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.
390 exam-style questions on OCR (MEI) A Level Maths 2.4 Probability Distributions, covering 2.4.1 Recognise binomial situations, 2.4.2 Probability of success p, 2.4.3 Calculate binomial probabilities, 2.4.4 Mean of the binomial distribution, 2.4.5 Expected frequencies for binomial, 2.4.6 Probability functions and discrete random variables, 2.4.7 Numerical probabilities for a simple distribution, 2.4.8 Normal distribution as a model (A-level only), 2.4.9 Shape of the Normal curve (A-level only), 2.4.10 Linear transformation and standardising (A-level only), 2.4.11 Symmetry and inflection of Normal curve (A-level only), 2.4.12 Calculate probabilities from a Normal distribution (A-level only), 2.4.13 Model with probability distributions, and 2.4 Probability Distributions. Each one has a worked solution and a mark scheme showing where the marks go.