Answer to Question #169483 in Statistics and Probability for Sunera

Question

Suppose that a random sample of five observations was taken from a normal population whose variance is 25. The results are 8, 15, 12, 6, 7. Find the 99% confidence interval estimate of the population mean.

Answer 1

$n=5, \sigma^2=25$

Sample mean

$$\bar{X}=\dfrac{8+15+12+6+7}{5}=9.6$$

Standard deviation

$$\sigma=\sqrt{\sigma^2}=\sqrt{25}=5$$

The critical value for $\alpha=0.01$ is $z_c=z_{1-\alpha/2}=2.576.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{X}-z_c\times \dfrac{\sigma}{\sqrt{n}},\bar{X}+z_c\times \dfrac{\sigma}{\sqrt{n}})$$

$$=(9.6-2.576\times \dfrac{5}{\sqrt{5}},9.6+2.576\times \dfrac{5}{\sqrt{5}})$$

$$=(3.84,15.36)$$