CAIE S2 (Statistics 2) 2022 March

1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
200
201
198
202
200
199
199
201
197
202
200
199

1. Find unbiased estimates of the population mean and variance.
2. Give a statistical reason why these estimates may not be reliable.

2 Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following hypotheses. $$\begin{aligned} & \mathrm { H } _ { 0 } : \mathrm { P } ( \text { the spinner lands on blue } ) = \frac { 1 } { 5 }
& \mathrm { H } _ { 1 } : \mathrm { P } ( \text { the spinner lands on blue } ) \neq \frac { 1 } { 5 } \end{aligned}$$ Harry spins the spinner 300 times. It lands on blue on 45 spins.
Use a suitable approximation to carry out Harry's test at the \(5 \%\) significance level.

3 A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, \(p\), of all households in that town that owned two or more cars was found to be \(0.355 < p < 0.445\). Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

4 In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes.

1. Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the \(1 \%\) significance level whether the population mean time for the new challenge is less than 25.5 minutes.
2. State, with a reason, whether it is possible that a Type I error was made in the test in part (a).

5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .

1. Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
2. Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .

6 In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable \(X\) with probability density function $$f ( x ) = \begin{cases} - k ( x - 1 ) ( x - 3 ) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Without calculation, explain why \(\mathrm { E } ( X ) = 2\).
2. Show that \(k = \frac { 3 } { 4 }\).
3. Find \(\operatorname { Var } ( X )\).
   One turn consists of rolling the ball 3 times and noting the largest value of \(X\) obtained. If this largest value is greater than 2.5, the player scores a point.
4. Find the probability that on a particular turn the player scores a point.

7

1. Two ponds, \(A\) and \(B\), each contain a large number of fish. It is known that \(2.4 \%\) of fish in pond \(A\) are carp and \(1.8 \%\) of fish in pond \(B\) are carp. Random samples of 50 fish from pond \(A\) and 60 fish from pond \(B\) are selected. Use appropriate Poisson approximations to find the following probabilities.
   1. The samples contain at least 2 carp from pond \(A\) and at least 2 carp from pond \(B\).
   2. The samples contain at least 4 carp altogether.
2. The random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. It is given that
   • \(\mathrm { P } ( X = 0 ) = [ \mathrm { P } ( Y = 0 ) ] ^ { 2 }\),
   • \(\mathrm { P } ( X = 2 ) = k [ \mathrm { P } ( Y = 1 ) ] ^ { 2 }\), where \(k\) is a non-zero constant.
   Find the value of \(k\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.