CAIE S2 (Statistics 2) 2021 November

1 It is known that the height \(H\), in metres, of trees of a certain kind has the distribution \(\mathrm { N } ( 12.5,10.24 )\). A scientist takes a random sample of 25 trees of this kind and finds the sample mean, \(\bar { H }\), of the heights.

1. State the distribution of \(\bar { H }\), giving the values of any parameters.
2. Find \(\mathrm { P } ( 12 < \bar { H } < 13 )\).

2 The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.

3 A random sample of 75 students at a large college was selected for a survey. 15 of these students said that they owned a car. From this result an approximate \(\alpha \%\) confidence interval for the proportion of all students at the college who own a car was calculated. The width of this interval was found to be 0.162 . Calculate the value of \(\alpha\) correct to 2 significant figures.

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X < 1.2 )\).
2. Find \(\mathrm { E } ( X )\).
   The median of \(X\) is \(m\).
3. Show that \(m ^ { 3 } - 27 m + 27 = 0\).

5

1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
   1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
   2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.

6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.

7 The masses, in grams, of apples from a certain farm have mean \(\mu\) and standard deviation 5.2. The farmer says that the value of \(\mu\) is 64.6. A quality control inspector claims that the value of \(\mu\) is actually less than 64.6. In order to test his claim he chooses a random sample of 100 apples from the farm.

1. The mean mass of the 100 apples is found to be 63.5 g . Carry out the test at the \(2.5 \%\) significance level.
2. Later another test of the same hypotheses at the \(2.5 \%\) significance level, with another random sample of 100 apples from the same farm, is carried out. Given that the value of \(\mu\) is in fact 62.7 , 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.