Edexcel S1 (Statistics 1) 2001 June

1. Each of the 25 students on a computer course recorded the number of minutes \(x\), to the nearest minute, spent surfing the internet during a given day. The results are summarised below.

$$\Sigma x = 1075 , \Sigma x ^ { 2 } = 44625 .$$

1. Find \(\mu\) and \(\sigma\) for these data. Two other students surfed the internet on the same day for 35 and 51 minutes respectively.
2. Without further calculation, explain the effect on the mean of including these two students.
   (2)

2. On a particular day in summer 1993 at 0800 hours the height above sea level, \(x\) metres, and the temperature, \(y ^ { \circ } \mathrm { C }\), were recorded in 10 Mediterranean towns. The following summary statistics were calculated from the results. $$\Sigma x = 7300 , \Sigma x ^ { 2 } = 6599600 , S _ { x y } = - 13060 , S _ { y y } = 140.9 .$$

1. Find \(S _ { x x }\).
2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(x\) and \(y\).
3. Give an interpretation of your coefficient.

3. The continuous random variable \(Y\) is normally distributed with mean 100 and variance 256 .

1. Find \(\mathrm { P } ( Y < 80 )\).
2. Find \(k\) such that \(\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516\).

4. The discrete random variable \(X\) has the probability function shown in the table below. Find

1. \(\alpha\),
2. \(\mathrm { P } ( - 1 < X \leq 2 )\),
3. \(\mathrm { F } ( - 0.4 )\),
4. \(\mathrm { E } ( 3 X + 4 )\),
5. \(\operatorname { Var } ( 2 X + 3 )\).

5. A market researcher asked 100 adults which of the three newspapers \(A , B , C\) they read. The results showed that \(30 \operatorname { read } A , 26\) read \(B , 21\) read \(C , 5\) read both \(A\) and \(B , 7\) read both \(B\) and \(C , 6\) read both \(C\) and \(A\) and 2 read all three.

1. Draw a Venn diagram to represent these data. One of the adults is then selected at random.
   Find the probability that she reads
2. at least one of the newspapers,
3. only \(A\),
4. only one of the newspapers,
5. \(A\) given that she reads only one newspaper.

6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.

1. Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.

   	Diane	Gopal
   Smallest value	35	25
   Lower quartile	37	34
   Median	42	42
   Upper quartile	53	50
   Largest value	65	57

2. Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
3. Compare and contrast the three box plots.

7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the
number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.

1. Plot these data on a scatter diagram.
2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). $$\text { (You may use } \left. \Sigma x ^ { 2 } = 746 , \Sigma x y = 749 . \right)$$
3. Give an interpretation of the slope and the intercept of your regression line.
4. State whether or not you think the regression model is reasonable
   1. for the range of \(x\)-values given in the table,
   2. for all possible \(x\)-values. In each case justify your answer either by giving a reason for accepting the model or by suggesting an alternative model. END