Edexcel FS1 (Further Statistics 1) Specimen

1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.

\section*{Q uestion 1 continued}

1. A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of a caller, chosen at random, being connected to the wrong agent is p.

The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent is 0.049 The call centre receives 1000 calls each day.

1. Find the mean and variance of the number of wrongly connected calls a day.
2. Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent.
3. Explain why the approximation used in part (b) is valid. The probability that more than 6 calls each day are connected to the wrong agent using the binomial distribution is 0.8711 to 4 decimal places.
4. Comment on the accuracy of your answer in part (b).

1. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins.

The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim B ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. He then calculates some of the expected frequencies, correct to 1 decimal place. \begin{table}[h] \end{table}

1. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]

   Number of fake coins in each bag	0	1	2	3	4 or more
   Observed frequency	43	62	26	13	6
   Expected frequency	44.5	55.7	33.2	12.5	4.1

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Explain why there are 2 degrees of freedom in this case.
3. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.

1. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3

Estimate the probability that the mean of the sample is greater than 2.5

1. The probability of Richard winning a prize in a game at the fair is 0.15

Richard plays a number of games.

1. Find the probability of Richard winning his second prize on his 8th game,
2. State two assumptions that have to be made, for the model used in part (a) to be valid. M ary plays the same game, but has a different probability of winning a prize. She plays until she has won r prizes. The random variable \(G\) represents the total number of games M ary plays.
3. Given that the mean and standard deviation of G are 18 and 6 respectively, determine whether Richard or Mary has the greater probability of winning a prize in a game.

1. The probability generating function of the discrete random variable \(X\) is given by

$$G _ { x } ( t ) = k \left( 3 + t + 2 t ^ { 2 } \right) ^ { 2 }$$

1. Show that \(\mathrm { k } = \frac { 1 } { 36 }\)
2. Find \(\mathrm { P } ( \mathrm { X } = 3 )\)
3. Show that \(\operatorname { Var } ( \mathrm { X } ) = \frac { 29 } { 18 }\)
4. Find the probability generating function of \(2 \mathrm { X } + 1\)
   \section*{Q uestion 6 continued} \section*{Q uestion 6 continued} \section*{Q uestion 6 continued}

1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

1. Find the critical region for Sam's test.
2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
3. Find the critical region for Tessa's test.
4. Find the size of Tessa's test.
5. 1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
   2. Find the power function for Tessa's test.
6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)