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.
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.