You have been asked to determine the probability that the contribution margin for a particular product line exceeds the fixed cost of $3410. 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 $10. The total number of units produced is a normally distributed random variable with a mean of 300 and a variance of 900 Y ~ N(300, 900). The variable production cost is $6 per unit. Production and sales have a positive correlation of 0.50.

Question

You have been asked to determine the probability that the contribution margin for a particular product line exceeds the fixed cost of ​$3410. 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 ​$10. The total number of units produced is a normally distributed random variable with a mean of 300 and a variance of 900 Y​ ~ ​N(300​, 900​). The variable production cost is ​$6 per unit. Production and sales have a positive correlation of 0.50.

Bot GPT 3.5 · Answer

To determine the probability that the contribution margin for the product line exceeds the fixed cost of $3410, we first need to calculate the contribution margin for each unit sold. The contribution margin per unit is calculated as the selling price per unit minus the variable production cost per unit: Contribution margin = Selling price - Variable production cost Contribution margin = $10 - $6 Contribution margin = $4 Next, we need to calculate the total contribution margin by multiplying the contribution margin per unit by the number of units sold. Since the number of units sold is a normally distributed random variable with a mean of 500 and a variance of 1600, we can calculate the mean and variance of the total contribution margin as follows: Mean total contribution margin = Mean number of units sold * Contribution margin per unit Mean total contribution margin = 500 * $4 Mean total contribution margin = $2000 Variance of total contribution margin = (Variance of number of units sold) * (Contribution margin per unit)^2 Variance of total contribution margin = 1600 * $4^2 Variance of total contribution margin = 1600 * $16 Variance of total contribution margin = $25600 Now, we can calculate the probability that the total contribution margin exceeds the fixed cost of $3410 by standardizing the random variable and using the standard normal distribution. We can standardize the variable as follows: Z = (Total contribution margin - Fixed cost) / sqrt(Variance of total contribution margin) Z = ($2000 - $3410) / sqrt($25600) Z = -$1410 / $160 Z = -8.8125 Using a standard normal distribution table or calculator, we can find the probability that Z is greater than -8.8125. This probability represents the probability that the total contribution margin exceeds the fixed cost of $3410.