Question

3. An electrical firm manufactures an equipment that has a lifetime that is normally distributed with mean 350 hours and standard deviation of 30 hours.

(a) The company is providing a warranty of 320 hours for their product. What is the proportion of product do you expect to be returned for repair during the warranty period? If the company is willing to repair only 2 % of his product, what warranty period should the company provide?

(b)A random sample of size 25 is drawn from the population, X1, ..., X25~ IID ~ N(350, 30²). Find the distribution of the sample mean (X̅) of the random sample. If 1000 random samples of size 25 are drawn from the population and the sample means are recorded. How many sample means out of the 1000 samples would you guess to fall below 340.

Answer 1

Let X represent the lifetime of the equipment

Then

a)

(i)

We need to compute Pr(X≤320). The corresponding z-value needed to be computed:

Therefore,

The following is obtained graphically:

Hence 15.87% we expect to be returned during the warranty period.

(ii)

We need to compute x such that:

Therefore, we get that

The following is obtained graphically:

b)

i.e.

We need to compute Pr(X≤340). The corresponding z-value needed to be computed:

Therefore,

The following is obtained graphically:

sample means out of the 1000 samples would you guess to fall below 340 = 1000 * 0.0478 = 47.8

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