Deriving sampling distribution

6. (a) Explain what you understand by the sampling distribution of a statistic. A factory produces beads in bags for craft shops. A small bag contains 40 beads, a medium bag contains 80 beads and a large bag contains 150 beads. The factory produces small, medium and large bags in the ratio 5:3:2 respectively. A random sample of 3 bags is taken from the factory.
(b) Find the sampling distribution for the range of the number of beads in the 3 bags in the sample. A random sample of \(n\) sets of 3 bags is taken. The random variable \(Y\) represents the number of these \(n\) sets of 3 bags that have a range of 70
(c) Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y = 0 ) < 0.2\)

6. A bag contains a large number of coins. Half of them are 1 p coins, one third are 2 p coins and the remainder are 5p coins.

1. Find the mean and variance of the value of the coins. A random sample of 2 coins is chosen from the bag.
2. List all the possible samples that can be drawn.
3. Find the sampling distribution of the mean value of these samples.

1. A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1 .

A random sample of 3 counters is selected.

1. List all possible samples.
2. Find the sampling distribution for the range.

7. A bag contains a large number of coins of which \(30 \%\) are 50 p coins, \(20 \%\) are 10 p coins and the rest are 2 p coins.

1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of this population of coins. A random sample of 2 coins is drawn from the bag one after the other.
2. List all possible samples that could be drawn.
3. Find the sampling distribution of \(\bar { X }\), the mean of the coins drawn.
4. Find \(\mathrm { P } ( 2 \leq \bar { X } < 7 )\).
5. Use the sampling distribution of \(\bar { X }\) to verify \(\mathrm { E } ( \bar { X } ) = \mu\) and \(\operatorname { Var } ( \bar { X } ) = \frac { 1 } { 2 } \sigma ^ { 2 }\). END