A hardware manufacturer produces 10-millimeter (mm) bolts. Although the diameters of the bolts can vary somewhat from 10 mm and also from each other, if the variation is too large, too many of the bolts produced will be unusable. The manufacturer must therefore ensure that the standard deviation, σ, of the bolt diameters is not too large. Let’s suppose that the manufacturer has set the tolerance specifications for the 10-mm bolts at ±0.3 mm; that is, a bolt’s diameter is considered satisfactory if it is between 9.7 mm and 10.3 mm. Further suppose that the manufacturer has decided that at most 0.1% (1 of 1000) of the bolts produced should be defective. Assume that the diameters of the bolts produced are normally distributed with a mean of 10 mm.
a. Let X denote the diameter of a randomly selected bolt. Show that the manufacturer’s production criteria can be expressed mathematically as P(9.7 ≤ X ≤ 10.3) ≥ 0.999.
b. Draw a normal-curve figure that illustrates the equation P(9.7 ≤ X ≤ 10.3) = 0.999. Include both an x-axis and a z-axis.
c. Deduce from your figure in part (b) that the manufacturer’s production criteria are equivalent to the condition that 0.3/σ ≥ z_0.0005.
d. Use part (c) to conclude that the manufacturer’s production criteria are equivalent to requiring that the standard deviation of bolt diameters be no more than 0.09 mm.