Sampling distribution theory

2

1. Write down the mean and variance of the distribution of the means of random samples of size \(n\) taken from a very large population having mean \(\mu\) and variance \(\sigma ^ { 2 }\).
2. What, if anything, can you say about the distribution of sample means
   (a) if \(n\) is large,
   (b) if \(n\) is small?

3. Explain what you understand by

1. a statistic,
2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
4. List all the possible samples.
5. Find the sampling distribution of \(\bar { Y }\)

6

1. Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio \(6 : 3 : 2\) respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
2. Find the sampling distribution for the total number, \(T\), of breakfast items bought by these 2 customers. Show your working clearly.
3. Find \(\mathrm { E } ( T )\)

4. The volume of milk, \(M\) litres, in cartons produced by a dairy, has distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu\) and \(\sigma\) are unknown. A random sample of 12 cartons is taken and the volume of milk in each carton is measured ( \(M _ { 1 } , M _ { 2 } , \ldots , M _ { 12 }\) ). A statistic \(X\) is based on this sample.

1. Explain what is meant by "a random sample" in this case.
2. State the population in this case.
3. Write down the distribution of \(\frac { M _ { 12 } - \mu } { \sigma }\)
4. Explain what you understand by the sampling distribution of \(X\).
5. State, giving a reason, which of the following is not a statistic based on this sample.
   (I) \(3 M _ { 1 } + \frac { 2 M _ { 11 } } { 6 }\)
   (II) \(\sum _ { i = 1 } ^ { 12 } \left( \frac { M _ { i } - \mu } { \sigma } \right) ^ { 2 }\)
   (III) \(\sum _ { i = 1 } ^ { 12 } \left( 2 M _ { i } - 3 \right)\)

3. Explain what you understand by

1. a statistic,
2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
4. List all the possible samples.
5. Find the sampling distribution of \(\bar { Y }\)
   

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4. Explain what you understand by

1. a sampling unit,
2. a sampling frame,
3. a sampling distribution.

3. A random sample \(X _ { 1 } , X _ { 2 } , \ldots X _ { n }\) is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\). A statistic \(Y\) is based on this sample.

1. Explain what you understand by the statistic \(Y\).
2. Explain what you understand by the sampling distribution of \(Y\).
3. State, giving a reason which of the following is not a statistic based on this sample.
   1. \(\sum _ { i = 1 } ^ { n } \frac { \left( X _ { i } - \bar { X } \right) ^ { 2 } } { n }\)
   2. \(\sum _ { i = 1 } ^ { n } \left( \frac { X _ { i } - \mu } { \sigma } \right) ^ { 2 }\)
   3. \(\sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }\)