Final answer:
The mean profit of the business is approximately $617 thousand.
Step-by-step explanation:
To find the mean profit of the business, we need to utilize the fact that the variance is equal to the cube of the mean. Let's define the mean profit as μ. The variance is then given by σ^2 = μ^3. We also know that the probability of the business not making a profit for the month is 10%, which means there is a 10% chance that the profit is less than or equal to 0. In a normal distribution, this corresponds to the z-score of -1.28.
Using a z-table, we can find the corresponding z-value for the 10th percentile, which is -1.28. From the z-table, we find that the area to the left of -1.28 is 0.1003, which is approximately 10%. Since the mean is equal to the 10th percentile, we have μ = -1.28σ. Substituting this into the variance formula, we get σ^2 = (-1.28σ)^3
Which simplifies to σ^2 = -2.0736σ^3.
Dividing both sides of the equation by σ^3, we get σ^-1 = -2.0736. Solving for σ, we find σ = -0.4825. It is not possible to have a negative standard deviation, so we take the positive value of σ which is approximately 0.4825. Substituting this back into μ = -1.28σ, we get μ = -1.28(0.4825) ≈ -0.617. Therefore, the mean profit of the business is approximately $0.617 thousand, which is rounded to $617.