CAIE S2 (Statistics 2) 2019 June

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 5 )\).

1. Find \(\mathrm { P } ( X = 2 )\).
   It is given that \(\mathrm { P } ( X = n ) = \mathrm { P } ( X = n + 1 )\).
2. Write down an equation in \(n\).
3. Hence or otherwise find the value of \(n\).

2 The random variable \(X\) has mean 372 and standard deviation 54 .

1. Describe fully the distribution of the mean of a random sample of 36 values of \(X\).
2. The distribution in part (i) might be either exact or approximate. State a condition under which the distribution is exact.

3 It is claimed that, on average, a particular train journey takes less than 1.9 hours. The times, \(t\) hours, taken for this journey on a random sample of 50 days were recorded. The results are summarised below. $$n = 50 \quad \Sigma t = 92.5 \quad \Sigma t ^ { 2 } = 175.25$$

1. Calculate unbiased estimates of the population mean and variance.
2. Test the claim at the \(5 \%\) significance level.

4 The heights of a certain variety of plant are normally distributed with mean 110 cm and variance \(1050 \mathrm {~cm} ^ { 2 }\). Two plants of this variety are chosen at random. Find the probability that the height of one of these plants is at least 1.5 times the height of the other.

5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.

1. State suitable hypotheses for the test.
2. Find the probability of a Type I error.
   On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
4. Use your confidence interval to comment on the manufacturer's claim.
   \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b
   0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(b = \frac { a } { a - 1 }\).
2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).

7 All the seats on a certain daily flight are always sold. The number of passengers who have bought seats but fail to arrive for this flight on a particular day is modelled by the distribution \(\mathbf { B } ( 320,0.005 )\).

1. Explain what the number 320 represents in this context.
2. The total number of passengers who have bought seats but fail to arrive for this flight on 2 randomly chosen days is denoted by \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( 2 < X < 6 )\).
3. Justify the use of your approximating distribution.
   After some changes, the airline wishes to test whether the mean number of passengers per day who fail to arrive for this flight has decreased.
4. During 5 randomly chosen days, a total of 2 passengers failed to arrive. Carry out the test at the 2.5\% significance level.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.