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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.