Financial Risk Manager Part 1

Explanation

For a 95% confidence interval for the mean μ with normal distribution, the formula is:

$\bar{X} \pm 1.96 \frac{\sigma}{\sqrt{n}}$

Given:

• Standard deviation σ = $200
• Margin of error = $50
• Confidence level = 95% (z-score = 1.96)

We need to find n such that:

$1.96 \times \frac{\sigma}{\sqrt{n}} = 50$

Substitute σ = 200:

$1.96 \times \frac{200}{\sqrt{n}} = 50$

$\frac{392}{\sqrt{n}} = 50$

$392 = 50\sqrt{n}$

$\sqrt{n} = \frac{392}{50} = 7.84$

$n = (7.84)^2 = 61.4656 \approx 62$

Therefore, a sample size of 62 is required to achieve a 95% confidence interval with a width of ±$50.