State conditions for Poisson approximation

1 A low-cost airline charges for breakfasts on its early morning flights. On average, \(10 \%\) of passengers order breakfast.

1. Find the probability that, out of 8 randomly selected passengers, exactly 1 orders breakfast.
2. Use a suitable Poisson approximating distribution to find the probability that the number of breakfasts ordered by 30 randomly selected passengers is
   (A) exactly 6,
   (B) at least 8 .
3. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
4. The aircraft carries 120 passengers and the flight is always full. Find the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of a Normal approximating distribution suitable for modelling the total number of passengers on the flight who order breakfast.
5. Use your Normal approximating distribution to calculate the probability that more than 15 breakfasts are ordered on a particular flight.
6. The airline wishes to be at least \(99 \%\) certain that the plane will have sufficient breakfasts for all passengers who order them. Find the minimum number of breakfasts which should be carried on each flight.

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
   (A) there are no faulty laptops in the batch,
   (B) there are more than the expected number of faulty laptops in the batch.
4. A large company buys a batch of 2000 of these laptops for its staff.
   (A) State the exact distribution of the number of faulty laptops in this batch.
   (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

1. (i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A factory produces tyres for bicycles and \(0.25 \%\) of the tyres produced are defective. A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company’s order.
(ii) At the company \(40 \%\) of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable \(C\) represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of \(C\) Louis uses a normal approximation to calculate the probability that at most \(\alpha\) employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668 Find, showing all stages of your working,
(c) the value of \(\alpha\)
(d) the correct probability.

1. (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A farmer supplies a bakery with eggs. The manager of the bakery claims that the proportion of eggs having a double yolk is 0.009 The farmer claims that the proportion of his eggs having a double yolk is more than 0.009
(b) State suitable hypotheses for testing these claims. In a batch of 500 eggs the baker records 9 eggs with a double yolk.
(c) Using a suitable approximation, test at the \(5 \%\) level of significance whether or not this supports the farmer's claim.

1. (a) Given \(n\) is large, state a condition for which the binomial distribution \(\mathrm { B } ( n , p )\) can be reasonably approximated by a Poisson distribution.

A manufacturer produces candles. Those candles that pass a quality inspection are suitable for sale. It is known that 2\% of the candles produced by the manufacturer are not suitable for sale. A random sample of 125 candles produced by the manufacturer is taken.
(b) Use a suitable approximation to find the probability that no more than 6 of the candles are not suitable for sale. The manufacturer also produces candle holders.
Charlie believes that 5\% of candle holders produced by the factory have minor defects.
The manufacturer claims that the true proportion is less than \(5 \%\)
To test the manufacturer's claim, a random sample of 30 candle holders is taken and none of them are found to contain minor defects.
(c) (i) Carry out a test of the manufacturer's claim using a \(5 \%\) level of significance. You should state your hypotheses clearly.
(ii) Give a reason why this is not an appropriate test. Ashley suggests changing the sample size to 50
(d) Comment on whether or not this change would make the test appropriate. Give a reason for your answer.

4. (a) Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. A researcher has suggested that 1 in 150 people is likely to catch a particular virus.
Assuming that a person catching the virus is independent of any other person catching it,
(b) find the probability that in a random sample of 12 people, exactly 2 of them catch the virus.
(c) Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus.

1. (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.

The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.
(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.

3. (a) Write down two conditions needed to approximate the binomial distribution by the Poisson distribution. A machine which manufactures bolts is known to produce \(3 \%\) defective bolts. The machine breaks down and a new machine is installed. A random sample of 200 bolts is taken from those produced by the new machine and 12 bolts were defective.
(b) Using a suitable approximation, test at the \(5 \%\) level of significance whether or not the proportion of defective bolts is higher with the new machine than with the old machine. State your hypotheses clearly.

6. A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008

1. Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery.
2. Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken.
3. State the conditions under which the binomial distribution can be approximated by the Poisson distribution.
4. Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer.