Edexcel S1 (Statistics 1) 2005 June

1. The scatter diagrams below were drawn by a student.

$$\begin{aligned} & y \underset { x } { \begin{array} { l l l l } & &
+ & & &
+ & + & + &
+ & + & + \end{array} } \end{aligned}$$ The student calculated the value of the product moment correlation coefficient for each of the sets of data. The values were $$\begin{array} { l l l } 0.68 & - 0.79 & 0.08 \end{array}$$ Write down, with a reason, which value corresponds to which scatter diagram.
(6)

2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.

1. Give a reason to justify the use of a histogram to represent these data.
2. Calculate the frequency densities needed to draw a histogram for these data.
   (DO NOT DRAW THE HISTOGRAM)
3. Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data. The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
4. Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
5. Evaluate this coefficient and comment on the skewness of these data.
6. Give another justification of your comment in part (e).

1. A long distance lorry driver recorded the distance travelled, \(m\) miles, and the amount of fuel used, \(f\) litres, each day. Summarised below are data from the driver's records for a random sample of 8 days.

The data are coded such that \(x = m - 250\) and \(y = f - 100\). $$\Sigma x = 130 \quad \Sigma y = 48 \quad \Sigma x y = 8880 \quad \mathrm {~S} _ { x x } = 20487.5$$

1. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
2. Hence find the equation of the regression line of \(f\) on \(m\).
3. Predict the amount of fuel used on a journey of 235 miles.

4. Aeroplanes fly from City \(A\) to City \(B\). Over a long period of time the number of minutes delay in take-off from City \(A\) was recorded. The minimum delay was 5 minutes and the maximum delay was 63 minutes. A quarter of all delays were at most 12 minutes, half were at most 17 minutes and \(75 \%\) were at most 28 minutes. Only one of the delays was longer than 45 minutes. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.

1. On the graph paper opposite draw a box plot to represent these data.
2. Comment on the distribution of delays. Justify your answer.
3. Suggest how the distribution might be interpreted by a passenger who frequently flies from City \(A\) to City \(B\).
   \includegraphics[max width=\textwidth, alt={}, center]{9698650f-ef85-468d-a703-1b40df7f9d02-07_1190_1487_278_223}

5. The random variable \(X\) has probability function $$P ( X = x ) = \begin{cases} k x , & x = 1,2,3
k ( x + 1 ) , & x = 4,5 \end{cases}$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the exact value of \(\mathrm { E } ( X )\).
3. Show that, to 3 significant figures, \(\operatorname { Var } ( X ) = 1.47\).
4. Find, to 1 decimal place, \(\operatorname { Var } ( 4 - 3 X )\).

1. A scientist found that the time taken, \(M\) minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.

Find

1. \(\mathrm { P } ( M > 160 )\).
2. \(\mathrm { P } ( 150 \leqslant M \leqslant 157 )\).
3. the value of \(m\), to 1 decimal place, such that \(\mathrm { P } ( M \leqslant m ) = 0.30\).

7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively. A student is chosen at random. Find the probability that this student

1. is studying Arts subjects,
2. does not wear glasses, given that the student is studying Arts subjects. Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are 75\% and 70\% respectively. A student is again chosen at random.
3. Find the probability that this student is right-handed.
4. Given that this student is right-handed, find the probability that the student is studying Science subjects.