Edexcel A Level Maths S2 · Statistics 2 June 2019 question paper 6684

Customers arrive at a bank at a rate of 10 customers per 5 minutes. John suggests that a Poisson distribution would be a suitable model to describe the number of customers arriving at the bank in a 1-minute interval.

Write down two assumptions that John has made.

Given that the Poisson distribution is a suitable model,

find the probability that in any given 1-minute interval

exactly 2 customers arrive,

more than 4 customers arrive.

When there is sufficient evidence that the rate of customers arriving at the bank is higher than 10 customers per 5 minutes, the manager must open another service till to avoid long queues. In a 4-minute interval, the manager observes 9 customers arriving.

Test, at the 5% significance level, whether or not the manager has sufficient evidence to open another service till. State your hypotheses clearly.

Using a 1% significance level, determine the smallest number of customers the manager must observe in a 1-minute interval in order to have sufficient evidence to open another service till.

A random variable X has probability density function

Use algebraic integration to show that E(X²) =

find, to 3 decimal places, the standard deviation of X

describe, giving a reason, the skewness of the distribution.

Find the cumulative distribution function F(x) for all values of x

Find P(2 X 5)

A company sells a variety of bars of chocolate including Pearchoc bars. In an attempt to increase the sales of Pearchoc bars, the company runs a promotion where any Pearchoc wrapper that has the word WINNER printed on the inside wins a prize. The company prints the word WINNER on 5% of Pearchoc bar wrappers. Pearchoc bars are sold in packets of 12

Find the probability that in a randomly selected packet of Pearchoc bars, the number of wrappers with the word WINNER printed on them will be

exactly one,

more than two.

A customer bought 4 packets of Pearchoc bars.

Find the probability that only 1 of the packets contained exactly 1 wrapper with the word WINNER printed on it.

Before the promotion, 4% of all bars of chocolate sold in a supermarket were Pearchoc bars. After the promotion, the company selects a random sample of 150 sales of bars of chocolate from the supermarket database and finds that 10 of these sales were Pearchoc bars.

Using a suitable approximation and stating your hypotheses clearly, test whether or not the company’s promotion has been successful. Use a 5% level of significance.

State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.

Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution.

Each weekend during the ski season, requests to hire pairs of skis from Skisea occur randomly. The mean number of requests to hire pairs of skis for a weekend is 205. During the ski season Skisea only has 220 pairs of skis that can be hired for a weekend.

Using a suitable approximation, find the probability that the requests to hire pairs of skis cannot all be met on a particular weekend during the ski season.

During the ski season there are 20 weekends on which a pair of skis can be hired.

Estimate the number of weekends that the company will not be able to meet all the requests to hire pairs of skis.

The continuous random variable X is uniformly distributed over the interval ≤ x where and are constants. The mean of X is and the standard deviation is

find the value of and the value of

calculate the exact value of P( X + )

Susie has 30 m of fencing. She wishes to form a rectangular enclosure of perimeter 30 m. The length, Y metres, of one side of the enclosure is uniformly distributed between 4 m and 12 m.

Find the probability that the length of the longer side of the enclosure is more than 8 m.

A random variable X has probability density function

where k and b are constants. Given that E(X í VKRZ WKDW b = 3