The continuous random variable XXX is normally distributed with X∼N(μ,42)X \sim \text{N}(\mu, 4^2)X∼N(μ,42). A random sample of 12 observations of XXX is taken and xˉ\bar{x}xˉ denotes the sample mean.
Show that a 95% confidence interval for μ\muμ, in terms of xˉ\bar{x}xˉ, is given by (xˉ−2.26,xˉ+2.26)(\bar{x} - 2.26, \bar{x} + 2.26)(xˉ−2.26,xˉ+2.26), correct to two decimal places.
The continuous random variable YYY is normally distributed with Y∼N(μ,22)Y \sim \text{N}(\mu, 2^2)Y∼N(μ,22). A random sample of 15 observations of YYY is taken and yˉ\bar{y}yˉ denotes the sample mean.
Find a 90% confidence interval for μ\muμ, in terms of yˉ\bar{y}yˉ.
Given that XXX and YYY are independent, (i) find 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.