Edexcel A Level Maths S2 · Statistics 2 June 2011 question paper 6684

A factory produces components. Each component has a unique identity number and it is assumed that 2% of the components are faulty. On a particular day, a quality control manager wishes to take a random sample of 50 components.

Identify a sampling frame.

The statistic F represents the number of faulty components in the random sample of size 50.

Specify the sampling distribution of F.

A traffic officer monitors the rate at which vehicles pass a fixed point on a motorway. When the rate exceeds 36 vehicles per minute he must switch on some speed restrictions to improve traffic flow.

Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval.

The traffic officer records 12 vehicles passing the fixed point in a 15 s interval.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not the traffic officer has sufficient evidence to switch on the speed restrictions.

Using a 5% level of significance, determine the smallest number of vehicles the traffic officer must observe in a 10 s interval in order to have sufficient evidence to switch on the speed restrictions.

Figure 1 shows a sketch of the probability density function f (x) of the random variable X. For 0 x 3, f (x) is represented by a curve OB with equation f (x) = kx², where k is a constant. For 3 x a, where a is a constant, f (x) is represented by a straight line passing through B and the point (a, 0). For all other values of x, f (x) = 0. Given that the mode of X = the median of X, find

the mode,

the value of k,

the value of a.

Without calculating E(X) and with reference to the skewness of the distribution

state, giving your reason, whether E(X) 3, E(X) 3 or E(X) 3.

In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm, of a randomly chosen stick has a continuous uniform distribution over the interval [7, 10]. A stick is selected at random from the box.

Find the probability that the stick is shorter than 9.5 cm.

To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm.

Find the probability of winning a bag of sweets.

To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm.

Find the probability of winning a soft toy.

Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm.

Find the probability that Jim’s plank contains at most 3 defects.

Shivani buys 6 planks each of length 100 cm.

Find the probability that fewer than 2 of Shivani’s planks contain at most 3 defects.

Using a suitable approximation, estimate the probability that the total number of defects on Shivani’s 6 planks is less than 18.

A shopkeeper knows, from past records, that 15% of customers buy an item from the display next to the till. After a refurbishment of the shop, he takes a random sample of 30 customers and finds that only 1 customer has bought an item from the display next to the till.

Stating your hypotheses clearly, and using a 5% level of significance, test whether or not there has been a change in the proportion of customers buying an item from the display next to the till.

During the refurbishment a new sandwich display was installed. Before the refurbishment 20% of customers bought sandwiches. The shopkeeper claims that the proportion of customers buying sandwiches has now increased. He selects a random sample of 120 customers and finds that 31 of them have bought sandwiches.

Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper’s claim. Use a 10% level of significance.

The continuous random variable X has probability density function given by

Sketch f (x) showing clearly the points where it meets the x-axis.

Write down the value of the mean, of X.

The cumulative distribution function of X is given by

where a is a constant.

Find the value of a.

Show that the lower quartile of X, q₁, lies between 2.29 and 2.31

Hence find the upper quartile of X, giving your answer to 1 decimal place.

Find, to 2 decimal places, the value of k so that