Confidence Intervals for One Mean: Known Population Variance

A confidence interval for a population mean $\mu$ is a range of plausible values for $\mu$ computed from a sample. Suppose we draw a random sample of size $n$ from a population with known standard deviation $\sigma$, and the sample mean is $\bar{x}$. If the population is normal (or $n$ is large enough for the Central Limit Theorem to apply), then a $(1-\alpha)\cdot 100\%$ confidence interval for $\mu$ is

$$\bar{x} \pm z_{\alpha/2}\cdot\dfrac{\sigma}{\sqrt{n}}.$$

Here, $z_{\alpha/2}$ is the critical value satisfying $P(Z>z_{\alpha/2})=\alpha/2$ for $Z\sim N(0,1)$, and $\sigma/\sqrt{n}$ is the standard error of $\bar{x}$.

For a $95\%$ confidence interval, $\alpha=0.05$, so $\alpha/2=0.025$ and $z_{0.025}=1.96$.

For example, if $\sigma=10$, $n=25$, and $\bar{x}=50$, then a $95\%$ CI for $\mu$ is

$$50 \pm 1.96 \cdot \dfrac{10}{\sqrt{25}} = 50 \pm 1.96 \cdot 2 = 50 \pm 3.92 = (46.08,\, 53.92).$$