GEA1000 Chap.5 Statistical Inference: Intervals and Tests
Statistical Inference: Intervals and Tests
Intervals show precision around an estimate
A confidence interval combines a sample estimate with a margin based on standard error. Its confidence level describes the long-run capture rate of the repeated procedure, while the population parameter remains fixed.
Width responds to variability, sample size and the chosen confidence level.
Higher confidence widens the interval because the procedure must capture the fixed parameter more often. Larger well-designed samples narrow it by reducing standard error. Greater data variability widens it.
None of those adjustments repairs systematic bias from a narrow frame or poor measurement, so the interpretation must keep the population supported by the design.
The procedure owns the confidence
Across repeated samples, about the stated percentage of intervals from the procedure capture the parameter. Once one interval is observed, it either contains that fixed value or it does not.
A clear sentence names the confidence level, parameter, population and endpoints without assigning a probability to the fixed parameter.
Tests measure evidence against a null model
A hypothesis test states null and alternative hypotheses, calculates how extreme the sample result is under the null, compares the p-value with a significance level and returns to the population question.
Failing to reject is not the same as accepting the null.
The alternative direction must be chosen from the research question before seeing the sample result. A one-sided test counts extremeness in one direction; a two-sided test counts both. The p-value assumes the null is true while asking how often the observed statistic or something more extreme would occur.
It is not the probability that the null is true.
Decision errors remain possible
A Type I error rejects a true null and is controlled in the long run by the significance level under the test assumptions. A Type II error does not reject a false null. Lowering the threshold reduces the first risk but can increase the second when sample size and effect are held fixed.
The consequences of the errors belong in the design, before the result is known.
Significance is not importance
Large samples can make tiny effects statistically detectable, while small samples can leave important effects uncertain. Report the estimate or confidence interval alongside the p-value, preserve the units, and compare the magnitude with what matters in the application.
The final sentence joins statistical evidence to the population claim without turning a threshold into a practical verdict.
What this chapter covers
- 01
Point estimates and standard errors
- 02
Confidence interval construction
- 03
Repeated-sampling interpretation
- 04
Null and alternative hypotheses
- 05
P-values and significance levels
- 06
Population conclusions and practical importance
Worked example · free
A proportion confidence interval
- 3Calculate the sample proportion and estimated standard error.
- 3Calculate endpoints and write the population interpretation.
Key terms
- Point Estimate
- A sample statistic used as a single best guess for a population parameter.
- Margin of Error
- The critical value multiplied by standard error when constructing an interval.
- Confidence Level
- The long-run share of intervals from a procedure expected to capture the parameter.
- Null Hypothesis
- The population claim used as the reference model for a hypothesis test.
- Alternative Hypothesis
- The population direction or difference investigated by the test.
- P-Value
- The probability under the null of the observed statistic or one more extreme in the alternative direction.
Statistical Inference: Intervals and Tests FAQ
What does 95% confidence mean?
It describes the interval-producing method: across repeated random samples, about 95% of intervals constructed in the same way contain the fixed population parameter. It does not mean that 95% of individual observations lie inside one confidence interval for a mean.
Why should the null not be accepted?
A large p-value shows that the sample did not provide sufficient evidence against the null at the chosen threshold. The data can remain compatible with several population values, and an imprecise study may fail to detect an important effect.
How are confidence intervals and tests connected?
For matching two-sided procedures, a null value outside a 95% confidence interval corresponds to rejection at the 5% significance level. A null value inside the interval corresponds to not rejecting. The interval also shows plausible effect sizes and precision.
Exam move
Write every inferential solution in a fixed sequence. Name the population parameter first, then record the estimate and its standard error with units. For an interval, calculate the margin, check that the endpoints sit symmetrically around the estimate and write a confidence sentence naming the population.
For a test, state null and alternative hypotheses before considering the sample direction, set the significance level, compute the statistic and p-value, compare p with the threshold and return to the research question. Keep a list of forbidden shortcuts: the parameter is not random after the interval is observed, a p-value is not the probability the null is true, and not rejecting is not accepting.
Use spreadsheet practice to verify every cell reference and reproduce the result by hand for a small example. End with an effect-size or interval sentence so statistical significance is not mistaken for practical importance. Recheck that the design from Chapter 1 supports the population scope of the conclusion.
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