Assume that Acme Tires sells their high performance tires for $180 each and their all weather tires for $135 each. Further assume that the cost of producing a high performance tire is $150 and the cost of producing an all weather tire is $120. Finally, assume that if a tire does not last 40,000 miles, Acme tires will replace it free of charge to the consumer; Acme will incur the cost of replacement, but will not receive any additional revenue.
(a) Calculate the profit earned/loss incurred on; a high performance tire that exceeds 40,000 miles, a high performance tire that does not exceed 40,000 miles, an all weather tire that exceeds 40,000 miles, and an all weather tire that fails to last 40,000 miles.
(b) What is the expected value (expected profit) of producing an all weather tire? Of producing a high performance tire?
(c) What is the variance and standard deviation of producing an all weather tire? Of producing a high performance tire?
To calculate the profit/loss on each type of tire, we need to subtract the cost of production from the selling price.
(a) Profit/Loss calculation:
1. High-performance tire that exceeds 40,000 miles:
Profit = Selling price - Cost of production
= $180 - $150
= $30
2. High-performance tire that does not exceed 40,000 miles:
Profit = Selling price (Acme incurs replacement cost, but no additional revenue)
= $0
3. All-weather tire that exceeds 40,000 miles:
Profit = Selling price - Cost of production
= $135 - $120
= $15
4. All-weather tire that fails to last 40,000 miles:
Profit = Selling price (Acme incurs replacement cost, but no additional revenue)
= $0
(b) Expected value (expected profit) calculation:
The expected value is the sum of the products of each outcome's profit and its respective probability.
Let's assume the probability of a high-performance tire lasting more than 40,000 miles is P1, and the probability of it not lasting this distance is (1 - P1). Similarly, let's assume the probability of an all-weather tire exceeding 40,000 miles is P2, and the probability of it not lasting this distance is (1 - P2).
Expected profit of producing a high-performance tire:
Expected profit = (Profit for exceeding 40,000 miles) x P1 + (Profit for not exceeding 40,000 miles) x (1 - P1)
= $30 x P1 + $0 x (1 - P1)
= $30P1
Expected profit of producing an all-weather tire:
Expected profit = (Profit for exceeding 40,000 miles) x P2 + (Profit for not exceeding 40,000 miles) x (1 - P2)
= $15 x P2 + $0 x (1 - P2)
= $15P2
(c) Variance and standard deviation calculation:
To calculate the variance and standard deviation, we need to know the probabilities of each outcome. Without that information, we cannot determine the variance and standard deviation accurately.
To calculate the profit/loss incurred for each scenario, we need to subtract the production cost from the selling price and consider the cost of replacement if applicable.
(a)
- For a high performance tire that exceeds 40,000 miles:
Profit = Selling price - Production cost = $180 - $150 = $30
- For a high performance tire that does not exceed 40,000 miles:
Profit = Selling price - Production cost = $180 - $150 = $30 (No additional cost incurred since Acme Tires will replace it for free)
- For an all-weather tire that exceeds 40,000 miles:
Profit = Selling price - Production cost = $135 - $120 = $15
- For an all-weather tire that fails to last 40,000 miles:
Profit = Selling price - Production cost - Replacement cost = $135 - $120 = $15 (Acme Tires incurs the cost of replacement)
(b)
To calculate the expected value (expected profit), we need to multiply each profit value by their respective probabilities and sum them up.
Expected value of producing an all-weather tire:
Expected profit = (Probability of exceeding 40,000 miles * Profit for exceeding 40,000 miles) + (Probability of failing to last 40,000 miles * Profit for failing to last 40,000 miles)
Expected profit = (P1 * $15) + (P2 * $15)
Expected value of producing a high-performance tire:
Expected profit = (Probability of exceeding 40,000 miles * Profit for exceeding 40,000 miles) + (Probability of not exceeding 40,000 miles * Profit for not exceeding 40,000 miles)
Expected profit = (P1 * $30) + (P2 * $30)
(c)
To calculate the variance and standard deviation for producing all-weather and high-performance tires, we need to calculate the squared differences of each profit from the expected profit, multiply them by their respective probabilities, and sum them up. The variance is the sum of these values, and the standard deviation is the square root of the variance.
Variance of producing an all-weather tire:
Variance = (P1 * (Profit for exceeding 40,000 miles - Expected profit)^2) + (P2 * (Profit for failing to last 40,000 miles - Expected profit)^2)
Variance of producing a high-performance tire:
Variance = (P1 * (Profit for exceeding 40,000 miles - Expected profit)^2) + (P2 * (Profit for not exceeding 40,000 miles - Expected profit)^2)
Standard deviation is the square root of the variance.