-1

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?

Michael Rozenberg
  • 194,933
NAA
  • 227
  • 1
    You don't use a $z$-test; it's an exponential distribution. (I'm assuming it's exponential because it's in your title and your tag, but I notice that it's not in the question. Can you give more context behind the question?) – Brian Tung Mar 20 '17 at 19:38

1 Answers1

1

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.

Brian Tung
  • 34,160