Construct a 95% confidence interval for the population mean if the standard deviation of the population is 12, the sample size is 70 and the sample mean is 100.

Answer :

Given :

Sample mean $(x\overline{)}=100$

Population standard deviation $(\sigma )=12$

Sample size $(n)=70$

Confidence level $=95$

Solution :

The level of significance :

$$\alpha =1–0.95=0.05$$

Critical value :

$$z_c=Z_{\frac{\alpha }{2}}=Z_{0.05/2}=Z_{0.025}=1.96$$

The 95% confidence interval :

$$\text{The 95\% confidence interval}=\overline{x}\pm z_c\times \frac{\sigma }{\sqrt{n}}$$ $$=100\pm 1.96\times \frac{12}{\sqrt{70}}$$ $$=100\pm 2.811$$ $$=(97.189,102.811)$$