One Sample Z Test For Population Mean

Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ and known variance $\sigma^2$. Let $\overline{x}=\frac{1}{n} \sum X_i$ be the sample mean.

The hypothesis testing problem can be setup as :

Situation Hypothesis Testing Problem
Situation A: $H_0: \mu=\mu_0$ against $H_a : \mu < \mu_0$ (Left-tailed)
Situation B: $H_0: \mu=\mu_0$ against $H_a : \mu > \mu_0$ (Right-tailed)
Situation C: $H_0: \mu=\mu_0$ against $H_a : \mu \neq \mu_0$ (Two-tailed)


The test statistic under $H_0:\mu=\mu_0$ is

$Z = \frac{\overline{x}-\mu_0}{SE(\overline{x})}$

The test statistic $Z$ follows standard normal distribution $N(0,1)$.


  • $\overline{x} =\dfrac{1}{n}\sum x_i$ is the sample mean,
  • $SE(\overline{x})=\dfrac{\sigma}{\sqrt{n}}$
  • $\sigma$ is the population standard deviation.

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