Edexcel S1 (Statistics 1)

1. The students in a class were each asked to write down how many CDs they owned. The student with the least number of CDs had 14 and all but one of the others owned 60 or fewer. The remaining student owned 65 . The quartiles for the class were 30,34 and 42 respectively.

Outliers are defined to be any values outside the limits of \(1.5 \left( Q _ { 3 } - Q _ { 1 } \right)\) below the lower quartile or above the upper quartile. On graph paper draw a box plot to represent these data, indicating clearly any outliers.
(7 marks)

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\).
   (4 marks)
   It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
   (2 marks)
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.
   (2 marks)

4. The employees of a company are classified as management, administration or production. The following table shows the number employed in each category and whether or not they live close to the company or some distance away. An employee is chosen at random.
Find the probability that this employee

1. is an administrator,
2. lives close to the company, given that the employee is a manager. Of the managers, \(90 \%\) are married, as are \(60 \%\) of the administrators and \(80 \%\) of the production employees.
3. Construct a tree diagram containing all the probabilities.
4. Find the probability that an employee chosen at random is married. (3 marks) An employee is selected at random and found to be married.
5. Find the probability that this employee is in production.

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

1. Using graph paper represent these data by a histogram.
2. Give a reason to justify the use of a histogram to represent these data.
3. Use interpolation to estimate the median of this distribution.
4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } } .$$
5. Evaluate this coefficient for the above data.
6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.

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. Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
2. Write down \(\mathrm { F } ( 0.8 )\).
   1. Evaluate \(\operatorname { Var } ( X )\).