Financial Risk Manager Part 1

Explanation

To construct a 99% confidence interval for the population mean, we use the formula:

$\mu = \bar{x} \pm Z_{\frac{\alpha}{2}} \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x} = 75$ (sample mean)
• $\sigma = 10$ (population standard deviation)
• $n = 72$ (sample size)
• $\alpha = 0.01$ (for 99% confidence level)

From the standard normal distribution table, $Z_{0.005} = 2.58$

Now calculate the margin of error:

$\text{Margin of Error} = 2.58 \times \frac{10}{\sqrt{72}} = 2.58 \times \frac{10}{8.4853} = 2.58 \times 1.1785 = 3.04$

Therefore, the confidence interval is:

$`$75` \pm 3.04 = (71.96, 78.04)$$

This means we are 99% confident that the true population mean score lies between 71.96 and 78.04.