Financial Risk Manager Part 1

Explanation

The confidence interval is constructed using the normal distribution, not the student's t-distribution because n is large (in line with the central limit theorem).

The interval is given by:

$\bar{X} - 1.96 * \left(\frac{s}{\sqrt{N}}\right), \bar{X} + 1.96 * \left(\frac{s}{\sqrt{N}}\right)$

Thus,

$CI = \$400,000 - 1.96\left(\frac{\$23,300}{\sqrt{100}}\right), \$400,000 + 1.96\left(\frac{\$23,300}{\sqrt{100}}\right)$

$= \$400,000 - 1.96(\$2,330), \$400,000 + 1.96(\$2,330)$

$= \$400,000 - \$4,566.8, \$400,000 + \$4,566.8$

$= (\$395,433.2, \$404,566.8)$

Key points:

• For large sample sizes (n ≥ 30), we use the normal distribution (z-distribution)
• The 95% confidence level corresponds to z = 1.96
• The standard error is calculated as s/√n = $23,300/√100 =$2,330
• The margin of error is 1.96 × $2,330 =$4,566.8