CAIE S2 (Statistics 2) 2020 Specimen

1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .

1. Calculate a 99\% confidence interval for the population mean length.
2. Write down the probability that the whole of a \(99 \%\) confidence interval will lie below the population mean.

2 Describe briefly how to use random numbers to choose a sample of 10 students from a year-group of 276 students.

3 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5-minute period.

1. Find the probability of exactly 4 calls in an 8 -minute period.
2. Find the probability of at least 3 calls in a 3-minute period.
   The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.

4 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.

1. Find the probability that the total of the lifetimes of five randomly chosen Longlive bulbs is less than 5200 hours.
2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least three times that of a randomly chosen Longlive bulb.

5 \includegraphics[max width=\textwidth, alt={}, center]{43403c12-93e6-44e4-b15e-e3c4363be5f9-08_254_634_260_717} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).

6 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked. A random sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the \(5 \%\) significance level.

1. Explain why the test is one-tailed and state suitable null and alternative hypotheses.
2. Use a binomial distribution to find the critical region, and find the probability of a Type I error.
3. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

7 The mean weight of bags of carrots is \(\mu\) kilograms. An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$ Carry out the test at the 10\% significance level.