Edexcel S1 (Statistics 1) 2001 January

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 \(1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) below the lower quartile or above the upper quartile. On graph paper draw a box plot to represent these data, indicating clearly any outliers.
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2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\). 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.
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.

3. A fair six-sided die is rolled. The random variable \(Y\) represents the score on the uppermost, face.

1. Write down the probability function of \(Y\).
2. State the name of the distribution of \(Y\). Find the value of
3. \(\mathrm { E } ( 6 Y + 2 )\),
4. \(\operatorname { Var } ( 4 Y - 2 )\).

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. 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 $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
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.

6. A local authority is investigating the cost of reconditioning its incinerators. Data from 10 randomly chosen incinerators were collected. The variables monitored were the operating time \(x\) (in thousands of hours) since last reconditioning and the reconditioning cost \(y\) (in \(\pounds 1000\) ). None of the incinerators had been used for more than 3000 hours since last reconditioning. The data are summarised below, $$\Sigma x = 25.0 , \Sigma x ^ { 2 } = 65.68 , \Sigma y = 50.0 , \Sigma y ^ { 2 } = 260.48 , \Sigma x y = 130.64 .$$

1. Find \(\mathrm { S } _ { x x } , \mathrm {~S} _ { x y } , \mathrm {~S} _ { y y }\).
2. Calculate the product moment correlation coefficient between \(x\) and \(y\).
3. Explain why this value might support the fitting of a linear regression model of the form \(y = a + b x\).
4. Find the values of \(a\) and \(b\).
5. Give an interpretation of \(a\).
6. Estimate
   1. the reconditioning cost for an operating time of 2400 hours,
   2. the financial effect of an increase of 1500 hours in operating time.
7. Suggest why the authority might be cautious about making a prediction of the reconditioning cost of an incinerator which had been operating for 4500 hours since its last reconditioning.