Edexcel FS1 (Further Statistics 1) 2020 June

1. The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.
Using this information,

1. 1. find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning,
   2. find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning. A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10
2. Test, at the 5\% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly. A further randomly selected 20 -minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,
3. calculate the probability of a Type II error.

1. The discrete random variables \(W , X\) and \(Y\) are distributed as follows

$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$

1. Explain whether or not \(\mathrm { Po } ( 4 )\) would be a good approximation to \(\mathrm { B } ( 10,0.4 )\)
2. State the assumption required for \(X + Y\) to be distributed as \(\operatorname { Po } ( 7 )\) Given the assumption in part (b) holds,
3. find \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)

1. Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.
Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

1. Find the probability that Suzanne selects the blue counter on her 4th selection.
2. Find the probability that the blue counter is first selected on or after Jon's third selection.
3. Find the mean and standard deviation of the number of selections made until the blue counter is selected.
4. Find the probability that Suzanne wins the game.

1. The discrete random variable \(X\) has the following probability distribution.

1. Find \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) is defined in terms of the discrete random variable \(X\)
   When \(X\) is negative, \(Y = X ^ { 2 }\)
   When \(X\) is positive, \(Y = 3 X - 2\)
2. Find \(\mathrm { P } ( Y < 9 )\)
3. Find \(\mathrm { E } ( X Y )\)

1. A factory produces pins.

An engineer selects 40 independent random samples of 6 pins produced at the factory and records the number of defective pins in each sample.

1. Show that the proportion of defective pins in the 40 samples is 0.15 The engineer suggests that the number of defective pins in a sample of 6 can be modelled using a binomial distribution. Using the information from the sample above, a test is to be carried out at the \(10 \%\) significance level, to see whether the data are consistent with the engineer's suggested model. The value of the test statistic for this test is 2.689
2. Justifying the degrees of freedom used, carry out the test, at the \(10 \%\) significance level, to see whether the data are consistent with the engineer's suggested model. State your hypotheses clearly. The engineer later discovers that the previously recorded information was incorrect. The data should have been as follows.

   Number of defective pins	0	1	2	3	4	5	6
   Observed frequency	19	11	6	3	1	0	0

3. Describe the effect this would have on the value of the test statistic that should be used for the hypothesis test.
   Give reasons for your answer.

1. A discrete random variable \(X\) has probability generating function given by

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 64 } \left( a + b t ^ { 2 } \right) ^ { 2 }$$ where \(a\) and \(b\) are positive constants.

1. Write down the value of \(\mathrm { P } ( X = 3 )\) Given that \(\mathrm { P } ( X = 4 ) = \frac { 25 } { 64 }\)
2. 1. find \(\mathrm { P } ( X = 2 )\)
   2. find \(\mathrm { E } ( X )\) The random variable \(Y = 3 X + 2\)
3. Find the probability generating function of \(Y\)

1. A six-sided die has sides labelled \(1,2,3,4,5\) and 6

The random variable \(S\) represents the score when the die is rolled.
Alicia rolls the die 45 times and the mean score, \(\bar { S }\), is calculated.
Assuming the die is fair and using a suitable approximation,

1. find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( \bar { S } < k ) = 0.05\)
2. Explain the relevance of the Central Limit Theorem in part (a). Alicia considers the following hypotheses:
   \(\mathrm { H } _ { 0 }\) : The die is fair
   \(\mathrm { H } _ { 1 }\) : The die is not fair
   If \(\bar { S } < 3.1\) or \(\bar { S } > 3.9\), then \(\mathrm { H } _ { 0 }\) will be rejected.
   Given that the true distribution of \(S\) has mean 4 and variance 3
3. find the power of this test.
4. Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die.
   Give a reason for your answer.