Reversing Central Limit Theorem?

Question

I have a question like this.

A company manufactures light bulbs. The life time of bulbs is assumed to be normally distributed. The CEO claims that an average light bulb lasts $300$ days. A researcher randomly selected $64$ bulbs for testing and found that the mean is $290$ days and the standard deviation is $50$ days. If the CEO's claim is true, what is the probability that a randomly selected bulb would have an average life of no more that $290$ days?

I went on like this;

According to the researcher, $N(290, 50)$

But as the sample size is $64$, the population SD is $\sqrt{50^2\times64}$

So according to CEO, $N(300, 400)$

Can I really assume the population standard deviation like that using a reverse method?

Answer 1

Since we are interested in the probability asosciated with a single light bulb the central limit theorem is not needed here. The sample standard deviation of $50$ is already the best estimate of the population standard deviation (look up the maximum liklihood principle for more information on this). The distribution you should use is $N(300,50)$.