The thickness of silicon wafers, XXX micrometres, produced by manufacturing line A is normally distributed with X∼N(μ,7.52)X \sim \text{N}(\mu, 7.5^2)X∼N(μ,7.52). A random sample of 25 wafers from line A is measured, and xˉ\bar{x}xˉ denotes the sample mean thickness.
Show that a 95% confidence interval for μ\muμ, in terms of xˉ\bar{x}xˉ, is given by (xˉ−2.94,xˉ+2.94)(\bar{x} - 2.94, \bar{x} + 2.94)(xˉ−2.94,xˉ+2.94).
The thickness of silicon wafers, YYY micrometres, produced by manufacturing line B is normally distributed with Y∼N(μ,4.82)Y \sim \text{N}(\mu, 4.8^2)Y∼N(μ,4.82). A random sample of 36 wafers from line B is measured, and yˉ\bar{y}yˉ denotes the sample mean thickness.
Find a 90% confidence interval for μ\muμ, in terms of yˉ\bar{y}yˉ, giving the limits to two decimal places.
Given that XXX and YYY are independent, (i) determine the distribution of Xˉ−Yˉ\bar{X} - \bar{Y}Xˉ−Yˉ; (ii) calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.
