The Chartered Financial Analyst (CFA) designation is fast becoming a requirement for serious investment professionals. Although it requires a successful completion of three levels of grueling exams, the designation often results in a promising career with a lucrative salary. A student of finance is curious about the average salary of a CFA charterholder. He takes a random sample of 49 recent charterholders and computes a mean salary of $100,000$with a standard deviation of $42,000.$ Use this sample information to determine the 90% confidence interval for the average salary of a CFA charterholder.

Answer:

we’ll calculate the 90% confidence interval for the average salary of CFA charterholders based on sample data.

Problem Statement

We are given the following sample data from a survey of recent CFA charterholders:

• Sample Mean Salary $(\overline{x})=100,000$
• Sample Standard Deviation $(s)=42,000$
• Sample Size $(n)=49$
• Confidence Level = 90%

The task is to calculate the 90% confidence interval for the average salary of a CFA charterholder.

Confidence Interval: What Does It Mean?

A confidence interval is a range of values, derived from the sample data, that is likely to contain the true population parameter (in this case, the average salary of CFA charterholders). A 90% confidence interval means that we are 90% confident the true average salary falls within the range we calculate.

Solution: 90% Confidence Interval for the Mean

To calculate the confidence interval, we use the following formula:

$\text{Confidence Interval}=\overline{x}\pm t_c\left(\frac{s}{\sqrt{n}}\right)$

Where:

• $\overline{x}$ is the sample mean,
• $t_c$ is the critical value from the t-distribution for a given confidence level,
• $s$ is the sample standard deviation,
• $n$ is the sample size.

Step-by-Step Calculation

1. Significance Level:

The confidence level is 90%, so the significance level $(\alpha )$ is calculated as:

$\alpha =1–0.90=0.10$

Since we are calculating a two-tailed confidence interval, we divide the significance level by 2:

$\frac{\alpha }{2}=0.05$

2. Degrees of Freedom:

The degrees of freedom $(df)$ are calculated as:

$df=n–1=49–1=48$

3. Critical Value:

We look up the critical value $(t_c)$ from the t-distribution table for 48 degrees of freedom at $\frac{\alpha }{2}=0.05$. From the table or using Excel, we find:

$t_c=t_{0.05,48}\approx 1.677$

4. Confidence Interval Calculation:

Now, we can calculate the confidence interval using the formula:

$\text{Confidence Interval}=100,000\pm 1.677\left(\frac{42,000}{\sqrt{49}}\right)$

First, calculate the standard error:

$\frac{42,000}{\sqrt{49}}=\frac{42,000}{7}=6,000$

Next, multiply the critical value by the standard error:

$1.677\times 6,000=10,062$

Finally, calculate the confidence interval:

$\text{Confidence Interval}=100,000\pm 10,062$

This gives us the range:

$\text{Confidence Interval}=[89,938,110,062]$

Conclusion

Based on the sample data, we can be 90% confident that the true average salary of CFA charterholders lies between $89,938 and $110,062. This confidence interval provides a valuable estimate for finance professionals, students, and anyone considering pursuing the CFA designation to understand the potential financial rewards.

Confidence intervals offer a statistically sound way to make inferences about population parameters based on sample data. For those exploring the CFA designation, understanding this range offers an insightful look into the expected salary outcomes.