Small problem finding the mean of a sample when I have the probability

Alex_Of_Darkness

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So I almost complete that one and I’m pretty sure for a) b) and c) but if someone could confirm that would be great. I also have a small problem in d).


A firm says that its batteries have a lifespan of 60 months with a standard deviation of 7 months. We suppose that a group of consumer decide to verify this declaration. They buy randomly 50 batteries and observe their lifespan. We suppose that the declaration was right and that n=50 is big enough for this kind of problem.


a) Describe the distribution of the empirical mean


X ~ N(μ,σ) it’s a normal distribution and because n is big enough (n>25) we can say that the empirical distribution is close of the normal distribution.


b) Which is the probability that X found by the consumers is small or equal to 58 months. Explain the result.


With P (X ≤ 58) with the z table and the formula I found that this probability is equal to 38,59 %. P ( (X - μ) / σ) ≤ (58 - μ) / σ)


c) If the consumers could add 150 new batteries in the sample (+ the 50 we already have). What would be the probability calculate in 2), was this predictable?


Yes because n is not in the formula used in b) and our sample was already big enough so it won’t affect the probability, it will still be 38,59%.


d) Suppose that the lifespan of the batteries follow a normal distribution N(60, 7^2), find the size m of a sample with a probability 0,90, the empirical mean has to +/- 2 months max of difference from the population mean (60).


This is where I have a problem. I know that P (58 ≤ X ≤ 62) but how do I find m? Maybe by using the binomial formula but I don’t think it follows a binomial distribution.

Also, I'm freaking out bad time in my other post with my estimator problem, if someone can help me there as well I would owe you my life!!! Here
 
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