Edexcel S1 (Statistics 1)

1. Briefly explain what is meant by a sample space. [2 marks]
2. State two properties which a function \(f(x)\) must have to be a probability function. [2 marks]

A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance 72.25 km\(^2\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance 169 km\(^2\). In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km. Find which model has the greater probability of completing this journey, and state the value of this probability. [8 marks]

\(A\), \(B\) and \(C\) are three events such that \(\text{P}(A) = x\), \(\text{P}(B) = y\) and \(\text{P}(C) = x + y\). It is known that \(\text{P}(A \cup B) = 0.6\) and \(\text{P}(B \mid A) = 0.2\).

1. Show that \(4x + 5y = 3\). [2 marks]

It is also known that \(B\) and \(C\) are mutually exclusive and that \(\text{P}(B \cup C) = 0.9\)

1. Obtain another equation in \(x\) and \(y\) and hence find the values of \(x\) and \(y\). [4 marks]
2. Deduce whether or not \(A\) and \(B\) are independent events. [2 marks]

The discrete random variable \(X\) has the following probability distribution:

\(x\)	0	1	2	3	4	5
\(\text{P}(X = x)\)	0.11	0.17	0.2	0.13	\(p\)	\(p^2\)

1. Find the value of \(p\). [4 marks]
2. Find
   1. \(\text{P}(0 < X \leq 2)\),
   2. \(\text{P}(X \geq 3)\).
   [3 marks]
3. Find the mean and the variance of \(X\). [3 marks]
4. Construct a table to represent the cumulative distribution function \(\text{F}(x)\). [2 marks]

The following marks out of 50 were given by two judges to the contestants in a talent contest:

Contestant	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Judge 1 (\(x\))	43	32	40	21	47	11	29	38
Judge 2 (\(y\))	39	25	40	22	36	13	27	32

Given that \(\sum x = 261\), \(\sum x^2 = 9529\) and \(\sum xy = 8373\),

1. calculate the product-moment correlation coefficient between the two judges' marks [5 marks]
2. Find an equation of the regression line of \(x\) on \(y\). [4 marks]

Contestant \(I\) was awarded 45 marks by Judge 2.

1. Estimate the mark that this contestant would have received from Judge 1. [2 marks]
2. Comment, with explanation, on the probable accuracy of your answer. [2 marks]

1000 houses were sold in a small town in a one-year period. The selling prices were as given in the following table:

Selling Price	Number of Houses	Selling Price	Number of Houses
Up to £50 000	60	Up to £150 000	642
Up to £75 000	227	Up to £200 000	805
Up to £100 000	305	Up to £500 000	849
Up to £125 000	414	Up to £750 000	1000

1. Name (do not draw) a suitable type of graph for illustrating this data. [1 mark]
2. Use interpolation to find estimates of the median and the quartiles. [6 marks]
3. Estimate the 37th percentile. [2 marks]

Given further that the lowest price was £42 000 and the range of the prices was £690 000,

1. draw a box plot to represent the data. Show your scale clearly. [4 marks]

In another town the median price was £149 000, and the interquartile range was £90 000.

1. Briefly compare the prices in the two towns using this information. [2 marks]

The random variable \(X\), which can take any value from \(\{1, 2, \ldots, n\}\), is modelled by the discrete uniform distribution with mean 10.

1. Show that \(n = 19\) and find the variance of \(X\). [4 marks]
2. Find \(\text{P}(3 < X \leq 6)\). [2 marks]

The random variable \(Y\) is defined by \(Y = 3(X - 10)\).

1. State the mean and the variance of \(Y\). [3 marks]

The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\text{f}(x) = \begin{cases} k(x + 1) & x = 1, 2, \ldots, 19, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{1}{209}\). [3 marks]
2. Find \(\text{P}(3 < X \leq 6)\) using this model. [3 marks]