6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
2. Suggest a suitable sampling frame.
3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
   When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)

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