MZB127 Chap.10 Confidence Intervals and Hypothesis Tests
Confidence Intervals and Hypothesis Tests
Confidence intervals quantify plausible parameter values through a repeated-sampling procedure. Hypothesis tests compare a sample statistic with a null value and convert its extremeness into a direction-specific p-value. The chapter links t intervals and one-sample t tests, distinguishes rejection from proof, and keeps the parameter, significance level, tail and context aligned throughout the calculation and final conclusion.
The alternative hypothesis determines whether extremeness lies in an upper tail, lower tail or both tails, so it must be specified before inspecting the statistic. A test statistic measures departure from the null in standard-error units, and the p-value describes how surprising an equally or more extreme result would be under that null model.
Comparing it with alpha gives a decision, not a probability that either hypothesis is true. A confidence interval centres on a sample estimate and extends by a critical value times its standard error. Matching two-sided intervals and tests provide consistent decisions when their levels and assumptions agree. Sample size, variability and design conditions govern precision and power.
Every final claim must concern the population parameter, preserve the tested direction and avoid treating non-rejection as proof of equality.
What this chapter covers
- 01
Confidence level and margin of error
- 02
One-sample t intervals
- 03
Null and alternative hypotheses
- 04
Test statistics and p-values
- 05
Decisions, errors and contextual conclusions
Worked example · free
Test a population mean
- 1State the null and upper-tail alternative.
- 1Calculate standard error one.
- 1Calculate t=2 with 15 degrees of freedom.
- 1Obtain the upper-tail p-value near 0.032.
- 1Reject and state evidence for a larger population mean.
Key terms
- Confidence level
- The long-run capture rate of intervals built by the stated procedure.
- Margin of error
- The critical value multiplied by the standard error in a symmetric interval.
- Null hypothesis
- The parameter claim used to generate the reference distribution for a test.
- Alternative hypothesis
- The direction or difference the test is designed to detect.
- P-value
- The null-model probability of a result at least as extreme in the alternative direction.
Confidence Intervals and Hypothesis Tests FAQ
Does a large p-value prove the null?
No. It means the sample does not provide enough evidence to reject the null at the chosen level. Limited sample size or high variability can also make departures difficult to detect.
How should a confidence interval be interpreted?
Treat the endpoints as plausible values produced by a procedure with the stated long-run confidence level. The fixed parameter and calculated endpoints are not random after the data have been observed.
Why must the alternative be stated first?
The alternative determines whether one tail or both tails count as at least as extreme. Choosing it after seeing the statistic changes the reference probability and undermines the planned test.
How can an interval calculation be checked?
Its endpoints should be symmetric around the point estimate when the standard form estimate plus or minus margin is used. Their midpoint must recover that estimate, and the half-width must equal the critical value times the standard error. Units should match the population parameter rather than the squared variance units.
Exam move
Write the parameter, null value, alternative direction and alpha before calculating. Decide whether z or t applies and show the standard error separately. Sketch the p-value region, compare p with alpha and use “reject” or “do not reject” precisely. For intervals, check centre, symmetry and units.
Practise translating every numerical decision into one restrained contextual sentence that neither proves the null nor assigns a probability to a fixed hypothesis.
Use a fixed response template while learning: name the population parameter, state both hypotheses, record alpha, verify assumptions, compute the standard error, calculate the statistic, identify the tail, obtain the p-value, make the decision and write the contextual conclusion. Keep the estimate, null value and standard error on separate lines so numerator signs can be audited.
Draw the relevant tail region before reading a table, especially for negative statistics or two-sided alternatives. For intervals, calculate the margin independently and verify that the endpoints average back to the point estimate. Check that interval units match the parameter and that a matched two-sided test gives the corresponding inclusion or exclusion of the null value.
Create correction exercises in which the arithmetic is right but the conclusion says accepts, proves, causes or refers to the sample statistic; rewrite each claim at the proper evidential strength. Compare Type I error, Type II error and power through concrete decisions rather than isolated definitions.
In timed work, reserve a final minute to confirm degrees of freedom, tail direction, p versus alpha and the population wording. These four checks catch many conclusions that otherwise look numerically polished.
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