Question

Let X₁,X₂ ,...,Xₙ be a random sample from a N(μ,σ²) with σ² = 25 and sample
mean X̅ = 60 . Find a 95% confidence interval for μ.

Answer 1

For a single mean from a normal distribution with known variance, a two-sided, 100(1 – α)% confidence interval is calculated by

$$\bar{X}-z_{\alpha/2}*{\sigma \over \sqrt{n}}\le\mu\le\bar{X}+z_{\alpha/2}*{\sigma \over \sqrt{n}}$$

For a 95% confidence interval for μ

$$z_{\alpha/2}=z_{0.025}=1.96$$

We have that

$$\bar{X}=60, \sigma^2=25, n$$

Then

$$60-1.96*{\sqrt{25} \over \sqrt{n}}\le\mu\le60+1.96*{\sqrt{25} \over \sqrt{n}}$$

$$60-{9.8 \over \sqrt{n}}\le\mu\le60+{9.8 \over \sqrt{n}}$$

$$95\% CI\ [60-{9.8 \over \sqrt{n}}, 60+{9.8 \over \sqrt{n}}]$$

If n=16

$$60-1.96*{\sqrt{25} \over \sqrt{16}}\le\mu\le60+1.96*{\sqrt{25} \over \sqrt{16}}$$

$$57.55\le\mu\le62.45$$

$$95\% CI\ [57.55, 62.45]$$