Edexcel S2 June/may 2011 exam answers review

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.

1. Identify a sampling frame.(1)

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

2. Specify the sampling distribution of F.(2)

2. 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.

1. Suggest a suitable model to describe the number of vehicles passing the fixed point in a 15 s interval.(1)

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

2. 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.(6)
3. 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.(3)

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

1. the mode,(1)
2. the value of k,(4)
3. the value of a.(3)

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

4. state, giving your reason, whether E(X )<3, E(X ) = 3 or E(X )<3.(2)

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.

1. Find the probability that the stick is shorter than 9.5 cm.(2)

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.

2. Find the probability of winning a bag of sweets.(2)

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.

3. Find the probability of winning a soft toy.(4)

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.

1. Find the probability that Jim’s plank contains at most 3 defects(2).

Shivani buys 6 planks each of length 100 cm.

2. Find the probability that fewer than 2 of Shivani’s planks contain at most 3 defects.(5)
3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani’s 6 planks is less than 18.(6)

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.

1. 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.(6)

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.

2. Using a suitable approximation and stating your hypotheses clearly, test the shopkeeper’s claim. Use a 10% level of significance.(8)

The continuous random variable X has probability density function given by

[IMAGE]

1. Sketch f (x) showing clearly the points where it meets the x-axis.
2. Write down the value of the mean, µ, of X.
3. Show that E(X²) = 9.8
4. Find the standard deviation, , of X.

The cumulative distribution function of X is given by

[IMAGE]

…where a is a constant.

5. Find the value of a.
6. Show that the lower quartile of X, q₁ , lies between 2.29 and 2.31
7. Hence find the upper quartile of X, giving your answer to 1 decimal place.
8. Find, to 2 decimal places, the value of k so that
   P(µ kσ < X < µ + kσ) = 0.5