Edexcel S2 (Statistics 2)

1. Explain what you understand by the term sampling frame when conducting a sample survey. [1 mark]
2. Suggest a suitable sampling frame and identify the sampling units when using a sample survey to study
   1. the frequency with which cars break down in the first 3 months after being serviced at a particular garage,
   2. the weight loss of people involved in trials of a new dieting programme.
   [4 marks]

An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.

1. Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
2. Explain why the parameter used with this model may need to be changed if
   1. 50 g of nuts are used instead of breadcrumbs,
   2. 100g of breadcrumbs are used.
   [2 marks]

A bird table in a rural garden has 50 g of breadcrumbs spread on it. Find the probability that

1. exactly 6 different species visit the table, [2 marks]
2. more than 2 different species visit the table. [4 marks]

In a test studying reaction times, white dots appear at random on a black rectangular screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen. The distribution of \(X\) is rectangular over the interval \([0, 20]\).

1. Find \(P(2 < X < 3.6)\). [2 marks]
2. Find the mean and variance of \(X\). [3 marks]

The continuous random variable \(Y\) represents the distance, in centimetres, of the dot from the bottom edge of the screen. The distribution of \(Y\) is rectangular over the interval \([0, 16]\). Find the probability that a dot appears

1. in a square of side 4 cm at the centre of the screen, [4 marks]
2. within 2 cm of the edge of the screen. [4 marks]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]

Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that

1. none of the dice show a score of 6, [3 marks]
2. more than one of the dice shows a score of 6, [4 marks]
3. there are equal numbers of odd and even scores showing on the dice. [3 marks]

One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]

The continuous random variable \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}x, & 0 \leq x \leq 2, \\ \frac{1}{12}(6-x), & 2 \leq x \leq 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [4 marks]
2. State the mode of \(X\). [1 mark]
3. Define fully the cumulative distribution function \(F(x)\) of \(X\). [9 marks]
4. Show that the median of \(X\) is 2.536, correct to 4 significant figures. [4 marks]