A relay microchip in a telecommunications satellite has a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of 3.5 months. When this computer-relay microchip malfunctions, the entire satellite is useless. A large London insurance company is going to insure the satellite for 50 million dollars. Assume that the only part of the satellite in question is the microchip. All other components will work indefinitely. In USE SALT (a) For how many months should the satellite be insured to be $94 \%$ confident that it will last beyond the insurance date? (Round your answer to the nearest month.) months (b) If the satellite is insured for 84 months, what is the probability that it will malfunction before the insurance coverage ends? (Round your answer to four decimal places.) (c) If the satellite is insured for 84 months, what is the expected loss to the insurance company? (Round your answer to the nearest dollar.) $\$$ (d) If the insurance company charges $\$ 3$ million for 84 months of insurance, how much profit does the company expect to make? (Round your answer to the nearest dollar.) \[ \$ \]
Solution
Step 1 ：We are given that the life expectancy of the relay microchip in a telecommunications satellite follows a normal distribution with a mean of 90 months and a standard deviation of 3.5 months.
Step 2 ：We are asked to find the time for which the satellite should be insured such that there is a 94% confidence that it will last beyond the insurance date.
Step 3 ：This is equivalent to finding the 94th percentile of the normal distribution with mean 90 and standard deviation 3.5.
Step 4 ：We can use the inverse of the cumulative distribution function (CDF) of the normal distribution to find this.
Step 5 ：Using the given mean of 90, standard deviation of 3.5, and confidence level of 0.94, we find the insurance time to be approximately 95.44170758108899 months.
Step 6 ：Rounding this to the nearest month, we get \(\boxed{95}\) months.
Step 7 ：Final Answer: The satellite should be insured for \(\boxed{95}\) months to be 94% confident that it will last beyond the insurance date.
