Quiz: Hypothesis Testing
Test your understanding of hypothesis testing, p-values, significance levels, Type I and II errors, and statistical inference with these review questions.
1. The null hypothesis (H₀) always contains which type of statement?
- An inequality (< or >)
- An equality (=)
- A range of values
- The sample statistic
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The correct answer is B. The null hypothesis always contains an equality statement, representing the status quo or "no effect" claim. Examples include H₀: p = 0.5 or H₀: p₁ - p₂ = 0. The alternative hypothesis (Hₐ) contains the inequality. Hypotheses are about population parameters, never sample statistics.
Concept Tested: Null Hypothesis
2. A researcher suspects that more than 60% of adults exercise regularly. What is the appropriate alternative hypothesis?
- Hₐ: p = 0.60
- Hₐ: p ≠ 0.60
- Hₐ: p > 0.60
- Hₐ: p < 0.60
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The correct answer is C. The researcher suspects the proportion is greater than 60%, so this is a one-sided test with Hₐ: p > 0.60. The phrase "more than" indicates a right-tailed test. If the researcher wanted to test whether the proportion differs from 60% in either direction, then Hₐ: p ≠ 0.60 would be appropriate.
Concept Tested: Alternative Hypothesis
3. What does a p-value of 0.03 mean?
- There is a 3% probability that the null hypothesis is true
- There is a 3% probability of obtaining results at least as extreme as observed, if H₀ is true
- The treatment is 97% effective
- 3% of the sample showed the effect
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The correct answer is B. The p-value is the probability of obtaining sample results at least as extreme as those observed, assuming the null hypothesis is true. A p-value of 0.03 means that if H₀ were true, we would see results this extreme only 3% of the time. It is NOT the probability that H₀ is true.
Concept Tested: Interpreting P-Values
4. If α = 0.05 and the p-value = 0.08, what is the correct conclusion?
- Reject H₀; the result is statistically significant
- Accept H₀; the null hypothesis is proven true
- Fail to reject H₀; there is not sufficient evidence against H₀
- The test is inconclusive and must be repeated
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The correct answer is C. Since p-value (0.08) > α (0.05), we fail to reject the null hypothesis. There is not sufficient evidence to conclude that the alternative hypothesis is true. We never "accept" H₀—we simply don't have enough evidence to reject it. Failing to reject does not prove H₀ is true.
Concept Tested: Making Conclusions
5. A Type I error occurs when we:
- Fail to reject a false null hypothesis
- Reject a true null hypothesis
- Correctly reject a false null hypothesis
- Correctly fail to reject a true null hypothesis
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The correct answer is B. A Type I error (false positive) occurs when we reject the null hypothesis even though it is actually true. The probability of a Type I error equals α, the significance level. In a medical context, this is like diagnosing a healthy patient with a disease.
Concept Tested: Type I Error
6. In a hypothesis test with H₀: p = 0.50, a sample of 200 yields 90 successes. What is the test statistic?
- z = -1.41
- z = -0.71
- z = 0.71
- z = 1.41
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The correct answer is A. First, calculate p̂ = 90/200 = 0.45. The test statistic is z = (p̂ - p₀)/√[p₀(1-p₀)/n] = (0.45 - 0.50)/√[0.50(0.50)/200] = -0.05/√0.00125 = -0.05/0.0354 ≈ -1.41. The sample proportion of 0.45 is about 1.41 standard errors below the hypothesized value of 0.50.
Concept Tested: Test Statistic
7. Which of the following increases the power of a hypothesis test?
- Decreasing the sample size
- Using a smaller significance level (α)
- Increasing the sample size
- Making the test two-sided instead of one-sided
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The correct answer is C. Larger sample sizes increase power because they provide more information and reduce sampling variability, making it easier to detect real effects. Decreasing α, decreasing sample size, or switching from one-sided to two-sided tests all decrease power.
Concept Tested: Power of a Test
8. A study finds that a new medication produces a statistically significant reduction in blood pressure (p < 0.001) of 1 mmHg. Which statement best describes this result?
- The medication is highly effective and should be prescribed
- The result is statistically significant but may lack practical significance
- The p-value must be calculated incorrectly
- The sample size was probably too small
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The correct answer is B. Statistical significance (small p-value) indicates the effect is unlikely due to chance, but practical significance considers whether the effect is meaningful in the real world. A 1 mmHg reduction in blood pressure, while statistically detectable with a large sample, is clinically negligible. The very small p-value likely results from a large sample size, not a large effect.
Concept Tested: Practical Significance
9. When comparing two proportions, the pooled proportion is calculated as:
- (p̂₁ + p̂₂)/2
- (x₁ + x₂)/(n₁ + n₂)
- (p̂₁ × p̂₂)
- (n₁p̂₁ + n₂p̂₂)/(n₁ + n₂)
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The correct answer is B. The pooled proportion combines the successes and sample sizes from both groups: p̂_pooled = (x₁ + x₂)/(n₁ + n₂), where x₁ and x₂ are the number of successes. This is equivalent to the weighted average in option D. The pooled proportion is used in hypothesis testing when H₀ assumes equal proportions.
Concept Tested: Test for Two Proportions
10. Which condition must be verified before conducting a z-test for proportions?
- The population standard deviation must be known
- The sample mean must be at least 30
- np₀ ≥ 10 and n(1-p₀) ≥ 10
- The samples must be paired
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The correct answer is C. The Large Counts condition requires that both np₀ ≥ 10 and n(1-p₀) ≥ 10 for the sampling distribution to be approximately normal. Note that for hypothesis tests, we use p₀ (the hypothesized value) rather than p̂ (the sample proportion) when checking this condition, because we calculate probabilities assuming H₀ is true.
Concept Tested: Conditions for Z-Test