CAIE S2 (Statistics 2) 2019 November

1 The random variable \(X\) has mean 2.4 and variance 3.1.

1. The random variable \(Y\) is the sum of four independent values of \(X\). Find the mean and variance of \(Y\).
2. The random variable \(Z\) is defined by \(Z = 4 X - 3\). Find the mean and variance of \(Z\).

2 Cars arrive at a filling station randomly and at a constant average rate of 2.4 cars per minute.

1. Calculate the probability that fewer than 4 cars arrive in a 2 -minute period.
2. Use a suitable approximating distribution to calculate the probability that at least 140 cars arrive in a 1-hour period.

3 The times, in minutes, taken by competitors to complete a puzzle have mean \(\mu\) and standard deviation 3 . The times taken by a random sample of 10 competitors are noted and the results are given below.
\(\begin{array} { l l l } 25.2 & 26.8 & 18.5 \end{array}\)
25.5
30.1
\(28.9 \quad 27.0\)
\(26.1 \quad 26.0\)
24.9

1. Stating a necessary assumption, calculate a \(97 \%\) confidence interval for \(\mu\).
2. Two more random samples, each of 10 competitors, are taken. Their times are used to calculate two more \(97 \%\) confidence intervals for \(\mu\). Find the probability that neither of these intervals contains the true value of \(\mu\).

4 A train company claims that \(92 \%\) of trains on a particular line arrive on time. Sanjeep suspects that the true percentage is less than \(92 \%\). He chooses a random sample of 20 trains on this line and finds that exactly 16 of them arrive on time. Making an assumption that should be stated, test at the 5\% significance level whether Sanjeep's suspicion is justified.
[0pt] [6]

5

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
   (a) Write down an inequality in \(n\).
   (b) Hence or otherwise find the largest possible value of \(n\).

6 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 2 } { 9 }\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
3. Find \(\operatorname { Var } ( X )\).

7 Bob is a self-employed builder. In the past his weekly income had mean \(
) 546\( and standard deviation \)\\( 120\). Following a change in Bob's working pattern, his mean weekly income for 40 randomly chosen weeks was \(
) 581\(. You should assume that the standard deviation remains unchanged at \)\\( 120\).

1. Test at the \(2.5 \%\) significance level whether Bob's mean weekly income has increased.
   Bob finds his mean weekly income for another random sample of 40 weeks and carries out a similar test at the \(2.5 \%\) significance level.
2. Given that Bob's mean weekly income is now in fact \(
   ) 595$, find 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.