CAIE S2 (Statistics 2) 2013 June

1 The mean and variance of the random variable \(X\) are 5.8 and 3.1 respectively. The random variable \(S\) is the sum of three independent values of \(X\). The independent random variable \(T\) is defined by \(T = 3 X + 2\).

1. Find the variance of \(S\).
2. Find the variance of \(T\).
3. Find the mean and variance of \(S - T\).

2 A hockey player found that she scored a goal on \(82 \%\) of her penalty shots. After attending a coaching course, she scored a goal on 19 out of 20 penalty shots. Making an assumption that should be stated, test at the 10\% significance level whether she has improved.

3 Each of a random sample of 15 students was asked how long they spent revising for an exam. The results, in minutes, were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 50 & 70 & 80 & 60 & 65 & 110 & 10 & 70 & 75 & 60 & 65 & 45 & 50 & 70 & 50 \end{array}$$ Assume that the times for all students are normally distributed with mean \(\mu\) minutes and standard deviation 12 minutes.

1. Calculate a \(92 \%\) confidence interval for \(\mu\).
2. Explain what is meant by a \(92 \%\) confidence interval for \(\mu\).
3. Explain what is meant by saying that a sample is 'random'.

4 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2 )\) and \(\operatorname { Po } ( 3 )\) respectively.

1. Given that \(X + Y = 5\), find the probability that \(X = 1\) and \(Y = 4\).
2. Given that \(\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )\), show that \(3 \times 2 ^ { r - 1 } = r\) ! and verify that \(r = 4\) satisfies this equation.

5 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { 3 } } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 2\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
3. Find \(\mathrm { E } ( X )\).

6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be

1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
2. fewer than 1000 calls in four weeks ( 672 hours).

7 In the past the weekly profit at a store had mean \(
) 34600\( and standard deviation \)\\( 4500\). Following a change of ownership, the mean weekly profit for 90 randomly chosen weeks was \(
) 35400$.

1. Stating a necessary assumption, test at the \(5 \%\) significance level whether the mean weekly profit has increased.
2. State, with a reason, whether it was necessary to use the Central Limit theorem in part (i). The mean weekly profit for another random sample of 90 weeks is found and the same test is carried out at the 5\% significance level.
3. State the probability of a Type I error.
4. Given that the population mean weekly profit is now \(
   ) 36500$, calculate the probability of a Type II error.