To assemble a piece of furniture, a wood peg must be inserted into a predrilled hole. Suppose that the diameter of a randomly selected peg is a random variable with mean 0.25 inch and standard deviation 0.006 inch and that the diameter of a randomly selected hole is a random variable with mean 0.253 inch and standard deviation 0.002 inch. Let x₁ = peg diameter, and let x₂ = hole diameter.

a. Why would the random variable y, defined as y = x₂ - x1, be of interest to the furniture manufacturer?

b. What is the mean value of the random variable y?

c. Assuming that x₁ and x₂ are independent, what is the standard deviation of y?

d. Based on your answers to Parts (b) and (c), do you think that finding a peg that is too big to fit in the predrilled hole would be a relatively common or a relatively rare occurrence? Explain.