Financial Risk Manager Part 1

Explanation

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

Confidence Interval = Sample Mean ± (Z-score × Standard Error)

Step 1: Calculate Standard Error

Standard Error = $\frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} = \frac{14}{\sqrt{100}} = \frac{14}{10} = 1.4$

Step 2: Find Z-score for 99% Confidence Level

For a 99% confidence interval, the Z-score (reliability factor) is 2.58

Step 3: Calculate Margin of Error

Margin of Error = Z-score × Standard Error = 2.58 × 1.4 = 3.612

Step 4: Calculate Confidence Interval

• Lower Limit = Sample Mean - Margin of Error = 18 - 3.612 = 14.388 ≈ 14.4 miles
• Upper Limit = Sample Mean + Margin of Error = 18 + 3.612 = 21.612 ≈ 21.6 miles

Therefore, the 99% confidence interval is [14.4 miles; 21.6 miles]

Key Points:

• We use Z-distribution because the population standard deviation is known
• The sample size (n=100) is sufficiently large
• The confidence interval represents the range where we are 99% confident the true population mean lies