Edexcel S2 (Statistics 2) 2009 January

1. A botanist is studying the distribution of daisies in a field. The field is divided into a number of equal sized squares. The mean number of daisies per square is assumed to be 3. The daisies are distributed randomly throughout the field.

Find the probability that, in a randomly chosen square there will be

1. more than 2 daisies,
2. either 5 or 6 daisies. The botanist decides to count the number of daisies, \(x\), in each of 80 randomly selected squares within the field. The results are summarised below $$\sum x = 295 \quad \sum x ^ { 2 } = 1386$$
3. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places.
4. Explain how the answers from part (c) support the choice of a Poisson distribution as a model.
5. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square.

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 2,7 ]\).
   1. Write down fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
   2. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\).
   Find
2. \(\mathrm { E } \left( X ^ { 2 } \right)\),
3. \(\mathrm { P } ( - 0.2 < X < 0.6 )\).

3. A single observation \(x\) is to be taken from a Binomial distribution \(\mathrm { B } ( 20 , p )\). This observation is used to test \(\mathrm { H } _ { 0 } : p = 0.3\) against \(\mathrm { H } _ { 1 } : p \neq 0.3\)

1. Using a \(5 \%\) level of significance, find the critical region for this test. The probability of rejecting either tail should be as close as possible to \(2.5 \%\).
2. State the actual significance level of this test. The actual value of \(x\) obtained is 3 .
3. State a conclusion that can be drawn based on this value giving a reason for your answer.

4. The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c l } k t & 0 \leqslant t \leqslant 10
0 & \text { otherwise } \end{array} \right.$$

1. Show that the value of \(k\) is \(\frac { 1 } { 50 }\).
2. Find \(\mathrm { P } ( T > 6 )\).
3. Calculate an exact value for \(\mathrm { E } ( T )\) and for \(\operatorname { Var } ( T )\).
4. Write down the mode of the distribution of \(T\). It is suggested that the probability density function, \(\mathrm { f } ( t )\), is not a good model for \(T\).
5. Sketch the graph of a more suitable probability density function for \(T\).

1. A factory produces components of which \(1 \%\) are defective. The components are packed in boxes of 10 . A box is selected at random.
   1. Find the probability that the box contains exactly one defective component.
   2. Find the probability that there are at least 2 defective components in the box.
   3. Using a suitable approximation, find the probability that a batch of 250 components contains between 1 and 4 (inclusive) defective components.
      
   4. A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.
   5. 1. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
      2. State the minimum number of visits required to obtain a significant result.
   6. State an assumption that has been made about the visits to the server.
   In a random two minute period on a Saturday the web server is visited 20 times.
2. Using a suitable approximation, test at the \(10 \%\) level of significance, whether or not the rate of visits is greater on a Saturday.

1. A random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c l } - \frac { 2 } { 9 } x + \frac { 8 } { 9 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{array} \right.$$

1. Show that the cumulative distribution function \(\mathrm { F } ( x )\) can be written in the form \(a x ^ { 2 } + b x + c\), for \(1 \leqslant x \leqslant 4\) where \(a , b\) and \(c\) are constants.
2. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. Given that the median of \(X\) is 1.88
4. describe the skewness of the distribution. Give a reason for your answer.