Confidence Intervals & Hypothesis Testing Practice Questions, Quiz, and Step-by-Step Lesson - improve your math skills with focused questions and clear explanations.
Confidence Intervals & Hypothesis Testing Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice confidence intervals and hypothesis testing with the most important statistics tools: confidence level \((1-\alpha)\), critical values (\(z^\*\), \(t^\*\), and \(\chi^2\) quantiles), and margin of error \(\text{ME}=z^\*\mathrm{SE}\); standard error and how sample size changes interval width; z confidence intervals and t confidence intervals for a mean \(\mu\) (including paired t methods); confidence intervals for a proportion \(\hat p\) and for a variance \(\sigma^2\) using the chi-square distribution; and the full hypothesis testing workflow: null and alternative hypotheses, test statistics (z, t, and \(\chi^2\)), p-values, significance level \(\alpha\), and decision-making that connects tests to confidence intervals. You will also strengthen core ideas like Type I vs. Type II error, statistical power, and when to use chi-square goodness-of-fit and chi-square independence tests. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
How this confidence intervals & hypothesis testing practice works
1. Take the quiz: answer the confidence intervals and hypothesis testing questions at the top of the page.
2. Open the lesson (optional): review confidence interval formulas, critical values, margin of error, and hypothesis testing steps with clear examples.
3. Retry: return to the quiz and apply the CI and hypothesis testing rules immediately.
What you will learn in the confidence intervals & hypothesis testing lesson
Confidence interval fundamentals
General CI structure: estimate \(\pm\) (critical value)\(\times\)(standard error)
Margin of error and standard error: how variability and \(n\) control precision
CI width: how confidence level and sample size affect interval width
Confidence intervals for means
z-interval for a mean (known \(\sigma\)): \(\bar x \pm z_{1-\alpha/2}\dfrac{\sigma}{\sqrt{n}}\)
t-interval for a mean (unknown \(\sigma\)): \(\bar x \pm t_{1-\alpha/2,\;n-1}\dfrac{s}{\sqrt{n}}\)
Paired t confidence intervals using differences \(d_i\) and \(df=n-1\)
Proportions and variance intervals
Proportion CI: \(\hat p \pm z_{1-\alpha/2}\sqrt{\hat p(1-\hat p)/n}\) (large-sample conditions)
Variance CI via chi-square: uses quantiles of \(\chi^2_{n-1}\) (normal population assumption)
Reading CI outputs and interpreting parameters \(\mu\), \(p\), and \(\sigma^2\) correctly
Hypothesis testing: z, t, and chi-square
Hypothesis testing steps: \(H_0\), \(H_1\), \(\alpha\), test statistic, p-value, conclusion
Common tests: one-sample z-test, one-sample/paired t-test, chi-square goodness-of-fit and independence tests
Errors and power: Type I error, Type II error, and how increasing \(n\) increases power
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing confidence intervals and hypothesis testing.
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Confidence Intervals
Hypothesis testing guide
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Confidence Intervals & Hypothesis Testing Lesson
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Lesson Overview
Confidence Intervals & Hypothesis Testing
Purpose: Build a clear understanding of confidence intervals and hypothesis testing so you can choose the right method, compute results correctly, and interpret conclusions responsibly. You will practice the “core loop” of inference: pick a parameter (like \(\mu\), \(p\), or \(\sigma^2\)), compute an estimate and a standard error, use a critical value to build a confidence interval, and use a test statistic and p-value to run a hypothesis test at significance level \(\alpha\).
Success criteria
Interpret a \(100(1-\alpha)\%\) confidence interval as a long-run capture rate (not a probability about \(\mu\) after observing data).
Use the general CI form: estimate \(\pm\) critical value \(\times\) standard error.
Compute margin of error: \(\text{ME}=(\text{critical value})\times \mathrm{SE}\).
Choose z vs t for a mean and identify degrees of freedom for t intervals/tests.
Build a confidence interval for a proportion using \(\hat p\) and \(\sqrt{\hat p(1-\hat p)/n}\).
Build a confidence interval for a variance \(\sigma^2\) using the chi-square distribution (normal population assumption).
Run a hypothesis test: write \(H_0\) and \(H_1\), compute a test statistic (z, t, or \(\chi^2\)), find a p-value, and decide at level \(\alpha\).
Use the key connection: for a two-sided test at level \(\alpha\), reject \(H_0\!:\theta=\theta_0\) if \(\theta_0\) is outside the \(100(1-\alpha)\%\) CI.
Explain Type I error, Type II error, and how increasing sample size affects power.
Key vocabulary
Parameter: an unknown population value (like \(\mu\), \(p\), or \(\sigma^2\)).
Statistic / estimate: a number computed from data (like \(\bar x\), \(\hat p\), \(s^2\)).
Standard error (SE): the standard deviation of an estimator (often estimated from data).
Critical value: a quantile such as \(z_{1-\alpha/2}\) or \(t_{1-\alpha/2,\;df}\).
Margin of error: \(\text{ME}=(\text{critical value})\times \mathrm{SE}\).
Null / alternative hypotheses: \(H_0\) vs \(H_1\) statements about a parameter.
p-value: probability (under \(H_0\)) of a result at least as extreme as what you observed.
Type I error: rejecting a true \(H_0\) (probability \(\alpha\)).
Type II error: failing to reject a false \(H_0\) (probability \(\beta\)); power is \(1-\beta\).
Quick pre-check
Pre-check 1: Which statement best describes the meaning of a 95% confidence interval for a population mean \(\mu\)?
Hint: The randomness is in the interval (because it comes from random samples), not in \(\mu\).
Pre-check 2: Which describes a Type I error?
Hint: Type I error is a “false positive”: you reject a true null hypothesis.
Confidence Interval Basics
Confidence intervals, critical values, standard error, and margin of error
Learning goal: Build any common confidence interval using the same structure and interpret it correctly.
Key idea
Most confidence intervals follow the same blueprint: \[\text{CI} = \text{estimate} \pm (\text{critical value})\times(\text{standard error}).\] The margin of error is: \[\text{ME}=(\text{critical value})\times \mathrm{SE}.\] A higher confidence level (like 99% vs 95%) uses a larger critical value, which makes the interval wider. Larger sample size usually makes \(\mathrm{SE}\) smaller (often proportional to \(1/\sqrt{n}\)), which makes the interval narrower.
Critical values you’ll see often
z critical value: \(z_{1-\alpha/2}\) for two-sided confidence intervals when the sampling distribution is (approximately) normal.
t critical value: \(t_{1-\alpha/2,\;df}\) for means when \(\sigma\) is unknown (common in practice).
One-sided bounds: for a lower bound with confidence \(1-\alpha\), the critical value is \(z_{1-\alpha}\) (or \(t_{1-\alpha,\;df}\)).
Worked example
Example: A sample has \(\bar x=72\), known \(\sigma=12\), and \(n=36\). Find the 95% confidence interval for \(\mu\).
For 95%, use \(z_{0.975}\approx 1.96\). The standard error is \(\sigma/\sqrt{n}=12/\sqrt{36}=12/6=2\). So the margin of error is: \[\text{ME}=1.96(2)=3.92.\] The CI is: \[72\pm 3.92 \Rightarrow (68.08,\;75.92).\]
Try it
Try it 1: Approximate the margin of error if \(z^\*=1.96\) and \(\mathrm{SE}=0.5\).
Hint: Margin of error \(=\) \(z^\*\times \mathrm{SE}\).
Try it 2: What is the critical \(z\)-value for an 80% confidence interval?
Hint: 80% means \(\alpha=0.20\), so each tail has \(\alpha/2=0.10\). Use \(z_{1-\alpha/2}=z_{0.90}\).
Summary
General CI: estimate \(\pm\) (critical value)\(\times\)(SE).
Margin of error: \(\text{ME}=(\text{critical value})\times \mathrm{SE}\).
CI Width & Planning
What changes confidence interval width? (and how to plan \(n\))
Learning goal: Predict how confidence level and sample size affect CI width and solve for a required sample size.
Key idea
Confidence interval width is controlled by two pieces:
Standard error: larger \(n\Rightarrow\) smaller \(\mathrm{SE}\Rightarrow\) narrower interval. For many estimators, \(\mathrm{SE}\propto 1/\sqrt{n}\).
A common planning formula comes from \(\text{ME}=z^\*\sigma/\sqrt{n}\) (mean with known \(\sigma\)): \[n=\left(\frac{z^\*\sigma}{\text{ME}}\right)^2.\] This “square rule” explains why reducing margin of error can require much larger samples.
Worked example: sample size for a target margin of error
Example: You want a 95% CI for \(\mu\) with known \(\sigma=10\) and margin of error at most \(2\). What sample size \(n\) is needed?
Use \(z^\*\approx 1.96\): \[n=\left(\frac{1.96(10)}{2}\right)^2=\left(9.8\right)^2=96.04.\] Round up: \(n=97\).
Try it
Try it 1: Increasing the confidence level from 95% to 99% has what effect on the confidence interval width (all else equal)?
Hint: Higher confidence uses a larger critical value (like 2.576 instead of 1.96 for z).
Try it 2: To double the width of a confidence interval (holding confidence level and \(\sigma\) fixed), by what factor must \(n\) change?
Hint: Width is proportional to \(1/\sqrt{n}\). To double width, \(\sqrt{n}\) must be cut in half.
Summary
Higher confidence \(\Rightarrow\) wider CI (bigger critical value).
Larger sample size \(\Rightarrow\) narrower CI (smaller SE, often \(1/\sqrt{n}\)).
For many problems, planning \(n\) comes from \(n=\left(\frac{\text{critical}\times \sigma}{\text{ME}}\right)^2\).
Mean CIs & Paired t
Confidence intervals for a mean: z vs t, and paired t ideas
Learning goal: Choose the correct distribution and compute confidence intervals for \(\mu\), including paired designs.
Key idea
For a population mean \(\mu\), the confidence interval depends on whether \(\sigma\) is known:
Known \(\sigma\) (z interval): \[\bar x \pm z_{1-\alpha/2}\frac{\sigma}{\sqrt{n}}.\]
Unknown \(\sigma\) (t interval): \[\bar x \pm t_{1-\alpha/2,\;n-1}\frac{s}{\sqrt{n}}.\] Here \(df=n-1\). As \(df\) increases, the t distribution approaches the standard normal.
Paired t intervals treat each pair as one observation by using differences \(d_i\) (for example, “after − before”). Compute \(\bar d\) and \(s_d\), then use: \[\bar d \pm t_{1-\alpha/2,\;n-1}\frac{s_d}{\sqrt{n}},\] where \(n\) is the number of pairs.
Worked example (t interval)
Example: A sample has \(\bar x=15\), \(s=4\), and \(n=16\). Write the 95% t-interval for \(\mu\).
Degrees of freedom: \(df=16-1=15\). The interval is: \[15 \pm t_{0.975,\;15}\frac{4}{\sqrt{16}} = 15 \pm t_{0.975,\;15}(1).\] Numerically, \(t_{0.975,\;15}\approx 2.13\), so the CI is approximately \(15\pm 2.13\).
Try it
Try it 1: Which formula gives the test statistic \(z\) for testing a mean when \(\sigma\) is known?
Hint: Known \(\sigma\) uses \(\sigma/\sqrt{n}\) in the denominator.
Try it 2: For a paired t-test with \(n\) pairs, the degrees of freedom equal:
Hint: Paired t uses the differences \(d_i\) as one sample of size \(n\).
Summary
Known \(\sigma\): z interval and z test use \(\sigma/\sqrt{n}\).
Unknown \(\sigma\): t interval and t test use \(s/\sqrt{n}\) with \(df=n-1\).
Paired t focuses on differences \(d_i\) and uses \(df=n-1\).
Proportion & Variance CIs
Confidence intervals for a proportion \(\,p\) and a variance \(\,\sigma^2\)
Learning goal: Build confidence intervals for proportions and variances, and know which distribution provides the critical values.
Proportion confidence interval (large-sample)
For a one-sample proportion, let \(\hat p=\dfrac{x}{n}\), where \(x\) is the number of successes. When conditions for a normal approximation hold (a common rule of thumb is \(n\hat p\ge 10\) and \(n(1-\hat p)\ge 10\)), an approximate \(100(1-\alpha)\%\) CI is: \[\hat p \pm z_{1-\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}.\]
Variance confidence interval (chi-square)
If the population is normally distributed, then \[\frac{(n-1)s^2}{\sigma^2}\sim \chi^2_{n-1}.\] A \(100(1-\alpha)\%\) CI for \(\sigma^2\) is: \[\left(\frac{(n-1)s^2}{\chi^2_{1-\alpha/2,\;n-1}},\;\frac{(n-1)s^2}{\chi^2_{\alpha/2,\;n-1}}\right).\]
Worked example (proportion CI setup)
Example: In a sample of \(n=120\) people, \(x=84\) prefer brand A. Find \(\hat p\) and write the 95% CI setup for \(p\).
\[\hat p=\frac{84}{120}=0.70.\] The 95% CI setup is: \[0.70 \pm 1.96\sqrt{\frac{0.70(0.30)}{120}}.\]
Try it
Try it 1: For a one-sample proportion CI with large \(n\), the approximate CI is \(\hat p \pm z_{1-\alpha/2}\sqrt{\hat p(1-\hat p)/n}\). What does \(\hat p\) represent?
Hint: \(\hat p\) is computed directly from your sample as a point estimate for \(p\).
Try it 2: A 95% confidence interval for a variance \(\sigma^2\) uses quantiles of which distribution?
Hint: \(\frac{(n-1)s^2}{\sigma^2}\) follows a \(\chi^2\) distribution when the population is normal.
Summary
Proportion CI uses \(\hat p=x/n\) and a z critical value (large-sample conditions).
Variance CI uses chi-square quantiles and assumes a normal population.
Hypothesis Testing Basics
Hypothesis testing: \(H_0\), \(H_1\), test statistic, p-value, and the CI connection
Learning goal: Run a correct hypothesis test and connect it to confidence intervals.
The standard testing workflow
1. State hypotheses: \(H_0:\theta=\theta_0\) vs \(H_1:\theta≠\theta_0\) (two-sided) or \(H_1:\theta>\theta_0\), \(H_1:\theta<\theta_0\) (one-sided).
3. Compute a test statistic: z, t, or \(\chi^2\) depending on the setting.
4. Compute a p-value and decide: reject \(H_0\) if p-value \(\le \alpha\); otherwise fail to reject.
Connection to confidence intervals
For a two-sided test at level \(\alpha\), there is a tight link: reject \(H_0:\theta=\theta_0\) if and only if \(\theta_0\) is outside the \(100(1-\alpha)\%\) confidence interval for \(\theta\).
One-sided critical values
A 95% lower confidence bound corresponds to \(\alpha=0.05\) one-sided and uses the critical value \(z_{0.95}\approx 1.645\) (or \(t_{0.95,df}\)). A common lower bound form for a mean with known \(\sigma\) is: \[L=\bar x - z_{0.95}\frac{\sigma}{\sqrt{n}}.\]
Worked example (z test setup)
Example: Test \(H_0:\mu=50\) vs \(H_1:\mu≠ 50\) with \(\bar x=52\), known \(\sigma=10\), \(n=25\).
Test statistic: \[z=\frac{\bar x-\mu_0}{\sigma/\sqrt{n}}=\frac{52-50}{10/5}=\frac{2}{2}=1.\] A two-sided p-value is \(2(1-\Phi(1))\approx 0.317\), so at \(\alpha=0.05\) we fail to reject \(H_0\).
Try it
Try it 1: If a 95% confidence interval for \(\mu\) excludes \(\mu_0\), what is the hypothesis test conclusion at \(\alpha=0.05\) (two-sided)?
Hint: Two-sided test at \(\alpha\) matches a \(100(1-\alpha)\%\) confidence interval.
Try it 2: What is the one-sided critical \(z\)-value for a 95% lower confidence bound?
Hint: For a one-sided 95% bound, use \(z_{1-\alpha}=z_{0.95}\).
Summary
Reject \(H_0\) if p-value \(\le \alpha\); otherwise fail to reject.
Two-sided test at \(\alpha\) matches a \(100(1-\alpha)\%\) CI: \(\theta_0\) outside the CI \(\Rightarrow\) reject \(H_0\).
One-sided 95% critical z value is \(z_{0.95}\approx 1.645\).
Common Tests
Which test should you use? (z, t, and chi-square)
Learning goal: Match a real problem to the correct test and know the core test statistic formulas.
Quick “which test?” guide
Mean vs known value, \(\sigma\) known:one-sample z test.
Mean vs known value, \(\sigma\) unknown:one-sample t test with \(df=n-1\).
Paired measurements:paired t test on differences \(d_i\).
Proportion vs known value:one-sample z test for a proportion (large-sample conditions).
Variance vs known value:chi-square test for one variance (normal population assumption).
Categorical counts:chi-square goodness-of-fit or chi-square test of independence.
One-sample variance test statistic
To test \(H_0:\sigma^2=\sigma_0^2\) using a normally distributed population: \[\chi^2=\frac{(n-1)s^2}{\sigma_0^2}\sim \chi^2_{n-1}\quad \text{under } H_0.\]
Worked example (test statistic only)
Example: A sample has \(n=21\) and \(s^2=16\). Under \(H_0:\sigma^2=9\), what is the chi-square test statistic?
Try it 1: Which test compares the mean of one group to a known value when \(\sigma\) is known?
Hint: Known \(\sigma\) \(\Rightarrow\) use z methods for a mean.
Try it 2: The test statistic for a one-sample variance test of \(H_0:\sigma^2=\sigma_0^2\) is:
Hint: One-variance inference uses \(\chi^2\) with \(df=n-1\) under a normal population model.
Summary
Known \(\sigma\): one-sample z test for \(\mu\).
Unknown \(\sigma\): one-sample t test for \(\mu\) (and paired t for differences).
Variance tests and variance CIs use the chi-square distribution under normality.
Categorical counts often use chi-square tests (goodness-of-fit or independence).
Power & Big Picture
Type I & Type II errors, power, and why sample size matters
Learning goal: Understand error tradeoffs and how sample size affects both confidence intervals and hypothesis tests — then finish with a final check.
Errors and power in one picture
Type I error (false positive): reject a true \(H_0\). Probability \(=\alpha\).
Type II error (false negative): fail to reject a false \(H_0\). Probability \(=\beta\).
Power: \(1-\beta\). This is the chance you correctly detect a real effect.
How sample size affects inference
Confidence intervals: larger \(n\Rightarrow\) smaller \(\mathrm{SE}\Rightarrow\) narrower CI (more precision).
Hypothesis tests: larger \(n\Rightarrow\) smaller \(\mathrm{SE}\Rightarrow\) larger test statistic magnitude (for a fixed effect) and therefore higher power.
Extra note: chi-square tests and survival curves
In categorical data, you’ll often see chi-square goodness-of-fit and chi-square independence tests. In survival analysis, a common test to compare survival curves between two groups is the log-rank test, which is typically reported using a chi-square reference distribution.
Try it
Try it 1: Increasing the sample size in a hypothesis test primarily increases which of the following?
Hint: Larger \(n\) reduces standard error, making it easier to detect real differences.
Try it 2: Which test assesses goodness-of-fit to a categorical distribution?
Hint: Goodness-of-fit compares observed counts to expected counts from a specified categorical distribution.
Final recap
CI blueprint: estimate \(\pm\) (critical value)\(\times\)(SE), with \(\text{ME}=(\text{critical})\times \mathrm{SE}\).
Mean CIs: z interval uses \(\sigma\); t interval uses \(s\) with \(df=n-1\).
Proportion CI: \(\hat p \pm z_{1-\alpha/2}\sqrt{\hat p(1-\hat p)/n}\) under large-sample conditions.
Variance CI: chi-square quantiles, \(\left(\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}},\frac{(n-1)s^2}{\chi^2_{\alpha/2}}\right)\) under normality.
Testing: set \(H_0\) and \(H_1\), choose \(\alpha\), compute test statistic and p-value, then decide.
Power: increasing \(n\) tends to increase power by reducing standard error.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the confidence interval or hypothesis testing skill you need.
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