MATH2209 Chap.6 One-Sample Tests and Statistical Power
One-Sample Tests and Statistical Power
One-sample inference tests a population claim using the distance between a sample estimate and a null value measured in standard-error units. This chapter writes research, null and alternative hypotheses, distinguishes one-sided from two-sided questions, and chooses z or t methods from knowledge of population spread. It also separates p-values from effect size and maps Type I error, Type II error and statistical power.
What this chapter covers
- 01
Research and statistical hypotheses
- 02
One-sided and two-sided tests
- 03
One-sample z and t methods
- 04
P-values and decision language
- 05
Type I and Type II errors
- 06
Effect size, sample size and power
Worked example · free
Test a reference concentration
- 1Use a one-sample z test because sigma is known.
- 1SE = 12 / sqrt(36) = 2 mg/L.
- 1z = (54 - 50) / 2 = 2.00.
- 1The magnitude slightly exceeds 1.96 at alpha 0.05.
- 1Conclude evidence of a difference, not proof of practical importance.
Key terms
- Null Hypothesis
- The reference claim about a population parameter evaluated by a statistical test.
- P Value
- A tail probability calculated under the null model for the observed or more extreme evidence.
- Statistical Power
- The probability of rejecting the null when the specified alternative is true.
One-Sample Tests and Statistical Power FAQ
What does a large p-value mean?
It means the observed evidence is not sufficiently unusual under the null model for rejection at the chosen threshold. It does not prove the null is true, establish equality or show that the study had enough power to detect a meaningful effect.
When should I use a t test instead of z?
For a one-sample mean, use t when population standard deviation is unknown and estimated from the sample. The t reference uses n minus one degrees of freedom and reflects the added uncertainty, especially in small samples.
How can power be increased?
Larger samples, larger true effects and lower unexplained variability generally increase power. Raising alpha also increases power but accepts more Type I error. Planning should define an effect that matters scientifically rather than only one that is easy to detect.
What belongs in a complete test conclusion?
State the population parameter and the null and alternative hypotheses before describing the evidence. Report the test statistic, direction and p-value or critical comparison, then use reject or fail to reject at the chosen alpha. Never translate a large p-value into proof that the null is true or exact equality.
Add the estimated effect and confidence interval so readers can judge direction, size and precision separately from the threshold decision. Mention whether the design and assumptions support the method and whether the study had enough planned sensitivity for a meaningful effect.
A final contextual sentence should answer the research question without claiming causation beyond the study design or treating statistical significance as practical importance.
Assessment move
Write hypotheses before any output and circle the direction of the alternative. Define the population parameter in words, place equality in the null and decide whether the research question is lower-tailed, upper-tailed or two-sided. This must happen before viewing a sample result, because changing the tail afterward changes the evidential rule. Next, choose z or t from the information supplied.
A known population standard deviation supports the one-sample z route; an unknown population spread estimated by s uses t with n-1 degrees of freedom. Record independence and the distribution or outlier condition before calculating. For each numerical exercise, show the difference between the sample estimate and null value, the standard error, the standardised statistic and the p-value or critical comparison.
Keep the sign so that it can be checked against the alternative direction. Then write a conclusion using reject or fail to reject, never accept the null. Practise explaining p-values as tail probabilities under the null model. Ban the phrases probability that the null is true and proof of no difference from your revision notes.
Pair the decision with an effect estimate and confidence interval so that statistical detectability is separated from practical importance. Recreate the decision-by-reality matrix from memory. Label Type I error, Type II error, alpha, beta and power, and explain which quantities are fixed by design and which depend on a specified alternative.
Run planning comparisons: increase sample size, reduce variability, move the true effect farther from the null or change alpha, and predict the effect on power. State trade-offs rather than treating power as a post-study verdict on the p-value. Finish with mixed prompts that include non-significant results.
Explain what the data fail to establish, which effect sizes remain plausible and whether the original design was sensitive enough to detect a meaningful departure. This creates conclusions that respect uncertainty without equating non-rejection with equality.
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