Sampling distributions

A bag contains a large number of counters with one of the numbers 4, 6 or 8 written on each of them in the ratio 5:3:2 respectively. A random sample of 2 counters is taken from the bag.

List all the possible samples of size 2 that can be taken.

The random variable M represents the mean value of the 2 counters.

find the sampling distribution for M.

A sample of n sets of 2 counters is taken. The random variable Y represents the number of these n sets that have a mean of 8

Calculate the minimum value of n such that P(Y 1) 0.9

A bag contains a large number of 10p, 20p and 50p coins in the ratio 1: 2: 2 A random sample of 3 coins is taken from the bag. Find the sampling distribution of the median of these samples.

A bag contains a large number of counters. Each counter has a single digit number on it and the mean of all the numbers in the bag is the unknown parameter μ. The number 2 is on 40% of the counters and the number 5 is on 25% of the counters. All the remaining counters have numbers greater than 5 on them. A random sample of 10 counters is taken from the bag.

State whether or not each of the following is a statistic

S = the sum of the numbers on the counters in the sample,

D = the difference between the highest number in the sample and μ,

F = the number of counters in the sample with a number 5 on them.

The random variable T represents the number of counters in a random sample of 10 with the number 2 on them.

Specify the sampling distribution of T.

The counters are selected one by one.

Find the probability that the third counter selected is the first counter with the number 2 on it.

A bag contains a large number of 1p, 2p and 5p coins. 50% are 1p coins 20% are 2p coins 30% are 5p coins A random sample of 3 coins is chosen from the bag.

List all the possible samples of size 3 with median 5p.

Find the probability that the median value of the sample is 5p.

Find the sampling distribution of the median of samples of size 3

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.

List all possible samples.

Find the sampling distribution for the range.

A factory produces components. Each component has a unique identity number and it is assumed that 2% of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.

Identify a sampling frame.

The statistic F represents the number of faulty components in the random sample of size 50.

Specify the sampling distribution of F.

Explain what you understand by

a population,

a statistic.

A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was 35%.

State the population and the statistic in this case.

Explain what you understand by the sampling distribution of this statistic.

A bag contains a large number of coins. It contains only 1p and 2p coins in the ratio 1:3

Find the mean μ and the variance σ² of the values of this population of coins.

A random sample of size 3 is taken from the bag.

List all the possible samples.

Find the sampling distribution of the mean value of the samples.

A random sample X₁, X₂,... X_n is taken from a population with unknown mean μ and unknown variance . A statistic Y is based on this sample.

Explain what you understand by the statistic Y.

Explain what you understand by the sampling distribution of Y.

State, giving a reason which of the following is not a statistic based on this sample.