I have the following question:
How do you estimate the population proportion of bulbs that survive at least 7 months without any assumption about the lifetime distribution? (with a mean of $0.4257$ years)
What does this exactly mean and how do I find the population proportion? Do I simply conduct a 1-sample Z-test? Or do I use a point estimator?
The mean value of an exponential distribution completely determines that distribution. Without it, it is impossible to say what proportion of light bulbs survive $7$ months.
If we know that the mean is $0.4257$ years, then the distribution must have the CDF
$$ F(t) = 1-e^{-\frac{t}{0.4257}} $$
$F(t)$ is the probability that the event of interest (in this case, the bulb failing) happens before time $t$, so if you plug in a $t$ of $7$ months ($\doteq 0.5833$ years), you should obtain the probability that a bulb fails within $7$ months. Subtracting that from $1$ gives you the probability that it survives $7$ months.