Edexcel S1 (Statistics 1)

Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.

1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution? [1 mark]
2. Making this modelling assumption, find the expectation and the variance of \(X\). [5 marks]

1. Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class. [2 marks]

In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was 15 - 20 (i.e. \(£x\), where \(15000 \leq x < 20000\)), with frequency 300. The highest frequency in a class was 400, for the class 30 - 40, and the bar representing this class was 8 cm high. The total area under the histogram was 50 cm\(^2\).

1. Find the height and the width of the bar representing the modal class. [7 marks]

The variable \(X\) represents the marks out of 150 scored by a group of students in an examination. The following ten values of \(X\) were obtained: 60, 66, 76, 80, 94, 106, 110, 116, 124, 140.

1. Write down the median, \(M\), of the ten marks. [1 mark]
2. Using the coding \(y = \frac{x - M}{2}\), and showing all your working clearly, find the mean and the standard deviation of the marks. [6 marks]
3. Find E\((3X - 5)\). [3 marks]

The discrete random variable \(X\) has probability function P\((X = x) = k(x + 4)\). Given that \(X\) can take any of the values \(-3, -2, -1, 0, 1, 2, 3, 4\),

1. find the value of the constant \(k\). [3 marks]
2. Find P\((X < 0)\). [2 marks]
3. Show that the cumulative distribution F\((x)\) is given by $$\text{F}(x) = \lambda(x + 4)(x + 5)$$ where \(\lambda\) is a constant to be found. [6 marks]

The events \(A\) and \(B\) are such that P\((A \cap B) = 0.24\), P\((A \cup B) = 0.88\) and P\((B) = 0.52\).

1. Find P\((A)\). [3 marks]
2. Determine, with reasons, whether \(A\) and \(B\) are
   1. mutually exclusive,
   2. independent.
   [4 marks]
3. Find P\((B | A)\). [2 marks]
4. Find P\((A' | B')\). [3 marks]

The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours. Use this model to calculate

1. the probability that a person chosen at random took between 5 and 9 hours to complete the task, [4 marks]
2. the range, symmetrical about the mean, within which 80% of the people's times lie. [5 marks]

It is found that, in fact, 80% of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.

1. Find the standard deviation of the times in the modified model. [3 marks]

The following data was collected for seven cars, showing their engine size, \(x\) litres, and their fuel consumption, \(y\) km per litre, on a long journey. \(\sum x = 12.905\), \(\sum x^2 = 26.8951\), \(\sum y = 108.2\), \(\sum y^2 = 1781.64\), \(\sum xy = 183.176\).

1. Calculate the equation of the regression line of \(x\) on \(y\), expressing your answer in the form \(x = ay + b\). [6 marks]
2. Calculate the product moment correlation coefficient between \(y\) and \(x\) and give a brief interpretation of its value. [4 marks]
3. Use the equation of the regression line to estimate the value of \(x\) when \(y = 12\). State, with a reason, how accurate you would expect this estimate to be. [3 marks]
4. Comment on the use of the line to find values of \(x\) as \(y\) gets very small. [2 marks]