- The continuous random variable \(X\) is normally distributed with
$$X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)$$
A random sample of 10 observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean.
- Show that a \(90 \%\) confidence interval for \(\mu\), in terms of \(\bar { x }\), is given by
$$( \bar { x } - 2.60 , \bar { x } + 2.60 )$$
The continuous random variable \(Y\) is normally distributed with
$$Y \sim \mathrm {~N} \left( \mu , 3 ^ { 2 } \right)$$
A random sample of 20 observations of \(Y\) are taken and \(\bar { Y }\) denotes the sample mean.
- Find a 95\% confidence interval for \(\mu\), in terms of \(\bar { y }\)
- Given that \(X\) and \(Y\) are independent,
- find the distribution of \(\bar { X } - \bar { Y }\)
- calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.