A lumber company plans to buy a machine that automatically cuts lumber. And the

Question

A lumber company plans to buy a machine that automatically cuts lumber. And the machine cuts lumber to a mean length of 96 inches with a standard deviation of 0.5 inch. Assume that the lengths of lumber are normally distributed.

a) What is the probability that a randomly chosen board will be at least 96.25 inches?

b) A random sample of 40 boards is chosen, denote the sample mean by x bar . Identify the mean and the standard deviation of x bar.

c) Find the probability that x bar is at least 96.25 inches.

Explanation / Answer

a)P(X > 96.25)

P(z>(96.25-96)/0.5)(for single sample)

P(z>0.5)=0.308

b)40 samples

P(X > 96.25) = P[z > (96.25 - 96) / (0.5 / sqrt 40)

P(z > 3.162)

Now use a Standard Normal table ...

P(z > 3.162) = 0.000783.(mean and sd as given)

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