We can use the central limit theorem to approximate the distribution of the sample mean. The sample mean follows a normal distribution with a mean of 800 hours and a standard deviation of 40/sqrt(16) = 10 hours.

To find the probability that the sample mean is less than 775 hours, we standardize the sample mean:

z = (775 - 800) / 10 = -2.5

Using a standard normal distribution table or calculator, we find that the probability of a standard normal variable being less than -2.5 is approximately 0.0062.

Therefore, the probability that a random sample of 16 bulbs has a mean lifetime of less than 775 hours is approximately 0.0062 or 0.62%.