Edexcel S3 (Statistics 3) 2003 June

1. Explain how to obtain a sample from a population using
   1. stratified sampling,
   2. quota sampling.
   Give one advantage and one disadvantage of each sampling method.

2. A random sample of 30 apples was taken from a batch. The mean weight of the sample was 124 g with standard deviation 20 g .

1. Find a \(99 \%\) confidence interval for the mean weight \(\mu\) grams of the population of apples. Write down any assumptions you made in your calculations. Given that the actual value of \(\mu\) is 140 ,
2. state, with a reason, what you can conclude about the sample of 30 apples.

3. Given the random variables \(X \sim \mathrm {~N} ( 20,5 )\) and \(Y \sim \mathrm {~N} ( 10,4 )\) where \(X\) and \(Y\) are independent, find

1. \(\mathrm { E } ( X - Y )\),
2. \(\operatorname { Var } ( X - Y )\),
3. \(\mathrm { P } ( 13 < X - Y < 16 )\).

4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below. Using a \(5 \%\) significance level, test whether or not the chance of catching a cold is affected by taking the new drug. State your hypotheses clearly.

5. A scientist monitored the levels of river pollution near a factory. Before the factory was closed down she took 100 random samples of water from different parts of the river and found an average weight of pollutants of \(10 \mathrm { mg } \mathrm { l } ^ { - 1 }\) with a standard deviation of \(2.64 \mathrm { mg } \mathrm { l } ^ { - 1 }\). After the factory was closed down the scientist collected a further 120 random samples and found that they contained \(8 \mathrm { mg } \mathrm { l } ^ { - 1 }\) of pollutants on average with a standard deviation of \(1.94 \mathrm { mg } \mathrm { l } ^ { - 1 }\). Test, at the \(5 \%\) level of significance, whether or not the mean river pollution fell after the factory closed down.

6. Two judges ranked 8 ice skaters in a competition according to the table below.

1. Evaluate Spearman's rank correlation coefficient between the ranks of the two judges.
2. Use a suitable test, at the \(5 \%\) level of significance, to interpret this result.

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