7 The diameter, in cm, of pistons made in a certain factory is denoted by \(X\), where \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). The diameters of a random sample of 100 pistons were measured, with the following results. $$n = 100 \quad \Sigma x = 208.7 \quad \Sigma x ^ { 2 } = 435.57$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\). The pistons are designed to fit into cylinders. The internal diameter, in cm , of the cylinders is denoted by \(Y\), where \(Y\) has an independent normal distribution with mean 2.12 and variance 0.000144 . A piston will not fit into a cylinder if \(Y - X < 0.01\).
2. Using your answers to part (i), find the probability that a randomly chosen piston will not fit into a randomly chosen cylinder.