Financial Risk Manager Part 1

Explanation

For a sample from a normally distributed population with known standard deviation, the confidence interval for the population mean is calculated as:

Formula: $\bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x} = 130$ (sample mean)
• $\sigma = 20$ (known population standard deviation)
• $n = 50$ (sample size)
• $Z_{\alpha/2} = 1.96$ (for 95% confidence level, two-tailed)

Calculation:

1. Standard error = $\frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{50}} = \frac{20}{7.071} \approx 2.828$
2. Margin of error = $1.96 \times 2.828 \approx 5.5437$
3. Confidence interval = $130 \pm 5.5437$
4. Lower bound = $130 - 5.5437 = 124.4563 \approx 124.5$
5. Upper bound = $130 + 5.5437 = 135.5437 \approx 135.5$

Therefore, the 95% confidence interval is approximately (124.5, 135.5).

Why other options are incorrect:

• A (125, 135): Too narrow; doesn't account for the full margin of error
• B (1.96 ± 5.5): This is not a proper confidence interval format for the mean IQ
• C (130, 135.5): Only provides upper bound, not a proper two-sided interval