We need to determine how long you can stay in school based on the remaining funds after deducting the amounts given to your sister and for the yearly vacations, taking into account the 8% interest rate. Here's how we will approach the problem:

1. Initial Fund: The total nest egg starts at $150,000.
2. Deduction for Sister: Your parents give your sister $24,000 to start her salon.
3. Vacation Costs: Your parents will take four vacations at $9,000 per year for the next four years.

After accounting for these deductions, we will figure out how long the remaining amount can support your graduate school costs of $25,620 per year, considering that the funds are earning 8% interest annually.

Step-by-Step Breakdown

1. Remaining Fund After Giving $24,000 to Courtney: $\text{Remaining fund} = 150,000 - 24,000 = 126,000$

2. Yearly Vacation Deduction for Four Years: Each year, $9,000 is deducted for vacations, but the remaining funds will also earn 8% interest annually. So, we need to calculate the future value of the fund after accounting for the vacations.

We will calculate the present value of the vacations over the four years, discounted at an 8% interest rate, and subtract it from the remaining fund. Let's compute that.

After giving $24,000 to your sister and accounting for the four yearly vacations (with a total present value of $29,809.14), the remaining fund is approximately $96,190.86.

Next, we'll calculate how long this remaining fund of $96,190.86 can cover your graduate school costs of $25,620 per year, considering an 8% annual interest rate. We'll use an annuity calculation for this.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the steps manually or help with any other questions. Let me know how you'd like to proceed!

In the meantime, here are some related questions that could help deepen your understanding of this topic:

1. How is the present value of future expenses calculated with interest rates?
2. What impact does compounding interest have on savings over multiple years?
3. How do annuities work when withdrawing from a fund at a fixed interest rate?
4. How would changing the interest rate affect the number of years you can stay in school?
5. If graduate school costs increased annually, how would that affect the duration of support?

Tip: When dealing with funds over time, consider both the costs and the interest gained, as it helps maximize the longevity of the funds.