Edexcel S1 (Statistics 1) 2005 January

1. A company assembles drills using components from two sources. Goodbuy supplies \(85 \%\) of the components and Amart supplies the rest. It is known that \(3 \%\) of the components supplied by Goodbuy are faulty and \(6 \%\) of those supplied by Amart are faulty.
   1. Represent this information on a tree diagram.
   An assembled drill is selected at random.
2. Find the probability that it is not faulty.

2. The number of caravans on Seaview caravan site on each night in August last year is summarised in the following stem and leaf diagram.

1. Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
2. On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
3. Compare and contrast these two box plots.
4. Give an interpretation to the upper quartiles of these two distributions.

3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.

1. On graph paper, draw a scatter diagram to represent these data.
2. Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
3. Find \(S _ { x y }\). The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
4. Find, to 3 decimal places, the value of \(b\).
5. Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
6. Find the values of \(\bar { y } \pm 1.96 s\).
7. Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.

4. The random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = k x , \quad x = 1,2 , \ldots , 5$$

1. Show that \(k = \frac { 1 } { 15 }\). Find
2. \(\mathrm { P } ( X < 4 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 3 X - 4 )\).

5. Articles made on a lathe are subject to three kinds of defect, \(A , B\) or \(C\). A sample of 1000 articles was inspected and the following results were obtained. \begin{displayquote} 31 had a type \(A\) defect
37 had a type \(B\) defect
42 had a type \(C\) defect
11 had both type \(A\) and type \(B\) defects
13 had both type \(B\) and type \(C\) defects
10 had both type \(A\) and type \(C\) defects
6 had all three types of defect.

1. Draw a Venn diagram to represent these data. \end{displayquote} Find the probability that a randomly selected article from this sample had
2. no defects,
3. no more than one of these defects. An article selected at random from this sample had only one defect.
4. Find the probability that it was a type \(B\) defect. Two different articles were selected at random from this sample.
5. Find the probability that both had type \(B\) defects.

6. A discrete random variable is such that each of its values is assumed to be equally likely.

1. Write down the name of the distribution that could be used to model this random variable.
2. Give an example of such a distribution.
3. Comment on the assumption that each value is equally likely.
4. Suggest how you might refine the model in part (a).

7. The random variable \(X\) is normally distributed with mean 79 and variance 144 . Find

1. \(\mathrm { P } ( X < 70 )\),
2. \(\mathrm { P } ( 64 < X < 96 )\). It is known that \(\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463\). This information is shown in the figure below.
   \includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590} Given that \(\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )\),
3. show that the area of the shaded region is 0.1179 .
4. Find the value of \(b\).