CAIE S2 (Statistics 2) 2018 June

1 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.

2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\)
Distribution of \(B\) : \(\_\_\_\_\)

3 The management of a factory wished to find a range within which the time taken to complete a particular task generally lies. It is given that the times, in minutes, have a normal distribution with mean \(\mu\) and standard deviation 6.5. A random sample of 15 employees was chosen and the mean time taken by these employees was found to be 52 minutes.

1. Calculate a \(95 \%\) confidence interval for \(\mu\).
   Later another \(95 \%\) confidence interval for \(\mu\) was found, based on a random sample of 30 employees.
2. State, with a reason, whether the width of this confidence interval was less than, equal to or greater than the width of the previous interval.

4 The mean mass of packets of sugar is supposed to be 505 g . A random sample of 10 packets filled by a certain machine was taken and the masses, in grams, were found to be as follows. $$\begin{array} { l l l l l l l l l l } 500 & 499 & 496 & 495 & 498 & 490 & 492 & 501 & 494 & 494 \end{array}$$

1. Find unbiased estimates of the population mean and variance.
   The mean mass of packets produced by this machine was found to be less than 505 g , so the machine was adjusted. Following the adjustment, the masses of a random sample of 150 packets from the machine were measured and the total mass was found to be 75660 g .
2. Given that the population standard deviation is 3.6 g , test at the \(2 \%\) significance level whether the machine is still producing packets with mean mass less than 505 g .
3. Explain why the use of the normal distribution is justified in carrying out the test in part (ii). [1]

5
\includegraphics[max width=\textwidth, alt={}, center]{b054d0a0-01b6-4785-807c-851551b90544-06_382_743_260_699} The diagram shows the probability density function, f , of a random variable \(X\), in terms of the constants \(a\) and \(b\).

1. Find \(b\) in terms of \(a\).
2. Show that \(\mathrm { f } ( x ) = \frac { 2 } { a } - \frac { 2 } { a ^ { 2 } } x\).
3. Given that \(\mathrm { E } ( X ) = 0.5\), find \(a\).

6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.

1. State, in context, one condition for the number of accidents in a given period to be modelled by a Poisson distribution.
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that exactly 4 accidents will occur during a randomly chosen 12-week period.
3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).

7 A ten-sided spinner has edges numbered \(1,2,3,4,5,6,7,8,9,10\). Sanjeev claims that the spinner is biased so that it lands on the 10 more often than it would if it were unbiased. In an experiment, the spinner landed on the 10 in 3 out of 9 spins.

1. Test at the \(1 \%\) significance level whether Sanjeev's claim is justified.
2. Explain why a Type I error cannot have been made.
   In fact the spinner is biased so that the probability that it will land on the 10 on any spin is 0.5 .
3. Another test at the \(1 \%\) significance level, also based on 9 spins, is carried out. Calculate the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.