Confidence intervals and hypothesis tests are two main tools for making conclusions from sample data. A confidence interval gives a range of plausible values for a population parameter, such as a mean or proportion. A hypothesis test checks whether the data provide strong enough evidence against a specific null value.
They matter because real data vary from sample to sample, so statistical conclusions must include uncertainty.
Key Facts
- A 95% confidence interval has the form estimate ± critical value × standard error.
- For a two-sided test, if a 95% confidence interval excludes the null value, then the test rejects H0 at α = 0.05.
- For a two-sided test, if a 99% confidence interval excludes the null value, then the test rejects H0 at α = 0.01.
- Test statistic = (estimate - null value) / standard error.
- A p-value is the probability, assuming H0 is true, of getting a result at least as extreme as the observed result.
- Confidence intervals show effect size and uncertainty, while hypothesis tests give a yes or no decision at a chosen significance level.
Vocabulary
- Confidence interval
- A range of values calculated from sample data that is used to estimate a population parameter with a stated confidence level.
- Null hypothesis
- A claim used as the starting assumption in a hypothesis test, often stating no effect, no difference, or a specific parameter value.
- Significance level
- The cutoff probability α for deciding whether sample evidence is strong enough to reject the null hypothesis.
- P-value
- The probability of observing a result as extreme as the sample result, or more extreme, if the null hypothesis is true.
- Standard error
- A measure of how much a sample estimate is expected to vary from sample to sample.
Common Mistakes to Avoid
- Saying a 95% confidence interval has a 95% chance of containing this fixed parameter is wrong because the parameter is not random in the usual frequentist interpretation. The method captures the true parameter in 95% of repeated samples.
- Using the wrong confidence level for the test is wrong because the connection depends on matching levels. A 95% confidence interval matches a two-sided test with α = 0.05, not α = 0.01.
- Rejecting H0 when the null value is inside the confidence interval is wrong for a matching two-sided test. If the null value is included, the interval does not show enough evidence against that value at the matching significance level.
- Reporting only the p-value is incomplete because it does not show the size or precision of the effect. A confidence interval helps show which parameter values are plausible.
Practice Questions
- 1 A sample gives a 95% confidence interval for a population mean of 12.4 to 18.6. For H0: μ = 10 versus Ha: μ ≠ 10 at α = 0.05, should you reject H0? Explain using the interval.
- 2 A study estimates a difference in proportions as 0.08 with a standard error of 0.03. Using the approximate 95% confidence interval estimate ± 1.96 × standard error, compute the interval and decide whether a two-sided test of H0: difference = 0 rejects at α = 0.05.
- 3 Two studies have the same p-value of 0.04, but one has a narrow confidence interval and the other has a wide confidence interval. Explain why reporting both the p-value and the confidence interval gives a better statistical summary.