Edexcel S2 (Statistics 2) 2009 January

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, [3]
2. either 5 or 6 daisies. [2]

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 $$\sum x = 295 \quad \sum x^2 = 1386$$

1. Calculate the mean and the variance of the number of daisies per square for the 80 squares. Give your answers to 2 decimal places. [3]
2. Explain how the answers from part (c) support the choice of a Poisson distribution as a model. [1]
3. Using your mean from part (c), estimate the probability that exactly 4 daisies will be found in a randomly selected square. [2]

The continuous random variable \(X\) is uniformly distributed over the interval \([-2, 7]\).

1. Write down fully the probability density function f(x) of \(X\). [2]
2. Sketch the probability density function f(x) of \(X\). [2]

Find

1. E(\(X^2\)), [3]
2. P(\(-0.2 < X < 0.6\)). [2]

A single observation \(x\) is to be taken from a Binomial distribution B(20, \(p\)). This observation is used to test \(H_0 : p = 0.3\) against \(H_1 : p \neq 0.3\)

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\%. [3]
2. State the actual significance level of this test. [2]

The actual value of \(x\) obtained is 3.

1. State a conclusion that can be drawn based on this value giving a reason for your answer. [2]

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 $$\text{f}(t) = \begin{cases} kt & 0 \leqslant t \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the value of \(k\) is \(\frac{1}{50}\). [3]
2. Find P(\(T > 6\)). [2]
3. Calculate an exact value for E(\(T\)) and for Var(\(T\)). [5]
4. Write down the mode of the distribution of \(T\). [1]

It is suggested that the probability density function, f(\(t\)), is not a good model for \(T\).

1. 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]
2. Find the probability that there are at least 2 defective components in the box. [3]
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.

1. 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.
   [7]
2. State an assumption that has been made about the visits to the server. [1]

In a random two minute period on a Saturday the web server is visited 20 times.

1. Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]

A random variable \(X\) has probability density function given by $$\text{f}(x) = \begin{cases} -\frac{2}{9}x + \frac{8}{9} & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the cumulative distribution function F(x) can be written in the form \(ax^2 + bx + c\), for \(1 \leqslant x \leqslant 4\) where \(a\), \(b\) and \(c\) are constants. [3]
2. Define fully the cumulative distribution function F(x). [2]
3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. [6]

Given that the median of \(X\) is 1.88

1. describe the skewness of the distribution. Give a reason for your answer. [2]