CAIE S2 (Statistics 2) 2024 November

2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag.
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6 The numbers of customers arriving at service desks \(A\) and \(B\) during a 10 -minute period have the independent distributions \(\operatorname { Po } ( 1.8 )\) and \(\operatorname { Po } ( 2.1 )\) respectively.

1. Find the probability that during a randomly chosen 15 -minute period more than 2 customers will arrive at \(\operatorname { desk } A\).
2. Find the probability that during a randomly chosen 5-minute period the total number of customers arriving at both desks is less than 4 .
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3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90 \%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute.

7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test.
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3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
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