Question

Following an exponential distribution, the average lifespan of a smartphone battery is 2.3 years. The battery manufacturer wants to offer a warranty for its customers to receive a free replacement if the battery fails during the first year. Each battery generates a profit of $10.85, and the replacement cost is $6.35. Develop a Monte Carlo simulation using Analysis ToolPak or R, both with a seed of 1, for 100 battery units sold. a. What is the expected total cost of this warranty program? b. In order to cover the cost of the warranty program, how many additional battery units does the company need to sell?

Answer 1

To calculate the expected total cost of the warranty program, we use the probability of battery failure within the first year. Then, we multiply the probability by the replacement cost and the profit from selling the battery. To cover the cost of the warranty program, the company needs to sell additional battery units.

Step-by-step explanation:

To calculate the expected total cost of the warranty program, we need to calculate the probability that a battery fails during the first year. The average lifespan of a smartphone battery is 2.3 years, which means the failure rate per year is 1/2.3. Using the exponential distribution formula, we can calculate the probability of failure within the first year as P(X < 1) = 1 - e^(-1/2.3).

Once we have the probability, we can calculate the expected cost by multiplying the probability by the replacement cost plus the profit from selling the battery. In this case, the expected cost is (1 - e^(-1/2.3)) * ($6.35 + $10.85).

To determine the number of additional battery units the company needs to sell to cover the cost of the warranty program, we divide the expected total cost by the profit per battery unit. The formula is (expected cost / profit per battery unit) * 100. In this case, the company needs to sell ((1 - e^(-1/2.3)) * ($6.35 + $10.85) / $10.85) * 100 additional battery units.

Answer 2

The expected total cost of the warranty program is $269.9785. The company needs to sell approximately 25 additional battery units to cover the cost of the warranty program.

Step-by-step explanation:

To calculate the expected total cost of the warranty program, we need to consider the probability of a battery failing during the first year. Since the lifetime of the batteries follows an exponential distribution with an average lifespan of 2.3 years, we can use the formula for the cumulative distribution function to calculate the probability of a battery failing within the first year:

P(X ≤ 1) = 1 - e^(-1/2.3) = 0.4259

Therefore, the probability of a battery failing within the first year is 0.4259. For 100 battery units sold, we can expect around 42.59 batteries to fail within the first year.

To calculate the expected total cost of the warranty program, we multiply the number of failing batteries by the replacement cost:

Expected total cost = 42.59 * $6.35 = $269.9785

In order to cover the cost of the warranty program, the company needs to sell enough additional battery units to generate a profit of at least $269.9785. Since each battery generates a profit of $10.85, we can calculate the number of additional battery units needed:

Additional battery units = $269.9785 / $10.85 = 24.871

Therefore, the company needs to sell approximately 25 additional battery units to cover the cost of the warranty program.