Many hand tools used by mechanics involve attachments that fit into sockets (e.g., a socket wrench). In the manufacturing of the tools, specifications require that the inside diameter of the socket be larger than the outside diameter of the extension. That is, there must be enough clearance so that the extensions actually fit in the sockets. To establish tolerances for the tools, independent random samples of 50 sockets and 50 attachments were selected from the production process and the diameters (inside for sockets and outside for extensions) were measured. An analysis revealed that the distributions for both dimensions were approximately normal. The means and standard deviations (in inches) for the two samples are given in the accompanying table.

a. Find a 95% tolerance interval for 99% of the socket diameters.

b. Find a 95% tolerance interval for 99% of the attachment diameters.

c. Specifications require that the clearance between attachment and socket (i.e., the difference between the inside socket diameter and outside attachment diameter) be at least .004 inch. Based on the tolerance limits from parts a and b, is it likely to find an extension and socket with less than the desired minimum clearance of .004 inch?

d. Specifications also require a maximum of .015-inch clearance between attachment and socket, to prevent fits that are too loose. Based on the tolerance intervals from parts a and b, would you expect to find some attachment and socket pairs that fit too loosely?

e. Refer to part d. Calculate the approximate probability of observing a loose fit. Use the fact that the difference between the inside socket diameter and outside attachment diameter is approximately normal (since the two distributions are normal) with mean μ₁ - μ₂ and variance σ₁² + σ₂² (from Theorem 6.6).

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sockets (1) Attachments (2) Sample Mean 5120 5005 Standard Deviation .0010 .0015