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.
