Worked Example 4

The height of a species of plant has a normal distribution with mean \(34\) cm and standard deviation of \(\sqrt{78}\) cm. A random sample of \(13\) plants is taken and the mean height is calculated. Find the probability that

1. The sample mean lies between \(29\) cm and \(35\) cm,
2. \(P\left( {{X}{1}}+{{X}{2}}+…+{{X}_{13}}>450\mathrm{cm} \right)\)