Final answer:
The probability that the average lifetime of a random sample of 45 light bulbs is at least 9.5 years is approximately 0.4083.
Step-by-step explanation:
To solve this problem, we need to use the exponential distribution and the properties of the mean. The lifetime of a brand of light bulb is exponentially distributed with a mean of 9 years. Let X be the average lifetime of a random sample of 45 light bulbs. We want to find the probability that X is at least 9.5 years.
To find this probability, we can use the exponential distribution formula P(X >= x) = e^(-lambda * x), where lambda is the decay parameter. In this case, the decay parameter is equal to 1/mean = 1/9. So, the probability we're looking for is P(X >= 9.5) = e^(-1/9 * 9.5).
Calculating this probability gives us P(X >= 9.5) ≈ 0.4083. Therefore, the probability that the average lifetime of a random sample of 45 light bulbs is at least 9.5 years is approximately 0.4083.