Hypothesis Testing & Confidence Intervals Cheat Sheet

A printable reference covering null and alternative hypotheses, p-values, confidence intervals, critical values, z-tests, t-tests, and errors for grades 11-12.

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Hypothesis testing and confidence intervals help students make decisions from sample data when the full population is unknown. This cheat sheet organizes the main ideas, formulas, and conditions used in high school statistics. It is useful for reviewing test setup, choosing a method, and interpreting results correctly. Students need it because many statistics problems require both calculation and careful wording. The most important ideas are the null hypothesis, the alternative hypothesis, the test statistic, the p-value, and the confidence interval. A hypothesis test asks whether sample evidence is strong enough to reject $H_0$ at a chosen significance level $\alpha$ . A confidence interval estimates a population parameter using the form $\text{estimate} \pm \text{margin of error}$ . Both methods depend on checking randomness, independence, and an appropriate sampling distribution.

Key Facts

The null hypothesis $H_0$ states the default claim, while the alternative hypothesis $H_a$ states the claim being tested.
A common test statistic formula is $z = \frac{\text{statistic} - \text{parameter}}{\text{standard error}}$ .
For a one-sample mean with unknown population standard deviation, use $t = \frac{\bar{x} - \mu_0}{s\/\sqrt{n}}$ with $df = n - 1$ .
For a one-proportion confidence interval, use $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ .
For a one-mean confidence interval, use $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$ .
If the p-value is less than or equal to $\alpha$ , reject $H_0$ ; if the p-value is greater than $\alpha$ , fail to reject $H_0$ .
A Type I error happens when you reject a true $H_0$ , and a Type II error happens when you fail to reject a false $H_0$ .
For a one-proportion procedure, the large counts condition is usually checked with $n\hat{p} \ge 10$ and $n(1 - \hat{p}) \ge 10$ for intervals.

Vocabulary

Null hypothesis: The null hypothesis $H_0$ is the default statement that there is no change, no effect, or a specific population value.
Alternative hypothesis: The alternative hypothesis $H_a$ is the statement that the data may support instead of $H_0$ .
P-value: The p-value is the probability of getting a result at least as extreme as the sample result, assuming $H_0$ is true.
Significance level: The significance level $\alpha$ is the cutoff probability used to decide whether evidence against $H_0$ is strong enough.
Confidence interval: A confidence interval is a range of plausible values for a population parameter based on sample data.
Margin of error: The margin of error is the amount added and subtracted from the sample estimate in a confidence interval.

Common Mistakes to Avoid

Saying that a $95\%$ confidence interval has a $95\%$ chance of containing the true parameter is wrong because the true parameter is fixed. The correct idea is that the method captures the true parameter in about $95\%$ of repeated samples.
Accepting $H_0$ after a large p-value is wrong because a test can only fail to reject $H_0$ . A large p-value means the sample does not give strong evidence against $H_0$ .
Using $z$ instead of $t$ for a mean when $\sigma$ is unknown is wrong because the sample standard deviation $s$ adds uncertainty. Use a $t$ distribution with $df = n - 1$ .
Forgetting to check conditions is wrong because formulas depend on assumptions about randomness, independence, and the sampling distribution. Always verify the conditions before interpreting results.
Confusing statistical significance with practical importance is wrong because a very small effect can be significant with a large sample size. Always interpret the size and context of the effect.

Practice Questions

1 A sample of $200$ students finds that $124$ prefer online homework. Find a $95\%$ confidence interval for the true proportion of students who prefer online homework using $z^* = 1.96$ .
2 A sample of $25$ test scores has $\bar{x} = 72$ and $s = 10$ . Test $H_0: \mu = 68$ against $H_a: \mu > 68$ by finding the test statistic $t = \frac{\bar{x} - \mu_0}{s\/\sqrt{n}}$ .
3 A test gives a p-value of $0.032$ at significance level $\alpha = 0.05$ . State the correct decision about $H_0$ and write one sentence interpreting the result in context.
4 Explain why a wider confidence interval can result from using a higher confidence level, even when the sample data stay the same.