Financial Risk Manager Part 1

Explanation

To calculate the test statistic for this hypothesis test, we use the formula for a one-sample t-test (or z-test when population standard deviation is known, but here we use sample standard deviation):

Step 1: Calculate the standard error of the sample mean $\text{Standard Error} = \frac{s}{\sqrt{n}}$ Where:

• s = \`4`,500$ (sample standard deviation)
• $n = 115$ (sample size)

$\text{Standard Error} = \frac{4500}{\sqrt{115}} = \frac{4500}{10.7238} \approx 419.6272$

Step 2: Calculate the test statistic $t = \frac{\bar{x} - \mu_0}{\text{Standard Error}}$ Where:

• \bar{x} = \`65`,000$ (sample mean)
• \mu_0 = \`57`,000$ (hypothesized population mean)

$t = \frac{65,000 - 57,000}{419.6272} = \frac{8,000}{419.6272} \approx 19.06$

Why this is correct:

1. The test statistic measures how many standard errors the sample mean is away from the hypothesized population mean
2. A value of 19.06 indicates the sample mean is 19.06 standard errors above the hypothesized mean of $57,000
3. This is a very large test statistic, suggesting strong evidence against the null hypothesis

Note: Although we're using sample standard deviation, with a sample size of 115, the t-distribution is very close to the normal distribution, so this test statistic would be highly significant.