Distribution of sample mean

1 It is known that the height \(H\), in metres, of trees of a certain kind has the distribution \(\mathrm { N } ( 12.5,10.24 )\). A scientist takes a random sample of 25 trees of this kind and finds the sample mean, \(\bar { H }\), of the heights.

1. State the distribution of \(\bar { H }\), giving the values of any parameters.
2. Find \(\mathrm { P } ( 12 < \bar { H } < 13 )\).

1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405.

2 The lengths of time people take to complete a certain type of puzzle are normally distributed with mean 48.8 minutes and standard deviation 15.6 minutes. The random variable \(X\) represents the time taken in minutes by a randomly chosen person to solve this type of puzzle. The times taken by random samples of 5 people are noted. The mean time \(\bar { X }\) is calculated for each sample.

1. State the distribution of \(\bar { X }\), giving the values of any parameters.
2. Find \(\mathrm { P } ( \bar { X } < 50 )\).

1 It is known that the number, \(N\), of words contained in the leading article each day in a certain newspaper can be modelled by a normal distribution with mean 352 and variance 29. A researcher takes a random sample of 10 leading articles and finds the sample mean, \(\bar { N }\), of \(N\).

1. State the distribution of \(\bar { N }\), giving the values of any parameters.
2. Find \(\mathrm { P } ( \bar { N } > 354 )\).