Final answer:
The cost-recovery method is least likely to overstate gross profit as it only recognizes profit after recovering the costs of the goods sold, thus being conservative in revenue recognition for customers with risky credit.
Step-by-step explanation:
When dealing with customers with questionable credit ratings, the concern for a manufacturer is the risk that the customers may default on their payments. Therefore, it's essential to choose a revenue recognition method that minimizes the chance of overstating gross profit. The completion of production method recognizes revenue when the product is completed, which might overstate profits since payment is uncertain. The sales method recognizes revenue at the point of delivery, which also can lead to overstated profits if the buyer defaults later. The installment-sales method recognizes revenue and profit as cash is received, which can spread the recognition over time but might not align with when the economic activities occurred. The cost-recovery method only recognizes profit after all the costs of the goods sold have been recovered through customer payments, which makes it the least likely to overstate profits.
In conclusion, out of the provided options, the cost-recovery method is least likely to overstate the amount of gross profit reported when selling to customers with questionable credit, as it provides a conservative approach to revenue and profit recognition.