Edexcel S1 (Statistics 1)

1. Explain briefly what is meant by a discrete random variable. [1 mark] A family has 3 cats and 4 dogs. Two of the family's animals are to be chosen at random. The random variable \(X\) represents the number of dogs chosen.
2. Copy and complete the table to show the probability distribution of \(X\):

   \(x\)	0	1	2
   P\((X = x)\)

   [4 marks]
3. Calculate
   1. E\((X)\),
   2. Var\((X)\),
   3. Var\((2X)\).
   [5 marks]

The discrete random variable \(X\) can take any value in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that P\((X = 1) = 0.125\) and that E\((X) = 4.5\). Beatrice finds that P\((X = 2) =\) P\((X = 3) =\) P\((X = 4) = p\). Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).

1. Calculate the values of \(p\) and \(q\). [7 marks]
2. Give the name for the distribution of \(X\). [1 mark]
3. Calculate the standard deviation of \(X\). [3 marks]

The marks obtained by ten students in a Geography test and a History test were as follows:

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Geography (\(x\))	34	57	49	21	84	53	10	77	61	85
History (\(y\))	40	49	55	40		71	39	47	65	73

1. Given that \(\sum y = 547\), calculate the mark obtained by student \(E\) in History. [1 mark] Given further that \(\sum x^2 = 34087\), \(\sum y^2 = 31575\) and \(\sum xy = 31342\), calculate
2. the product moment correlation coefficient between \(x\) and \(y\), [4 marks]
3. an equation of the regression line of \(y\) on \(x\), [4 marks]
4. an estimate of the History mark of student \(K\), who scored 70 in Geography. [2 marks]
5. State, with a reason, whether you would expect your answer to part (d) to be reliable. [2 marks]

The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\).

1. If \(2\mu = 3\sigma\), find P\((X < 2\mu)\). [5 marks]
2. If, instead, P\((X < 3\mu) = 0.86\),
   1. find \(\mu\) in terms of \(\sigma\), [4 marks]
   2. calculate P\((X > 0)\). [4 marks]

The stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).

\(A\)		\(B\)
8, 7, 4, 1, 0	1	1, 1, 2, 5, 6, 8, 9
9, 8, 7, 6, 6, 5, 2	2	0, 3, 4, 6, 7, 7, 9
9, 7, 6, 4, 2, 1, 0	3	1, 4, 5, 5, 8
8, 6, 3, 2, 2	4	0, 2, 6, 6, 9, 9
6, 4, 0	5	2, 3, 5, 7
5, 3, 1	6	0, 1

Key: 3 | 1 | 2 means \(A = 13\), \(B = 12\)

1. For each set of data, calculate estimates of the median and the quartiles. [6 marks]
2. Calculate the 42nd percentile for \(A\). [2 marks]
3. On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data. [4 marks]
4. Describe the skewness of the distribution of \(A\) and of \(B\). [2 marks]

The values of the two variables \(A\) and \(B\) given in the table in Question 5 are written on cards and placed in two separate packs, which are labelled \(A\) and \(B\). One card is selected from Pack \(A\). Let \(A_i\) represent the event that the first digit on this card is \(i\).

1. Write down the value of P\((A_2)\). [1 mark] The card taken from Pack \(A\) is now transferred into Pack \(B\), and one card is picked at random from Pack \(B\). Let \(B_i\) represent the event that the first digit on this card is \(i\).
2. Show that P\((A_1 \cap B_1) = \frac{1}{24}\). [3 marks]
3. Show that P\((A_6 | B_2) = \frac{4}{41}\). [5 marks]
4. Find the value of P\((A_1 \cup B_4)\). [5 marks]