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 at a chosen significance level . A confidence interval estimates a population parameter using the form .
Both methods depend on checking randomness, independence, and an appropriate sampling distribution.
Key Facts
- The null hypothesis states the default claim, while the alternative hypothesis states the claim being tested.
- A common test statistic formula is .
- For a one-sample mean with unknown population standard deviation, use with .
- For a one-proportion confidence interval, use .
- For a one-mean confidence interval, use .
- If the p-value is less than or equal to , reject ; if the p-value is greater than , fail to reject .
- A Type I error happens when you reject a true , and a Type II error happens when you fail to reject a false .
- For a one-proportion procedure, the large counts condition is usually checked with and for intervals.
Vocabulary
- Null hypothesis
- The null hypothesis is the default statement that there is no change, no effect, or a specific population value.
- Alternative hypothesis
- The alternative hypothesis is the statement that the data may support instead of .
- P-value
- The p-value is the probability of getting a result at least as extreme as the sample result, assuming is true.
- Significance level
- The significance level is the cutoff probability used to decide whether evidence against 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 confidence interval has a 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 of repeated samples.
- Accepting after a large p-value is wrong because a test can only fail to reject . A large p-value means the sample does not give strong evidence against .
- Using instead of for a mean when is unknown is wrong because the sample standard deviation adds uncertainty. Use a distribution with .
- 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 students finds that prefer online homework. Find a confidence interval for the true proportion of students who prefer online homework using .
- 2 A sample of test scores has and . Test against by finding the test statistic .
- 3 A test gives a p-value of at significance level . State the correct decision about 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.
Understanding Hypothesis Testing & Confidence Intervals
A test begins by turning a real claim into a parameter statement. Suppose a school says that at least 80 percent of students support a later start time. The parameter is the true proportion of all students at that school who support it, not the proportion in one survey.
The direction of the claim matters. A claim about greater than, less than, or different from leads to different evidence rules. A test for different from looks for unusual results on either side of the claimed value.
A test for greater than looks only above it. Students often lose points by choosing a two sided test when the stated claim has a direction.
The p value is not the chance that the null hypothesis is true. It measures how surprising the sample result would be if the null hypothesis were true. A small p value means the observed result, or one more extreme in the direction of the alternative, would be rare under the default claim.
This is evidence against that claim, but it is not proof. Random samples can produce unusual results.
A large p value does not show that the null hypothesis has been proven. It only means the sample did not provide strong enough evidence to reject it at the chosen significance level.
Confidence intervals tell a related story from a different angle. A 95 percent confidence interval is built by a method that captures the true parameter in about 95 percent of many repeated random samples. Once one particular interval has been calculated, the parameter is fixed.
The interval either contains it or does not. Wider intervals give less precise estimates. They become narrower when the sample size grows because the standard error becomes smaller.
Higher confidence makes an interval wider because the method needs a larger safety margin. In many common two sided situations, a null value outside a 95 percent confidence interval would be rejected by a test at the 5 percent significance level.
Good calculations cannot repair bad data. Random selection helps reduce bias, while a larger sample mainly reduces random variation. A survey of 5,000 volunteers can still be misleading if the volunteers differ from the population of interest.
Independence matters too. Sampling more than about 10 percent of a population without replacement can make observations too connected for simple procedures. For means, students should examine graphs for strong skewness or outliers, especially with small samples.
The choice between a z procedure and a t procedure depends on what is known about population variation. In practice, tests guide decisions about medicine, manufacturing, polling, and school policies. A Type I error can cause an unnecessary action.
A Type II error can miss a real problem. Lowering the significance level reduces false alarms, but it can make real effects harder to detect. Larger, well designed samples improve the ability to find meaningful effects.