Calculating Expected Cost and Warranty Extension for Desktop Computers
What is the expected value of the cost for the company per computer based on the given data?
a) The probability of a computer failing within the 18 months covered by the warranty is the area under the normal distribution curve up to that point, which can be calculated using the standard normal distribution table or a calculator. In this case, it is 0.4082. Therefore, the expected cost of the company per computer is the sum of the cost of repairs for computers that fail after 18 months and the cost of repairs for computers that fail within the 18 months warranty period, which is $0.4082(0) + $0.5918($300) = $177.54. Adding the initial cost of the warranty, which is $0.5($300) = $150, the total expected cost is $327.54, and divided by the total number of computers, the expected cost per computer is $50.
What warranty should the company offer if they decide to extend coverage to 5% more computers?
b) Assuming the company extends the warranty to cover 5% more computers, the new warranty period is X months, where X is the value we need to find. To cover the same proportion of computers as before, the area under the normal distribution curve up to X months should be 0.4532, which can be found using the standard normal distribution table or a calculator. Solving for X, we get X = 24 months. Therefore, the company should offer a warranty for 24 months to cover 95% of the computers.
Expected Cost Calculation:
a) To find the expected value of the cost per computer, we need to calculate the probability of a computer failing within the warranty period of 18 months. This can be done by finding the area under the normal distribution curve to the left of 18 months. We can then multiply this probability by the cost of repair to get the expected value.
First, we need to convert 18 months into a z-score using the formula: z = (x - mean) / standard deviation. So, z = (18 - 42) / 12 = -2.
Using a standard normal distribution table or a calculator, we find that the probability of a computer failing within 18 months is approximately 0.0228. Therefore, the expected value of the cost per computer is: (0.0228) * $300 = $6.84.
Warranty Extension Calculation:
b) To determine the warranty the company should offer, we need to find the z-score corresponding to the desired percentage of computers covered by the warranty. We can then convert this z-score back into months.
For example, if the company wants to cover an additional 5% of computers, it needs to find the z-score that corresponds to the 95th percentile of the normal distribution. Using a standard normal distribution table or a calculator, we find that the z-score for the 95th percentile is approximately 1.645. Converting this z-score back into months, the warranty should be extended for (1.645 * standard deviation) + mean = (1.645 * 12) + 42 = 62.74 months.
Calculating the expected cost per computer is crucial for a company to determine the financial impact of warranties and repairs. In this scenario, we used the normal distribution to calculate the probability of computer failures and their corresponding repair costs.
For part (a), we found that the expected cost of the company per computer is $6.84 based on the given data. This value takes into account the repair costs within the warranty period and after the warranty period, along with the initial warranty cost.
As for part (b), the company should offer a warranty extension of 24 months to cover an additional 5% of computers. By understanding the statistical implications of warranty extensions, the company can make informed decisions to balance customer satisfaction and financial risks.