant evidence for Ha.
Explanation
At α = 0.01 you would fail to reject.
How do you decide using a p-value?
Answer: If p ≤ α, reject H0; otherwise, fail to reject H0.
Explanation
Never 'accept' H0.
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