Hypothesis Testing and Confidence Intervals: Practice Exercises

Claudius Gräbner-Radkowitsch

Hypothesis Testing and Confidence Intervals: Practice Exercises

statistics

exercise

Conceptual and computational exercises to test and reinforce your understanding of hypothesis testing, p-values, confidence intervals, and the connections between them.

Author

Claudius Gräbner-Radkowitsch

Published

June 16, 2026

Modified

June 16, 2026

Companion resource

These exercises accompany the conceptual resource Hypothesis Testing and Confidence Intervals. Work through that page first, then use these exercises to check your understanding. Solutions are hidden — try each exercise before revealing the answer.

1 Conceptual exercises

1.1 Exercise 1 — Interpreting a confidence interval

You survey 200 participants and find an average rating of 7.8 out of 10, with a 95% confidence interval of [7.5, 8.1]. A colleague says: “We know that 95% of all participants have ratings between 7.5 and 8.1.”

What is wrong with this interpretation? What should your colleague say instead?

Solution

What is wrong: The colleague is confusing the confidence interval for the population mean with the range where individual observations fall. The interval [7.5, 8.1] estimates where the true average rating lies across the whole population — not where individual participants score.

Correct interpretation: “We used a procedure that, over repeated sampling, would contain the true average rating in 95% of constructed intervals. Based on this sample, a plausible range for the true population mean is 7.5 to 8.1.”

The distinction: If you wanted a range containing 95% of individual scores, you would need something like $\bar{x} \pm 2s$, which uses the standard deviation (not the standard error) and would produce a much wider interval.

1.2 Exercise 2 — Interpreting a p-value

A study yields $p = 0.08$ at a pre-specified significance level of $\alpha = 0.05$. A colleague concludes: “The test failed — there’s no effect.”

What is problematic about this conclusion? What would be a more appropriate interpretation?

Solution

Problems with the conclusion:

“The test failed” — The test worked as intended; it simply did not produce sufficient evidence to reject $H_0$ at the chosen threshold.
“There’s no effect” — Failing to find statistical significance is not evidence of absence. It could mean: (a) no effect exists, (b) an effect exists but the sample was too small to detect it, or (c) the effect exists but is smaller than this study was powered to find.

Better interpretation: “At $\alpha = 0.05$, we do not have sufficient evidence to reject the null hypothesis. With $p = 0.08$, the result is borderline. We should consider the effect size and confidence interval, and avoid concluding that no effect exists.”

Key reminder: $p > \alpha$ means “insufficient evidence to reject $H_0$” — not “$H_0$ is true.”

1.3 Exercise 3 — Statistical vs. practical significance

Two independent studies test the same intervention:

Study A: $n = 50$, observed effect = 15%, $p = 0.08$
Study B: $n = 50{,}000$, observed effect = 0.5%, $p = 0.001$

Which finding is more likely to be decision-relevant? Why?

Solution

Study A is more likely to be decision-relevant, despite not crossing the conventional significance threshold.

Reasoning:

Study A reports a 15% effect. The non-significant $p$-value is most likely a consequence of the small sample ($n = 50$), not an absence of effect. The effect size is large enough that it would almost certainly matter in practice if confirmed with more data.
Study B reports a 0.5% effect that is “highly significant” only because $n = 50{,}000$ gives the test enormous power to detect even negligible differences. A 0.5% improvement rarely justifies the costs of implementation and change.

The core lesson: The $p$-value is a function of effect size, sample size, and variability. With large enough samples, trivially small effects become statistically significant. Always evaluate effect size and practical relevance alongside the $p$-value.

1.4 Exercise 4 — Type I and Type II errors

A quality control process tests whether a production batch meets safety standards.

$H_0$: The batch meets standards
$H_1$: The batch is below standards

For each of the following contexts, decide which type of error is more serious and explain why:

A manufacturer of medical devices
A manufacturer of children’s toys
A car manufacturer testing brake components

Solution

In all three cases, the error types map as follows:

Type I error (false positive): Concluding the batch fails when it actually meets standards → unnecessary recall or production halt
Type II error (false negative): Concluding the batch passes when it is actually below standards → defective products reach consumers

a) Medical devices — Type II error is more serious. A missed defect could directly harm patients, lead to regulatory action, and create massive liability. The cost of a false alarm (halting production unnecessarily) is high but safe. Use a more liberal $\alpha$ (e.g., 0.10) to reduce the Type II error rate.

b) Children’s toys — Type II error is more serious, though the stakes are lower than medical devices. Defective toys reaching children creates safety risk, liability, and reputational damage. A standard or slightly elevated $\alpha$ is appropriate.

c) Brake components — Type II error is more serious. Missing a safety defect in braking systems risks accidents and fatalities. The cost of an unnecessary production halt is high but preferable. Use a more liberal $\alpha$ for safety-critical components.

General principle: When safety is at stake, minimise false negatives even at the cost of more false positives. When only money is at risk, balance error costs more evenly.

2 Computational exercises

2.1 Exercise 5 — Constructing a confidence interval

A sample of 64 measurements yields:

Sample mean: $\bar{x} = 85$ units
Sample standard deviation: $s = 16$ units

Calculate the standard error.
Construct and interpret a 95% confidence interval for the population mean.

The critical value for a 95% CI with $df = 63$ is $t^* \approx 2.00$.

Solution

a) Standard error: \[SE = \frac{s}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2 \text{ units}\]

b) 95% confidence interval:

Margin of error: \[ME = t^* \times SE = 2.00 \times 2 = 4 \text{ units}\]

Confidence interval: \[\bar{x} \pm ME = 85 \pm 4 \implies [81,\ 89] \text{ units}\]

Interpretation: “We used a procedure that, if repeated many times, would contain the true population mean in 95% of constructed intervals. Based on this sample, the plausible range for the population mean is between 81 and 89 units.”

In R:

# With raw data
result <- t.test(your_data, conf.level = 0.95)
result$conf.int

# From summary statistics
n <- 64; x_bar <- 85; s <- 16
se <- s / sqrt(n)
t_star <- qt(0.975, df = n - 1)
c(x_bar - t_star * se, x_bar + t_star * se)

2.2 Exercise 6 — Running a hypothesis test

A service provider claims its average response time is 3 minutes. You observe 36 service interactions and find:

Sample mean: $\bar{x} = 3.5$ minutes
Sample standard deviation: $s = 1.2$ minutes

State appropriate two-sided hypotheses.
Calculate the test statistic.
Would you reject $H_0$ at $\alpha = 0.05$? The critical value for a two-sided test with $df = 35$ is $|t^*| = 2.03$.
State your conclusion in plain language.

Solution

a) Hypotheses:

\[H_0: \mu = 3 \quad \text{(claimed response time)}\] \[H_1: \mu \neq 3 \quad \text{(two-sided: could be faster or slower)}\]

b) Test statistic: \[SE = \frac{s}{\sqrt{n}} = \frac{1.2}{\sqrt{36}} = 0.2\] \[t = \frac{\bar{x} - \mu_0}{SE} = \frac{3.5 - 3.0}{0.2} = 2.5\]

c) Decision: $|t| = 2.5 > 2.03$ (critical value), so we reject $H_0$ at $\alpha = 0.05$.

Equivalently, the $p$-value ($\approx 0.017$) is below $\alpha$.

d) Conclusion: There is statistically significant evidence that the true average response time differs from the claimed 3 minutes. The sample suggests response times average 3.5 minutes — about 30 seconds longer than claimed. A 95% confidence interval for the true mean is approximately [3.1, 3.9] minutes, consistent with the rejection of $H_0 = 3$.

In R:

# With raw data
t.test(your_data, mu = 3, alternative = "two.sided")

2.3 Exercise 7 — Interpreting a two-sample comparison

Two training methods are compared on the same outcome measure:

Method A: $n = 40$, mean = 78, SD = 12
Method B: $n = 40$, mean = 82, SD = 12

The 95% confidence interval for the difference (B − A) is $[-1,\ 9]$.

What does this interval tell you about statistical significance?
Given that the methods cost the same to implement, what would you recommend?
What additional information would most improve this decision?

Solution

a) Statistical significance: The interval [−1, 9] includes zero, which means the difference is not statistically significant at $\alpha = 0.05$. We cannot confidently conclude that Method B outperforms Method A based on this data alone.

b) Recommendation: Two reasonable positions:

Conservative: Continue with Method A (current practice) until there is clearer evidence of improvement.
Pragmatic: Adopt Method B, since the point estimate favours it by 4 points and the downside is at most −1 (minimal risk), while the upside could be +9. Given equal costs and no meaningful downside, this is defensible.

Neither position is wrong — the interval shows genuine uncertainty, and context determines which error (acting too soon vs. waiting too long) is more costly.

c) Additional information that would help:

More data. With $n = 40$ per group and SD = 12, this study may have been underpowered to detect a 4-point difference. A larger sample would narrow the interval and clarify the result.
Effect size context. Is a 4-point difference meaningful for the outcome you care about? Statistical significance does not determine practical importance.
Secondary outcomes. Do the methods differ on other relevant dimensions (time, retention, scalability)?

3 Comments and questions

Citation

BibTeX citation:

@inreference{graebner2026-hypothesis-testing-exercises,
  author = {Gräbner-Radkowitsch, Claudius},
  editor = {Gräbner-Radkowitsch, Claudius},
  title = {Hypothesis {Testing} and {Confidence} {Intervals:} {Practice}
    {Exercises}},
  booktitle = {Method and Inquiry},
  date = {2026-06-16},
  url = {https://euf-methodology-hub.netlify.app/resources/hypothesis-testing-exercises},
  langid = {en}
}

For attribution, please cite this work as:

Gräbner-Radkowitsch, Claudius. 2026. “Hypothesis Testing and Confidence Intervals: Practice Exercises.” In Method and Inquiry, edited by Claudius Gräbner-Radkowitsch. June 16. https://euf-methodology-hub.netlify.app/resources/hypothesis-testing-exercises.

--- title: "Hypothesis Testing and Confidence Intervals: Practice Exercises" description: "Conceptual and computational exercises to test and reinforce your understanding of hypothesis testing, p-values, confidence intervals, and the connections between them." author: "Claudius Gräbner-Radkowitsch" date: 2026-06-16 date-modified: 2026-06-16 image: ../assets/images/statistics.png learning-objectives: - "Apply the five-step hypothesis testing procedure to a new problem" - "Diagnose common misinterpretations of p-values and confidence intervals" - "Distinguish between statistical significance and practical importance" - "Choose between Type I and Type II error priorities based on context" categories: - statistics - R - exercise - BA - MA level: - BA - MA content-type: exercise citation: id: graebner2026-hypothesis-testing-exercises type: entry-encyclopedia container-title: "Method and Inquiry" title: "Hypothesis Testing and Confidence Intervals: Practice Exercises" editor: - name: "Claudius Gräbner-Radkowitsch" url: https://euf-methodology-hub.netlify.app/resources/hypothesis-testing-exercises --- ::: {.callout-note} ## Companion resource These exercises accompany the conceptual resource [Hypothesis Testing and Confidence Intervals](hypothesis-testing-and-confidence-intervals.qmd). Work through that page first, then use these exercises to check your understanding. Solutions are hidden — try each exercise before revealing the answer. ::: # Conceptual exercises ## Exercise 1 — Interpreting a confidence interval You survey 200 participants and find an average rating of 7.8 out of 10, with a 95% confidence interval of [7.5, 8.1]. A colleague says: "We know that 95% of all participants have ratings between 7.5 and 8.1." What is wrong with this interpretation? What should your colleague say instead? ::: {.callout-tip} ## Solution **What is wrong:** The colleague is confusing the confidence interval for the *population mean* with the range where *individual observations* fall. The interval [7.5, 8.1] estimates where the true *average* rating lies across the whole population — not where individual participants score. **Correct interpretation:** "We used a procedure that, over repeated sampling, would contain the true average rating in 95% of constructed intervals. Based on this sample, a plausible range for the true population mean is 7.5 to 8.1." **The distinction:** If you wanted a range containing 95% of individual scores, you would need something like $\bar{x} \pm 2s$, which uses the standard deviation (not the standard error) and would produce a much wider interval. ::: ## Exercise 2 — Interpreting a p-value A study yields $p = 0.08$ at a pre-specified significance level of $\alpha = 0.05$. A colleague concludes: "The test failed — there's no effect." What is problematic about this conclusion? What would be a more appropriate interpretation? ::: {.callout-tip} ## Solution **Problems with the conclusion:** 1. "The test failed" — The test worked as intended; it simply did not produce sufficient evidence to reject $H_0$ at the chosen threshold. 2. "There's no effect" — Failing to find statistical significance is not evidence of absence. It could mean: (a) no effect exists, (b) an effect exists but the sample was too small to detect it, or (c) the effect exists but is smaller than this study was powered to find. **Better interpretation:** "At $\alpha = 0.05$, we do not have sufficient evidence to reject the null hypothesis. With $p = 0.08$, the result is borderline. We should consider the effect size and confidence interval, and avoid concluding that no effect exists." **Key reminder:** $p > \alpha$ means "insufficient evidence to reject $H_0$" — not "$H_0$ is true." ::: ## Exercise 3 — Statistical vs. practical significance Two independent studies test the same intervention: - **Study A:** $n = 50$, observed effect = 15%, $p = 0.08$ - **Study B:** $n = 50{,}000$, observed effect = 0.5%, $p = 0.001$ Which finding is more likely to be decision-relevant? Why? ::: {.callout-tip} ## Solution **Study A is more likely to be decision-relevant**, despite not crossing the conventional significance threshold. **Reasoning:** - Study A reports a 15% effect. The non-significant $p$-value is most likely a consequence of the small sample ($n = 50$), not an absence of effect. The effect size is large enough that it would almost certainly matter in practice if confirmed with more data. - Study B reports a 0.5% effect that is "highly significant" only because $n = 50{,}000$ gives the test enormous power to detect even negligible differences. A 0.5% improvement rarely justifies the costs of implementation and change. **The core lesson:** The $p$-value is a function of effect size, sample size, and variability. With large enough samples, trivially small effects become statistically significant. Always evaluate effect size and practical relevance alongside the $p$-value. ::: ## Exercise 4 — Type I and Type II errors A quality control process tests whether a production batch meets safety standards. - $H_0$: The batch meets standards - $H_1$: The batch is below standards For each of the following contexts, decide which type of error is more serious and explain why: a) A manufacturer of medical devices b) A manufacturer of children's toys c) A car manufacturer testing brake components ::: {.callout-tip} ## Solution In all three cases, the error types map as follows: - **Type I error (false positive):** Concluding the batch fails when it actually meets standards → unnecessary recall or production halt - **Type II error (false negative):** Concluding the batch passes when it is actually below standards → defective products reach consumers **a) Medical devices — Type II error is more serious.** A missed defect could directly harm patients, lead to regulatory action, and create massive liability. The cost of a false alarm (halting production unnecessarily) is high but safe. Use a more liberal $\alpha$ (e.g., 0.10) to reduce the Type II error rate. **b) Children's toys — Type II error is more serious**, though the stakes are lower than medical devices. Defective toys reaching children creates safety risk, liability, and reputational damage. A standard or slightly elevated $\alpha$ is appropriate. **c) Brake components — Type II error is more serious.** Missing a safety defect in braking systems risks accidents and fatalities. The cost of an unnecessary production halt is high but preferable. Use a more liberal $\alpha$ for safety-critical components. **General principle:** When safety is at stake, minimise false negatives even at the cost of more false positives. When only money is at risk, balance error costs more evenly. ::: # Computational exercises ## Exercise 5 — Constructing a confidence interval A sample of 64 measurements yields: - Sample mean: $\bar{x} = 85$ units - Sample standard deviation: $s = 16$ units a) Calculate the standard error. b) Construct and interpret a 95% confidence interval for the population mean. The critical value for a 95% CI with $df = 63$ is $t^* \approx 2.00$. ::: {.callout-tip} ## Solution **a) Standard error:** $$SE = \frac{s}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2 \text{ units}$$ **b) 95% confidence interval:** Margin of error: $$ME = t^* \times SE = 2.00 \times 2 = 4 \text{ units}$$ Confidence interval: $$\bar{x} \pm ME = 85 \pm 4 \implies [81,\ 89] \text{ units}$$ **Interpretation:** "We used a procedure that, if repeated many times, would contain the true population mean in 95% of constructed intervals. Based on this sample, the plausible range for the population mean is between 81 and 89 units." **In R:** ```r # With raw data result <- t.test(your_data, conf.level = 0.95) result$conf.int # From summary statistics n <- 64; x_bar <- 85; s <- 16 se <- s / sqrt(n) t_star <- qt(0.975, df = n - 1) c(x_bar - t_star * se, x_bar + t_star * se) ``` ::: ## Exercise 6 — Running a hypothesis test A service provider claims its average response time is 3 minutes. You observe 36 service interactions and find: - Sample mean: $\bar{x} = 3.5$ minutes - Sample standard deviation: $s = 1.2$ minutes a) State appropriate two-sided hypotheses. b) Calculate the test statistic. c) Would you reject $H_0$ at $\alpha = 0.05$? The critical value for a two-sided test with $df = 35$ is $|t^*| = 2.03$. d) State your conclusion in plain language. ::: {.callout-tip} ## Solution **a) Hypotheses:** $$H_0: \mu = 3 \quad \text{(claimed response time)}$$ $$H_1: \mu \neq 3 \quad \text{(two-sided: could be faster or slower)}$$ **b) Test statistic:** $$SE = \frac{s}{\sqrt{n}} = \frac{1.2}{\sqrt{36}} = 0.2$$ $$t = \frac{\bar{x} - \mu_0}{SE} = \frac{3.5 - 3.0}{0.2} = 2.5$$ **c) Decision:** $|t| = 2.5 > 2.03$ (critical value), so we **reject $H_0$** at $\alpha = 0.05$. Equivalently, the $p$-value ($\approx 0.017$) is below $\alpha$. **d) Conclusion:** There is statistically significant evidence that the true average response time differs from the claimed 3 minutes. The sample suggests response times average 3.5 minutes — about 30 seconds longer than claimed. A 95% confidence interval for the true mean is approximately [3.1, 3.9] minutes, consistent with the rejection of $H_0 = 3$. **In R:** ```r # With raw data t.test(your_data, mu = 3, alternative = "two.sided") ``` ::: ## Exercise 7 — Interpreting a two-sample comparison Two training methods are compared on the same outcome measure: - Method A: $n = 40$, mean = 78, SD = 12 - Method B: $n = 40$, mean = 82, SD = 12 The 95% confidence interval for the difference (B − A) is $[-1,\ 9]$. a) What does this interval tell you about statistical significance? b) Given that the methods cost the same to implement, what would you recommend? c) What additional information would most improve this decision? ::: {.callout-tip} ## Solution **a) Statistical significance:** The interval [−1, 9] **includes zero**, which means the difference is not statistically significant at $\alpha = 0.05$. We cannot confidently conclude that Method B outperforms Method A based on this data alone. **b) Recommendation:** Two reasonable positions: - **Conservative:** Continue with Method A (current practice) until there is clearer evidence of improvement. - **Pragmatic:** Adopt Method B, since the point estimate favours it by 4 points and the downside is at most −1 (minimal risk), while the upside could be +9. Given equal costs and no meaningful downside, this is defensible. Neither position is wrong — the interval shows genuine uncertainty, and context determines which error (acting too soon vs. waiting too long) is more costly. **c) Additional information that would help:** 1. **More data.** With $n = 40$ per group and SD = 12, this study may have been underpowered to detect a 4-point difference. A larger sample would narrow the interval and clarify the result. 2. **Effect size context.** Is a 4-point difference meaningful for the outcome you care about? Statistical significance does not determine practical importance. 3. **Secondary outcomes.** Do the methods differ on other relevant dimensions (time, retention, scalability)? ::: # Comments and questions <script src="https://giscus.app/client.js" data-repo="graebnerc/research-methodology-hub" data-repo-id="R_kgDOS7J-4w" data-category="Comments" data-category-id="DIC_kwDOS7J-484C_N51" data-mapping="pathname" data-strict="0" data-reactions-enabled="1" data-emit-metadata="0" data-input-position="bottom" data-theme="preferred_color_scheme" data-lang="en" crossorigin="anonymous" async> </script>