2.04d Normal approximation to binomial

A manufacturer produces large quantities of coloured mugs. It is known from previous records that 6\% of the production will be green. A random sample of 10 mugs was taken from the production line.

1. Define a suitable distribution to model the number of green mugs in this sample. [1]
2. Find the probability that there were exactly 3 green mugs in the sample. [3]

A random sample of 125 mugs was taken.

1. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
   1. a Poisson approximation, [3]
   2. a Normal approximation. [6]

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games, [3]
2. fewer than half of the games. [2]

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

1. Calculate the mean and variance for the number of these 60 games that Bhim loses. [2]
2. Using a suitable approximation calculate the probability that Bhim loses more than 4 games. [3]

A company has a large number of regular users logging onto its website. On average 4 users every hour fail to connect to the company's website at their first attempt.

1. Explain why the Poisson distribution may be a suitable model in this case. [1]

Find the probability that, in a randomly chosen 2 hour period,

1. 1. all users connect at their first attempt,
   2. at least 4 users fail to connect at their first attempt.
   [5]

The company suffered from a virus infecting its computer system. During this infection it was found that the number of users failing to connect at their first attempt, over a 12 hour period, was 60.

1. Using a suitable approximation, test whether or not the mean number of users per hour who failed to connect at their first attempt had increased. Use a 5\% level of significance and state your hypotheses clearly. [9]

In a survey it is found that barn owls occur randomly at a rate of 9 per 1000 km\(^2\).

1. Find the probability that in a randomly selected area of 1000 km\(^2\) there are at least 10 barn owls. [2]
2. Find the probability that in a randomly selected area of 200 km\(^2\) there are exactly 2 barn owls. [3]
3. Using a suitable approximation, find the probability that in a randomly selected area of 50000 km\(^2\) there are at least 470 barn owls. [6]

The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval [0, 0.5]

1. Find P(\(L < 0.4\)). [1]
2. Write down E(\(L\)). [1]
3. Calculate Var(\(L\)). [2]

A random sample of 30 poles cut by this machine is taken.

1. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. [3]

When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function F(\(x\)) where $$\text{F}(x) = \begin{cases} 0 & x < 0 \\ 4x - 4x^2 & 0 \leq x \leq 0.5 \\ 1 & \text{otherwise} \end{cases}$$

1. Using this model, find P(\(X > 0.4\)) [2]

A random sample of 100 poles cut by this new machine is taken.

1. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres. [3]

In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]

Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.

1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. State the significance level of this test giving your answer to 2 significant figures. [1]

A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]

Calculate the probability that a randomly selected sample square contains

1. no sheep, [1]
2. more than 2 sheep. [4]

A sheepdog has been sent into the field to round up the sheep.

1. Explain why the model may no longer be applicable. [1]

In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.

1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]

On average, 35\% of the candidates in a certain subject get an A or B grade in their exam. In a class of 20 students, find the probability that

1. less than 5 get A or B grades, [2 marks]
2. exactly 8 get A or B grades. [2 marks]

Five such classes of 20 students are combined to sit the exam.

1. Use a suitable approximation to find the probability that less than a quarter of the total get A or B grades. [6 marks]

A greengrocer sells apples from a barrel in his shop. He claims that no more than 5\% of the apples are of poor quality. When he takes 10 apples out for a customer, 2 of them are bad.

1. Stating your hypotheses clearly, test his claim at the 1\% significance level. [5 marks]
2. State an assumption that has been made about the selection of the apples. [1 mark]
3. When five other customers also buy 10 apples each, the numbers of bad apples they get are 1, 3, 1, 2 and 1 respectively. By combining all six customers' results, and using a suitable approximation, test at the 1\% significance level whether the combined results provide evidence that the proportion of bad apples in the barrel is greater than 5\%. [5 marks]
4. Comment briefly on your results in parts (a) and (c). [1 mark]

A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,

1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page. [1 mark]
2. Find the probability that a particular page has more than 2 misprints. [3 marks]
3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. [2 marks]

Chapter 2 is longer, at 20 pages.

1. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. [7 marks]

A certain Sixth Former is late for school once a week, on average. In a particular week of 5 days, find the probability that

1. he is not late at all, [2 marks]
2. he is late more than twice. [3 marks]

In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.

1. Use the Normal approximation to estimate the probability that he loses his privileges. [7 marks]

When a park is redeveloped, it is claimed that 70\% of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,

1. 6 or more approve, [3 marks]
2. exactly 7 approve. [3 marks]

A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.

1. Use this information to carry out a hypothesis test on the original claim, working at the 5\% significance level. State your conclusion clearly. [5 marks]

If the conservationists are right, and only 45\% approve of the new park,

1. use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve. [7 marks]

In an orchard, all the trees are either apple or pear trees. There are four times as many apple trees as pear trees. Find the probability that, in a random sample of 10 trees, there are

1. equal numbers of apple and pear trees, [3 marks]
2. more than 7 apple trees. [3 marks]

In a sample of 60 trees in the orchard,

1. find the expected number of pear trees. [1 mark]
2. Calculate the standard deviation of the number of pear trees and compare this result with the standard deviation of the number of apple trees. [2 marks]
3. Find the probability that exactly 35 in the sample of 60 trees are pear trees. [4 marks]
4. Find an approximate value for the probability that more than 15 of the 60 trees are pear trees. [5 marks]

A small opinion poll shows that the Trendies have a \(10\%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10\%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.

1. At the \(5\%\) significance level, test which is right, stating your null hypothesis carefully. [6 marks]
2. If it is indeed true that the Trendies are supported by \(55\%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes. [8 marks]

In a large town, 35% of the inhabitants have access to television channel \(C\). A random sample of 60 inhabitants is obtained. Use a suitable approximation to find the probability that 18 or fewer inhabitants in the sample have access to channel \(C\). [6]

Buttercups in a meadow are distributed independently of one another and at a constant average incidence of 3 buttercups per square metre.

1. Find the probability that in 1 square metre there are more than 7 buttercups. [2]
2. Find the probability that in 4 square metres there are either 13 or 14 buttercups. [3]
3. Use a suitable approximation to find the probability that there are no more than 69 buttercups in 20 square metres. [5]
4. 1. Without using an approximation, find an expression for the probability that in \(m\) square metres there are at least 2 buttercups. [2]
   2. It is given that the probability that there are at least 2 buttercups in \(m\) square metres is 0.9. Using your answer to part (a), show numerically that \(m\) lies between 1.29 and 1.3. [4]

The random variable \(Y\) has the distribution B(140, 0.03). Use a suitable approximation to find P(\(Y = 5\)). Justify your approximation. [5]

The discrete random variable \(H\) takes values 1, 2, 3 and 4. It is given that E(\(H\)) = 2.5 and Var(\(H\)) = 1.25. The mean of a random sample of 50 observations of \(H\) is denoted by \(\bar{H}\). Use a suitable approximation to find P(\(\bar{H} < 2.6\)). [5]

In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.

1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. [2]

Assume now that a Poisson model is appropriate.

1. Find the probability that in 10 ml there are at least 5 bacteria. [2]
2. Find the probability that in 3.7 ml there are exactly 2 bacteria. [3]
3. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation. [7]

The random variable \(F\) has the distribution B\((40, 0.65)\). Use a suitable approximation to find P\((F \leq 30)\), justifying your approximation. [7]

A teacher wants to investigate the sports played by students at her school in their free time. She decides to ask a random sample of 120 pupils to complete a short questionnaire.

1. Give two reasons why the teacher might choose to use a sample survey rather than a census. [2 marks]
2. Suggest a suitable sampling frame that she could use. [1 mark]

The teacher believes that 1 in 20 of the students play tennis in their free time. She uses the data collected from her sample to test if the proportion is different from this.

1. Using a suitable approximation and stating the hypotheses that she should use, find the critical region for this test. The probability for each tail of the region should be as close as possible to 5\%. [6 marks]
2. State the significance level of this test. [1 mark]

As part of a business studies project, 8 groups of students are each randomly allocated 10 different shares from a listing of over 300 share prices in a newspaper. Each group has to follow the changes in the price of their shares over a 3-month period. At the end of the 3 months, 35\% of all the shares in the listing have increased in price and the rest have decreased.

1. Find the probability that, for the 10 shares of one group,
   1. exactly 6 have gone up in price,
   2. more than 5 have gone down in price. [5 marks]
2. Using a suitable approximation, find the probability that of the 80 shares allocated in total to the groups, more than 35 will have decreased in value. [6 marks]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]

The sales staff at an insurance company make house calls to prospective clients. Past records show that 30% of the people visited will take out a new policy with the company. On a particular day, one salesperson visits 8 people. Find the probability that, of these,

1. exactly 2 take out new policies, [3 marks]
2. more than 4 take out new policies. [2 marks]

The company awards a bonus to any salesperson who sells more than 50 policies in a month.

1. Using a suitable approximation, find the probability that a salesperson gets a bonus in a month in which he visits 150 prospective clients. [5 marks]