Question

The lengths of lumber a machine cuts are normally distributed with a mean of

106106

inches and a standard deviation of

0.40.4

inch.​(a) What is the probability that a randomly selected board cut by the machine has a length greater than

106.15106.15

​inches?​(b) A sample of

3535

boards is randomly selected. What is the probability that their mean length is greater than

106.15106.15

​inches?

​(a) The probability is

nothing.

​(Round to four decimal places as​ needed.)

Answer 1

Solution :

Let X be a random variable which represents the lengths of lumber a machine cuts.

Given that, X ~ N(106, 0.4²)

μ = 106 inches and  σ = 0.4 inch

a) We have to find P(X > 106.15).

We know that X ~ N(μ, σ²) then,

Using "pnorm" function of R we get, P(Z > 0.375) = 0.3538

Hence, the probability that a randomly selected board cut by the machine has a length greater than 106.15 inches is 0.3538.

b) We have to find P(x̅ > 106.15 inches).

We know that if X ~ N(μ, σ²) then, x̅ ~ N(μ, σ²/n).

And if x̅ ~ N(μ, σ²/n) then,

Using "pnorm" function of R we get, P(Z > 2.2185) = 0.0133

Hence, the probability that their mean length is greater than 106.15 ​inches is 0.0133.

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