Hypothesis testing and normal distribution theoretically derived medium coverage connects probability models to decision-making rules: the standard normal distribution supplies the reference scale for z statistics, critical values, and p-values.
Normal model behind a z test
A common setting assumes an independent random sample \(X_1, X_2, \dots, X_n\) drawn from a population with mean \(\mu\) and known standard deviation \(\sigma\). When the population distribution is normal, or when \(n\) is large enough for a normal approximation to be accurate, the sample mean \(\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i\) is (approximately) normally distributed.
Under the null hypothesis \(H_0:\mu=\mu_0\), the sampling distribution of \(\bar{X}\) satisfies \[ \bar{X}\ \sim\ N\!\left(\mu_0,\ \frac{\sigma^2}{n}\right), \] so variability around \(\mu_0\) is measured by the standard error \(\sigma/\sqrt{n}\).
Test statistic and standardization
Standardization converts \(\bar{X}\) into a unitless quantity measured in standard deviations. The z test statistic is \[ Z=\frac{\bar{X}-\mu_0}{\sigma/\sqrt{n}}. \] Under \(H_0\), this statistic satisfies \(Z \sim N(0,1)\), the standard normal distribution.
The standard normal cumulative distribution function is denoted \(\Phi(z)=P(Z \le z)\). Tail probabilities derived from \(\Phi\) produce both critical values and p-values.
Critical region and p-value
The significance level \(\alpha\) fixes an allowed long-run Type I error rate: rejecting \(H_0\) even though \(H_0\) is true occurs with probability \(\alpha\). For a chosen alternative hypothesis, the rejection region is placed in the tail(s) where the data would look most inconsistent with \(H_0\).
| Test form | Alternative hypothesis | Rejection region in z | p-value (tail area under \(N(0,1)\)) |
|---|---|---|---|
| Right-tailed | \(H_1:\mu>\mu_0\) | \(z \ge z_{\alpha}\) | \(p = 1-\Phi(z)\) |
| Left-tailed | \(H_1:\mu<\mu_0\) | \(z \le -z_{\alpha}\) | \(p = \Phi(z)\) |
| Two-tailed | \(H_1:\mu\ne\mu_0\) | \(|z| \ge z_{\alpha/2}\) | \(p = 2 \cdot \bigl(1-\Phi(|z|)\bigr)\) |
The decision rule aligns the two viewpoints: rejecting \(H_0\) at level \(\alpha\) occurs exactly when the p-value is less than or equal to \(\alpha\).
Worked example (medium)
A quality-control process has known \(\sigma=12\). A random sample of size \(n=36\) gives \(\bar{x}=54.8\). The null hypothesis is \(H_0:\mu=50\) and the alternative is \(H_1:\mu \ne 50\) with \(\alpha=0.05\).
The standardized statistic is \[ z=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}} =\frac{54.8-50}{12/\sqrt{36}} =\frac{4.8}{2} =2.4. \] For a two-tailed test, the critical values are approximately \(\pm 1.96\) because \(z_{0.025}\approx 1.96\).
The p-value is the two-tailed standard normal area beyond \(|z|=2.4\): \[ p = 2 \cdot \bigl(1-\Phi(2.4)\bigr) \approx 0.0164. \] Since \(p \le 0.05\) (equivalently, \(|z| \ge 1.96\)), the result falls in the rejection region for \(H_0\).
Visualization of rejection regions on the standard normal curve
Common pitfalls and checks
- Mismatch between the alternative hypothesis direction and the tail area used for the p-value.
- Confusion between \(\sigma\) (known) and \(s\) (estimated); replacing \(\sigma\) with \(s\) typically shifts the reference distribution from normal to t.
- Interpretation of the p-value as \(P(H_0\ \text{true})\) rather than a tail probability under the assumption that \(H_0\) is true.
- Normal approximation conditions ignored when \(n\) is small and the population distribution is strongly non-normal.