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.