Construct a 95% confidence interval to estimate the population mean using the data below:$x̄=48,\sigma =10,n=50$With 95% confidence, when $n=50$, the population mean is between a lower limit of _ and an upper limit of _.

Answer :

Given :

Sample mean $(x\overline{)}=48$

Population standard deviation $(\sigma )=10$

Sample size $(n)=50$

Confidence interval level $(CI)=95$

Solution :

The level of significance is calculated as:

$$\alpha =1–0.95=0.05$$

The critical value is calculated as:

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

The confidence interval is calculated as:

$$CI=\overline{x}\pm z_c\cdot \frac{\sigma }{\sqrt{n}}$$ $$=48\pm 1.96\cdot \frac{10}{\sqrt{50}}$$ $$=(45.24,50.76)$$