OCR S1 (Statistics 1) 2016 June

1 The table shows the probability distribution of a random variable \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three values of \(X\) are chosen at random. Find the probability that \(X\) takes the value 2 at least twice.

2

1. The table shows the amount, \(x\), in hundreds of pounds, spent on heating and the number of absences, \(y\), at a factory during each month in 2014.

   Amount, \(x\), spent on heating (£ hundreds)	21	23	19	15	14	5	2	10	9	20	18	23
   Number of absences, \(y\)	23	25	18	18	12	10	4	9	11	15	20	26

   \(n = 12 \quad \Sigma x = 179 \quad \Sigma x ^ { 2 } = 3215 \quad \Sigma y = 191 \quad \Sigma y ^ { 2 } = 3565 \quad \Sigma x y = 3343\)
   (a) Calculate \(r\), the product moment correlation coefficient, showing that \(r > 0.92\).
   (b) A manager says, 'The value of \(r\) shows that spending more money on heating causes more absences, so we should spend less on heating.' Comment on this claim.
2. The months in 2014 were numbered \(1,2,3 , \ldots , 12\). The output, \(z\), in suitable units was recorded along with the month number, \(n\), for each month in 2014. The equation of the regression line of \(z\) on \(n\) was found to be \(z = 0.6 n + 17\).
   (a) Use this equation to explain whether output generally increased or decreased over these months.
   (b) Find the mean of \(n\) and use the equation of the regression line to calculate the mean of \(z\).
   (c) Hence calculate the total output in 2014.

3 The masses, \(m\) grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$

1. Find the mean and variance of the masses of these 52 apples.
2. Use your answers from part (i) to find the exact value of \(\Sigma m ^ { 2 }\). The masses of the apples are illustrated in the box-and-whisker plot below.
   \includegraphics[max width=\textwidth, alt={}, center]{b5ce3230-7528-439c-9e85-ef159a49cba3-3_250_1310_662_383}
3. How many apples have masses in the interval \(130 \leqslant m < 140\) ?
4. An 'outlier' is a data item that lies more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile. Explain whether any of the masses of these apples are outliers.

4 In this question the product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).

1. The scatter diagram in Fig. 1 shows the results of an experiment involving some bivariate data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_597_595_434_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Write down the value of \(r _ { s }\) for these data.
2. On the diagram in the Answer Booklet, draw five points such that \(r _ { s } = 1\) and \(r \neq 1\).
3. The scatter diagram in Fig. 2 shows the results of another experiment involving 5 items of bivariate data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_604_608_1484_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Calculate the value of \(r _ { s }\).
4. A random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.6 )\). Find
   (a) \(\mathrm { P } ( X \leqslant 14 )\),
   (b) \(\mathrm { P } ( X = 14 )\),
   (c) \(\quad \operatorname { Var } ( X )\).
5. A random variable \(Y\) has the distribution \(\mathrm { B } ( 24,0.3 )\). Write down an expression for \(\mathrm { P } ( Y = y )\) and evaluate this probability in the case where \(y = 8\).
6. A random variable \(Z\) has the distribution \(\mathrm { B } ( 2,0.2 )\). Find the probability that two randomly chosen values of \(Z\) are equal.
   (a) Find the number of ways in which 12 people can be divided into three groups containing 5 people, 4 people and 3 people, without regard to order.
   (b) The diagram shows 7 cards, each with a letter on it. $$\mathrm { A } \mathrm {~A} \mathrm {~A} \mathrm {~B} \text { } \mathrm { B } \text { } \mathrm { R } \text { } \mathrm { R }$$ The 7 cards are arranged in a random order in a straight line.
7. Find the number of possible arrangements of the 7 letters.
8. Find the probability that the 7 letters form the name BARBARA. The 7 cards are shuffled. Now 4 of the 7 cards are chosen at random and arranged in a random order in a straight line.
9. Find the probability that the letters form the word ABBA .