Two high-precision mass spectrometers, Alpha and Beta, are used to determine the mass, μ\muμ picograms, of a synthetic protein. The readings from Alpha are modeled by the continuous random variable X∼N(μ,62)X \sim \text{N}(\mu, 6^2)X∼N(μ,62). A sample of 18 observations is taken from spectrometer Alpha, with a sample mean denoted by xˉ\bar{x}xˉ.
Show that a 95% confidence interval for μ\muμ, based on the Alpha sample, is given by (xˉ−2.77,xˉ+2.77)(\bar{x} - 2.77, \bar{x} + 2.77)(xˉ−2.77,xˉ+2.77), correct to two decimal places.
The readings from spectrometer Beta are modeled by the continuous random variable Y∼N(μ,32)Y \sim \text{N}(\mu, 3^2)Y∼N(μ,32). A sample of 12 observations is taken from spectrometer Beta, with a sample mean denoted by yˉ\bar{y}yˉ.
Determine a 98% confidence interval for μ\muμ in terms of yˉ\bar{y}yˉ.
Assuming that the measurements from the two spectrometers are independent: (i) state the distribution of Xˉ−Yˉ\bar{X} - \bar{Y}Xˉ−Yˉ; (ii) calculate the probability that the two confidence intervals calculated in 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.