PSYB06H3 Chap.6 Hypothesis Testing with z and One-Sample t
Hypothesis Testing with z and One-Sample t
Start from the position of no difference
Null hypothesis significance testing begins by assuming H0, the claim that any observed difference is no larger than random noise would create. This is not pessimism. The distribution of outcomes under no effect is known, while the distribution under an effect is not, so only the null can supply a probability.
The test then asks how likely the observed sample mean would be if the null were true, and compares that probability with alpha, set at .05 by convention.
Reject or fail to reject, never accept
A p value below alpha leads to rejecting H0; a larger one means failing to reject it. Neither outcome is proof.
A rejection can still be a Type I error, which happens at the alpha rate when the null is true, and a failure to reject can be a Type II error caused by low power or a weak design.
Conclusions are therefore written as evidence for or against a difference, with the study design deciding whether a causal verb is justified.
Choosing between z and t
When the population standard deviation is known, a z test divides the gap between the sample mean and μ by the standard error σ / √n and compares the result with ±1.96 for a two-tailed test or 1.64 for a one-tailed test.
When σ is unknown, the sample standard deviation replaces it, and the one-sample t test reads its critical value from the t table at n − 1 degrees of freedom. The t distribution has thicker tails because s tends to underestimate σ, especially in small samples.
Power and sample size
Power is the probability of rejecting H0 when an effect truly exists, and 0.80 is the usual minimum.
It rises when measurement noise falls, when the true difference grows and when the sample gets larger. Loosening alpha also raises power but inflates false positives, so it is not a legitimate fix. Because larger samples shrink the standard error, the same mean difference can be ordinary with a few participants and highly unlikely with many.
What this chapter covers
- 01
Null and Alternative Hypotheses
- 02
Alpha and p Values
- 03
One-Tailed and Two-Tailed Tests
- 04
Type I and Type II Errors
- 05
Statistical Power
- 06
One-Sample z and t Tests
Worked example · free
A one-tailed z test on reading comprehension
- 1Hypotheses: H0: μ = 20; H1: μ > 20, a one-tailed prediction.
- 1Standard error: σx̄ = 6 / √36 = 6 / 6 = 1.
- 1Obtained z = (21.8 − 20) / 1 = 1.80.
- 1Compare with the one-tailed critical value 1.64: 1.80 is larger, so reject H0; the smaller portion for z = 1.80 is .0359.
Key terms
- Alternative Hypothesis
- The claim, written H1, that a difference exceeds what random noise alone would be expected to produce.
- Alpha Level
- The rejection criterion, .05 by default, that a p value must fall below before the null hypothesis is rejected.
- Critical Value
- The boundary of the rejection region, such as z* = 1.96 for a two-tailed z test at alpha .05.
- Type II Error
- Failing to reject a null hypothesis that is actually false, a false negative whose probability is beta.
- One-Sample t Test
- A test comparing a sample mean with a known population mean when σ is unknown and estimated by s.
- Two-Tailed Test
- A test that splits alpha across both ends of the distribution because a difference in either direction counts.
Hypothesis Testing with z and One-Sample t FAQ
Why can a study never accept the null hypothesis?
Testing assumes the null from the outset and only measures how surprising the data would be under it. A non-significant result may come from insufficient power or a poor design rather than a true absence of effect. The defensible decision is to fail to reject, and any claim that the null is likely true has to be argued separately.
What is the difference between a p value and power?
A p value is the probability of the observed results if the null hypothesis is true, computed entirely from the null distribution. Power is the probability of rejecting the null when it is actually false. One describes the data under no effect; the other describes the design's chance of detecting a real effect.
When should a t test replace a z test?
Use the one-sample t test when the population mean is known but the population standard deviation is not. The sample standard deviation stands in for σ, and because it tends to underestimate σ, critical values come from the t table at n minus 1 degrees of freedom instead of the fixed z values.
Is raising alpha a valid way to find significance?
No. Moving alpha from .05 to .10 makes rejection easier and does raise power, but it also doubles the long-run false positive rate when the null is true. Legitimate ways to gain power are reducing measurement noise, studying a larger true difference and collecting a larger sample.
Exam move
For each practice question, write H0 and H1 in symbols, state the tails and alpha, and only then compute. Keep a one-page decision map: σ known leads to z, σ unknown leads to t at n − 1 degrees of freedom. Recompute the same data set with a smaller n to watch the standard error and the decision change.
Finish every answer with an interpretation sentence about the population, and practise explaining in two sentences why a non-significant result is not proof that no effect exists.
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