OCR S1 (Statistics 1) 2011 June

1 Five salesmen from a certain firm were selected at random for a survey. For each salesman, the annual income, \(x\) thousand pounds, and the distance driven last year, \(y\) thousand miles, were recorded. The results were summarised as follows. $$n = 5 \quad \Sigma x = 251 \quad \Sigma x ^ { 2 } = 14323 \quad \Sigma y = 65 \quad \Sigma y ^ { 2 } = 855 \quad \Sigma x y = 3247$$

1. (a) Show that the product moment correlation coefficient, \(r\), between \(x\) and \(y\) is - 0.122 , correct to 3 significant figures.
   (b) State what this value of \(r\) shows about the relationship between annual income and distance driven last year for these five salesmen.
   (c) It was decided to recalculate \(r\) with the distances measured in kilometres instead of miles. State what effect, if any, this would have on the value of \(r\).
2. Another salesman from the firm is selected at random. His annual income is known to be \(\pounds 52000\), but the distance that he drove last year is unknown. In order to estimate this distance, a regression line based on the above data is used. Comment on the reliability of such an estimate.

2 The orders in which 4 contestants, \(P , Q , R\) and \(S\), were placed in two competitions are shown in the table. Calculate Spearman's rank correlation coefficient between these two orders.

3

1. A random variable, \(X\), has the distribution \(\mathrm { B } ( 12,0.85 )\). Find
   (a) \(\mathrm { P } ( X > 10 )\),
   (b) \(\mathrm { P } ( X = 10 )\),
   (c) \(\operatorname { Var } ( X )\).
2. A random variable, \(Y\), has the distribution \(\mathrm { B } \left( 2 , \frac { 1 } { 4 } \right)\). Two independent values of \(Y\) are found. Find the probability that the sum of these two values is 1 .

4 The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.

1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
3. Calculate estimates of the mean and standard deviation of the data.
4. It was found that the winner's time had been incorrectly recorded and that it was actually less than 27 minutes 30 seconds. State whether each of the following will increase, decrease or remain the same:
   (a) the mean,
   (b) the standard deviation,
   (c) the median,
   (d) the interquartile range.

5 A bag contains 4 blue discs and 6 red discs. Chloe takes a disc from the bag. If this disc is red, she takes 2 more discs. If not, she takes 1 more disc. Each disc is taken at random and no discs are replaced.

1. Complete the probability tree diagram in your Answer Book, showing all the probabilities.
   \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-4_730_1203_529_511} The total number of blue discs that Chloe takes is denoted by \(X\).
2. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 5 }\). The complete probability distribution of \(X\) is given below.

   \(x\)	0	1	2
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 6 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 7 } { 30 }\)

3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 A group of 7 students sit in random order on a bench.

1. (a) Find the number of orders in which they can sit.
   (b) The 7 students include Tom and Jerry. Find the probability that Tom and Jerry sit next to each other.
2. The students consist of 3 girls and 4 boys. Find the probability that
   (a) no two boys sit next to each other,
   (b) all three girls sit next to each other.

7 The diagram shows the results of an experiment involving some bivariate data. The least squares regression line of \(y\) on \(x\) for these results is also shown.
\includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-5_748_919_390_612}

1. Given that the least squares regression line of \(y\) on \(x\) is used for an estimation, state which of \(x\) or \(y\) is treated as the independent variable.
2. Use the diagram to explain what is meant by 'least squares'.
3. State, with a reason, the value of Spearman's rank correlation coefficient for these data.
4. What can be said about the value of the product moment correlation coefficient for these data?

8 Ann, Bill, Chris and Dipak play a game with a fair cubical die. Starting with Ann they take turns, in alphabetical order, to throw the die. This process is repeated as many times as necessary until a player throws a 6 . When this happens, the game stops and this player is the winner. Find the probability that

1. Chris wins on his first throw,
2. Dipak wins on his second throw,
3. Ann gets a third throw,
4. Bill throws the die exactly three times.