The lengths of lumber a machine cuts are normally distributed with a mean of 106 inches and a standard deviation of 0.3 inch. (a) What is the probability that a randomly selected board cut by the machine has a length greater than 106.11 inches? (b) A sample of 44 boards is randomly selected. What is the probability that their mean length is greater than 106.11 inches? (a) The probability is nothing.

a)

And we can find this probability with the complement rule:

b)

And if we use the z score we got:

Step-by-step explanation:

Let X the random variable that represent the lengths of a population, and for this case we know the distribution for X is given by:

Where and

Part a

We are interested on this probability

And we can use the z score formula given by:

And using this formula we got:

And we can find this probability with the complement rule:

Part b

For this case we select a sample of n =44 and the new z score formula is given by:

And if we find the z score we got:

And if we use the z score we got:

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.