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 \(\mathrm { P } ( 12 < \bar { H } < 13 )\).

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 \(\mathrm { P } ( \bar { N } > 354 )\).

A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.

1. The mass \(W\), in kg, of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation 0.6. Given that, for containers of wheat, 10\% store less than 19 kg, find the value of \(\mu\). [3]

The mass \(X\), in kg, of corn stored in each individual container is normally distributed with mean 20.1 and standard deviation 1.2.

1. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg. [3]

The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean 22.2 and standard deviation 1.5. The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.

1. Calculate the probability that the farmer's wife will move
   1. the einkorn,
   2. the corn. [5]
2. The mass \(E\), in kg, of emmer stored in each individual container is normally distributed with mean 10.5 and standard deviation \(\sigma\). The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208.
   1. Find the value of \(\sigma\) that the farmer's son used.
   2. Explain why the value of \(\sigma\) that he used is unreasonable. [8]