A credit union entered a lease contract valued at $\$ 6900$. The contract provides for payments at the end of each quarter for 4 years. If interest is $5.4 \%$ compounded quarterly, what is the size of the quarterly payment? The payment is $\$$ (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Step 1 ：The problem is asking for the size of the quarterly payment for a lease contract. This is a problem of annuity where we need to find the periodic payment for a loan of a certain amount, given the interest rate and the number of periods.

Step 2 ：The formula for the payment of an ordinary annuity is: \(P = \frac{PV}{\left(1 - (1 + r)^{-n}\right) / r}\) where: \(P\) is the payment, \(PV\) is the present value or the total amount of the loan, \(r\) is the interest rate per period, and \(n\) is the number of periods.

Step 3 ：In this case, the present value \(PV\) is $6900, the interest rate \(r\) is 5.4% per year compounded quarterly, so the quarterly interest rate is 5.4% / 4 = 1.35% or 0.0135 in decimal form, and the number of periods \(n\) is 4 years * 4 quarters = 16 quarters.

Step 4 ：Let's plug these values into the formula and calculate the payment.

Step 5 ：\(PV = 6900\)

Step 6 ：\(r = 0.0135\)

Step 7 ：\(n = 16\)

Step 8 ：\(P = 482.39\)

Step 9 ：Final Answer: The size of the quarterly payment is \(\boxed{\$482.39}\)