AQA S2 (Statistics 2) 2016 June

The water in a pond contains three different species of a spherical green algae: Volvox globator, at an average rate of 4.5 spheres per 1 cm³; Volvox aureus, at an average rate of 2.3 spheres per 1 cm³; Volvox tertius, at an average rate of 1.2 spheres per 1 cm³. Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond. Find the probability that:

1. a 1 cm³ sample contains no more than 5 Volvox globator spheres; [1 mark]
2. a 1 cm³ sample contains at least 2 Volvox aureus spheres; [3 marks]
3. a 5 cm³ sample contains more than 8 but fewer than 12 Volvox tertius spheres; [3 marks]
4. a 0.1 cm³ sample contains a total of exactly 2 Volvox spheres; [3 marks]
5. a 1 cm³ sample contains at least 1 sphere of each of the three different species of algae. [3 marks]

A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be $$(30.47, 32.93)$$ Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\). [4 marks]

Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.

\(x\)	0	1	2	3	4	5	6
P(X = x)	0	0.19	0.26	0.20	0.13	0.07	0.15

1. Find the probability that a member borrows more than 3 books. [1 mark]
2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
   1. the mean amount that will be paid by a member;
   2. the standard deviation of the amount that will be paid by a member.
   [3 marks]

A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2. The error, \(X\) °C, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f(x) = \begin{cases} k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text{otherwise} \end{cases}$$

1. State the value of \(k\). [1 mark]
2. Find the probability that the error resulting from this rounding down is greater than 0.03 °C. [1 mark]
3. 1. State the value for E(\(X\)).
   2. Use integration to find the value for E(\(X^2\)).
   3. Hence find the value for the standard deviation of \(X\).
   [5 marks]

A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.

	Problems during first 3 months	Problems during first year but after first 3 months	No problems during first year	Total
Petrol engine	10	35	170	215
Diesel engine	4	8	23	35
Total	14	43	193	250

1. Use a \(\chi^2\)-test to investigate, at the 10% significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine. [11 marks]
2. Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa. [2 marks]

Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm. Gerald visited three islands, A, B and C, and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.

Island	\(\sum x\)	\(n\)
A	1384.5	78
B	116.9	7
C	394.6	20

1. Carry out a hypothesis test to investigate whether the data from Island A provide support for Gerald's belief at the 2% significance level. Assume that the standard deviation of the lengths of sand lizards on Island A is 1.8 cm. [7 marks]
2. For Island B, it is also given that $$\sum(x - \bar{x})^2 = 22.64$$
   1. Construct a 95% confidence interval for \(\mu_B\), where \(\mu_B\) centimetres is the mean length of sand lizards on Island B. Assume that the lengths of sand lizards on Island B are normally distributed with unknown standard deviation.
   2. Comment on whether your confidence interval provides support for Gerald's belief.
   [7 marks]
3. Comment on whether the data from Island C provide support for Gerald's belief. [2 marks]

The continuous random variable \(X\) has a cumulative distribution function F(\(x\)), where $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\ \frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$

1. Sketch the probability density function, f(\(x\)), on the grid below. [5 marks]
2. Find the mean value of \(X\). [4 marks]