7 The random variable \(X\) has distribution \(\mathrm { N } ( \mu , 1 )\). A random sample of 4 observations of \(X\) is taken. The sample mean is denoted by \(\bar { X }\).
- Find the value of the constant \(a\) for which ( \(\bar { X } - a , \bar { X } + a\) ) is a \(98 \%\) confidence interval for \(\mu\).
The independent random variable \(Y\) has distribution \(\mathrm { N } ( \mu , 9 )\). A random sample of 16 observations of \(Y\) is taken. The sample mean is denoted by \(\bar { Y }\).
- Write down the distribution of \(\bar { X } - \bar { Y }\).
- A \(90 \%\) confidence interval for \(\mu\) based on \(\bar { Y }\) is given by ( \(\bar { Y } - 1.234 , \bar { Y } + 1.234\) ). Find the probability that this interval does not overlap with the interval in part (i).