ou have been asked to determine the probability that the contribution margin for a particular product line exceeds the fixed cost of $3800. The total number of units sold is a normally distributed random variable with a mean of 500 and a variance of 1600 X ~ N(500, 1600). The selling price per unit is $12. The total number of units produced is a normally distributed random variable with a mean of 500 and a variance of 2500 Y ~ N(500, 2500). The variable production cost is $4 per unit. Production and sales have a positive correlation of 0.50.
To find the probability that the contribution margin exceeds the fixed cost of $3800, we first need to calculate the contribution margin and then determine the probability that it is greater than $3800.
The contribution margin is calculated as the selling price per unit minus the variable production cost per unit: $12 - $4 = $8.
Given that the total number of units sold X ~ N(500, 1600), we can calculate the mean and variance of the total contribution margin as follows:
Mean total contribution margin = 500 units * $8 = $4000
Variance of total contribution margin = 500 units * 1600 units * $8^2 = $64000
Next, we need to consider the total fixed cost of $3800. To find the probability that the contribution margin exceeds the fixed cost, we will calculate the difference between the total contribution margin and the fixed cost and then find the probability of this difference being positive.
Let Z represent the difference between the total contribution margin and the fixed cost:
Z = $4000 - $3800 = $200
Since Z follows a normal distribution with mean $200 and variance $64000, we can calculate the probability P(Z > 0) using the Z-score formula:
Z = (0 - 200) / sqrt(64000) = -200 / 252.98 = -0.791
Looking up the Z-score in a standard normal distribution table, we find that P(Z > 0) ≈ 0.7867.
Therefore, the probability that the contribution margin for this product line exceeds the fixed cost of $3800 is approximately 0.7867, or 78.67%.