2.04d Normal approximation to binomial

2 The discrete random variable \(X\) has a Poisson distribution with mean 12.25.

1. Calculate \(\mathrm { P } ( X \leqslant 5 )\).
2. Calculate an approximate value for \(\mathrm { P } ( X \leqslant 5 )\) using a normal approximation to the Poisson distribution.
3. Comment, giving a reason, on the accuracy of using a normal approximation to the Poisson distribution in this case.

5

1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space
   1. at least 15 times,
   2. no more than 12 times.
   3. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).

5 Each year a college has a large fixed number, \(n\), of places to fill. The probability, \(p\), that a randomly chosen student comes from abroad is constant. Using a suitable normal approximation and applying a continuity correction, it is calculated that the probability of more than 60 students coming from abroad is 0.0187 and the probability of fewer than 40 students coming from abroad is 0.0783 . Find the values of \(n\) and \(p\).

2

1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).

1

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 200,0.2 )\). Use a suitable approximation to find \(\mathrm { P } ( X \leqslant 30 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 200,0.02 )\). Use a suitable approximation to find \(\mathrm { P } ( Y \leqslant 3 )\).

In Marumbo, three quarters of the adults own a cell phone.

1. A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. [3]
2. A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. [5]
3. Justify the use of your approximation in part (ii). [1]

The number of accidents per year on a certain road has the distribution \(\text{Po}(\lambda)\). In the past the value of \(\lambda\) was \(3.3\). Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5\%\) significance level.

1. Calculate the probability of a Type I error. [4]
2. Given that \(X = 2\), carry out the test. [3]
3. The council decides to carry out another similar test at the \(5\%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is \(0.6\), calculate the probability of a Type II error. [3]
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than \(10\) accidents in \(30\) years. [4]

\(X\) is a random variable having the distribution \(\text{B}(12, \frac{1}{4})\). A random sample of 60 values of \(X\) is taken. Find the probability that the sample mean is less than 2.8. [5]

1\% of adults in a certain country own a yellow car.

1. Use a suitable approximating distribution to find the probability that a random sample of 240 adults includes more than 2 who own a yellow car. [4]
2. Justify your approximation. [2]

1.5% of the population of the UK can be classified as 'very tall'.

1. The random variable \(X\) denotes the number of people in a sample of \(n\) people who are classified as very tall. Given that E\((X) = 2.55\), find \(n\). [2]
2. By using the Poisson distribution as an approximation to a binomial distribution, calculate an approximate value for the probability that a sample of size 210 will contain fewer than 3 people who are classified as very tall. [3]

The number of calls received at a large call centre has a Poisson distribution with mean 4 calls per 5 minute period.

1. [(c)] Use an approximation to find the probability that the number of calls received in a 5 minute period is between 4 and 9 inclusive. [4]

Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.

1. 1. Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
   2. Evaluate your expression, giving your answer to 3 significant figures. [3]
2. Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
3. Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
4. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]

A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that 20\% of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3, [2]
2. fewer than 2. [2]

One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,

1. use a suitable approximation to find the probability that there are enough first class stamps, [7]
2. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid. [1]

An airline knows that overall 3\% of passengers do not turn up for flights. The airline decides to adopt a policy of selling more tickets than there are seats on a flight. For an aircraft with 196 seats, the airline sold 200 tickets for a particular flight.

1. Write down a suitable model for the number of passengers who do not turn up for this flight after buying a ticket. [2]

By using a suitable approximation, find the probability that

1. more than 196 passengers turn up for this flight, [3]
2. there is at least one empty seat on this flight. [2]

An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.

1. Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour. [2]

Find the probability that in a randomly chosen hour

1. all Internet users connect at their first attempt, [2]
2. more than 4 users fail to connect at their first attempt. [2]

1. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period. [1]
2. Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period. [6]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length L where L ~ N(\(\mu\), 0.3²).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m.

1. Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. [2]

The twine is usually sold in balls of length 100 m. A customer buys three balls of twine.

1. Find the probability that only one of them will have fewer than 6 faults. [6]

As a special order a ball of twine containing 500 m is produced.

1. Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive. [6]

A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.

1. Write down two conditions that must apply for this model to be applicable. [2]

Assuming this model and a mean of 0.7 weeds per m², find

1. the probability that in a randomly chosen plot of size 4 m² there will be fewer than 3 of these weeds, [4]
2. Using a suitable approximation, find the probability that in a plot of 100 m² there will be more than 66 of these weeds. [6]

A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05. Eggs are packed in boxes of 12. Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one, [3]
2. more than three. [2]

A customer bought three boxes.

1. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. [3]

The farmer delivered 10 boxes to a local shop.

1. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. [4]

The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g.

1. Find the probability that a randomly chosen egg weighs more than 68 g. [3]

The discrete random variable \(X\) is distributed B(\(n\), \(p\)).

1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution. [1]
2. Give a reason to support your value. [1]
3. Given that \(n = 200\) and \(p = 0.48\), find P(\(90 \leq X < 105\)). [7]

1. Write down two conditions needed to be able to approximate the binomial distribution by the Poisson distribution. [2]

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,

1. find the probability that in a random sample of 12 people, exactly 2 of them catch the virus. [4]
2. Estimate the probability that in a random sample of 1200 people fewer than 7 catch the virus. [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 disease occurs in 3\% of a population.

1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution. [2]
2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people. [3]
3. Find the mean and variance of the number of people with the disease in a random sample of 100 people. [2]

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

1. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination. [3]

An estate agent sells properties at a mean rate of 7 per week.

1. Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model. [3]
2. Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties. [2]
3. Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties. [6]