Suppose the lifetime, in years, of a motherboard is modeled by a Gamma distribution with parameters α=80 and λ=4. Use the Central Limit Theorem to approximate the probability that the motherboard of a new computer will last for at least the next 15 years.

Therefore there's a 99.99% probability the motherboard of your new computer will last for at least 15 years.

Step-by-step explanation:

This is the general idea to solve the problem.

Suppose that the mean and variance of the your distribution are . respectively. Then, according to the problem you are looking for the probability.

Consider then the following random variable.

Using the central limit theorem   distribution will be close to normal, and its mean and variance will be    , respectively. Therefore you just have to find the probability that a normally distributed  random variable with that mean and that variance which I just mentioned is less than 14.

For this case we have that

Then you have that

and we have that if   is a normally distributed random variables with mean 280 and variance 70 we have that

the actual probability we are looking for is

Therefore there's a 99.99% probability the motherboard of your new computer will last for at least 15 years.

table 3 represents a direct variation function. : )