2.04d Normal approximation to binomial

4. Pieces of ribbon are cut to length \(L \mathrm {~cm}\) where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\)

1. Given that \(30 \%\) of the pieces of ribbon have length more than 100 cm , find the value of \(\mu\) to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
2. Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
3. Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .

7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .

1. Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
2. Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length \(x \mathrm {~m}\) is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
3. Find the value of \(x\).

2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8

1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.

6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a) to be suitable. The probability that fewer than \(r\) customers buy insurance is less than 0.05
3. Find the largest possible value of \(r\). A second random sample, of 100 customers, is taken.
   The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places.
4. Using a suitable approximation, find the value of \(t\). The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
5. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.

1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.

The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75

1. Find the largest possible value of \(v\)
2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
   

2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)

1. 1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
   2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)

7. Last year \(4 \%\) of cars tested in a large chain of garages failed an emissions test. A random sample of \(n\) of these cars is taken. The number of cars that fail the test is represented by \(X\) Given that the standard deviation of \(X\) is 1.44

1. 1. find the value of \(n\)
   2. find \(\mathrm { E } ( X )\) A random sample of 20 of the cars tested is taken.
2. Find the probability that all of these cars passed the emissions test. Given that at least 1 of these cars failed the emissions test,
3. find the probability that exactly 3 of these cars failed the emissions test. A car mechanic claims that more than \(4 \%\) of the cars tested at the garage chain this year are failing the emissions test. A random sample of 125 of these cars is taken and 10 of these cars fail the emissions test.
4. Using a suitable approximation, test whether or not there is evidence to support the mechanic's claim. Use a \(5 \%\) level of significance and state your hypotheses clearly.
   

2. Crispy-crisps produces packets of crisps. During a promotion, a prize is placed in \(25 \%\) of the packets. No more than 1 prize is placed in any packet. A box contains 6 packets of crisps.

1. 1. Write down a suitable distribution to model the number of prizes found in a box.
   2. Write down one assumption required for the model.
2. Find the probability that in 2 randomly selected boxes, only 1 box contains exactly 1 prize.
3. Find the probability that a randomly selected box contains at least 2 prizes. Neha buys 80 boxes of crisps.
4. Using a normal approximation, find the probability that no more than 30 of the boxes contain at least 2 prizes.

4. In a large population, past records show that 1 in 200 adults has a particular allergy. In a random sample of 700 adults selected from the population, estimate

1. 1. the mean number of adults with the allergy,
   2. the standard deviation of the number of adults with the allergy. Give your answer to 3 decimal places. A doctor claims that the past records are out of date and the proportion of adults with the allergy is higher than the records indicate. A random sample of 500 adults is taken from the population and 5 are found to have the allergy. A test of the doctor's claim is to be carried out at the 5\% level of significance.
2. 1. State the hypotheses for this test.
   2. Using a suitable approximation, carry out the test.
      (6) It is also claimed that \(30 \%\) of those with the allergy take medication for it daily. To test this claim, a random sample of \(n\) people with the allergy is taken. The random variable \(Y\) represents the number of people in the sample who take medication for the allergy daily. A two-tailed test, at the \(1 \%\) level of significance, is carried out to see if the proportion differs from 30\% The critical region for the test is \(Y = 0\) or \(Y \geqslant w\)
3. Find the smallest possible value of \(n\) and the corresponding value of \(w\)

5. Cars stop at a service station randomly at a rate of 3 every 5 minutes.

1. Calculate the probability that in a randomly selected 10 minute period,
   1. exactly 7 cars will stop at the service station,
   2. more than 7 cars will stop at the service station. Using a normal approximation, the probability that more than 40 cars will stop at the service station during a randomly selected \(n\) minute period is 0.2266 correct to 4 significant figures.
2. Find the value of \(n\).

3. A manufacturer produces plates. The proportion of plates that are flawed is \(45 \%\), with flawed plates occurring independently. A random sample of 10 of these plates is selected.

1. Find the probability that the sample contains
   1. fewer than 2 flawed plates,
   2. at least 6 flawed plates.
      (4) George believes that the proportion of flawed plates is not \(45 \%\). To assess his belief George takes a random sample of 120 plates. The random variable \(F\) represents the number of flawed plates found in the sample.
2. Using a normal approximation, find the maximum number of plates, \(c\), and the minimum number of plates, \(d\), such that $$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$ where \(F \sim \mathrm {~B} ( 120,0.45 )\) The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
3. Use a suitable hypothesis test, at the \(5 \%\) level of significance, to assess the manufacturer's claim. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-11_2255_50_314_34}

4. In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)

1. Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per \(\mathrm { m } ^ { 2 }\) of the peat bog to follow a Poisson distribution.
   (1) Given that the number of Common Spotted-orchids in \(1 \mathrm {~m} ^ { 2 }\) of the peat bog can be modelled by a Poisson distribution,
2. find the probability that in a randomly selected \(1 \mathrm {~m} ^ { 2 }\) of the peat bog
   1. there are exactly 6 Common Spotted-orchids,
   2. there are fewer than 10 but more than 4 Common Spotted-orchids.
      (4) Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected \(2 \mathrm {~m} ^ { 2 }\) of the peat bog contains 11 Common Spotted-orchids.
3. Using a \(5 \%\) significance level assess Juan's belief. State your hypotheses clearly. Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per \(\mathrm { m } ^ { 2 }\)
4. use a normal approximation to find the probability that in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog there are fewer than 70 Common Spotted-orchids. Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected \(20 \mathrm {~m} ^ { 2 }\) of the peat bog is 0.012 A random sample of 200 non-overlapping \(20 \mathrm {~m} ^ { 2 }\) areas of the peat bog is taken.
5. Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-15_2255_50_314_34}

1. A research project into food purchases found that \(35 \%\) of people who buy eggs do not buy free range eggs.

A random sample of 30 people who bought eggs is taken. The random variable \(F\) denotes the number of people who do not buy free range eggs.

1. Find \(\mathrm { P } ( F \geqslant 12 )\)
2. Find \(\mathrm { P } ( 8 \leqslant F < 15 )\) A farm shop gives 3 loyalty points with every purchase of free range eggs. With every purchase of eggs that are not free range the farm shop gives 1 loyalty point. A random sample of 30 customers who buy eggs from the farm shop is taken.
3. Find the probability that the total number of points given to these customers is less than 70 The manager of the farm shop believes that the proportion of customers who buy eggs but do not buy free range eggs is more than \(35 \%\) In a survey of 200 customers who buy eggs, 86 do not buy free range eggs. Using a suitable test and a normal approximation,
4. determine, at the \(5 \%\) level of significance, whether there is evidence to support the manager's belief. State your hypotheses clearly.

1. A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.

A packet is selected at random.

1. Using a \(5 \%\) level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to \(2.5 \%\), for each tail of your critical region.
2. Write down the actual significance level of this test. Past records suggest that \(2.8 \%\) of the company's sunflower seeds grow to a height of more than 3 metres.
   A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
3. Using a suitable approximation test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than \(2.8 \%\) State your hypotheses clearly.

The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, \(X\), represents the number of people who complete the task in less than 15 minutes.

1. Find \(\mathrm { P } ( 5 \leqslant X < 8 )\) Mia takes a random sample of 140 people.
   Using a normal approximation, the probability that fewer than \(n\) of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.
2. Find the value of \(n\) Show your working clearly.

A supermarket receives complaints at a mean rate of 6 per week.

1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
   1. fewer than 3 complaints received by the supermarket,
   2. at least 6 complaints received by the supermarket. In a randomly selected week, the supermarket received 12 complaints.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
4. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.

The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8

1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera. A car has been caught speeding by this camera.
2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined £60
4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a \(10 \%\) level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.

1. A multiple choice examination paper has \(n\) questions where \(n > 30\)

Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.
7.

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

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. Find the probability that, in a randomly chosen \(\mathbf { 2 }\) hour period,
2. 1. all users connect at their first attempt,
   2. at least 4 users fail to connect at their first attempt. 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 .
3. 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.
   

3. 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. By using a suitable approximation, find the probability that
2. more than 196 passengers turn up for this flight,
3. there is at least one empty seat on this flight.

5. 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. more than 4 users fail to connect at their first attempt.
3. Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period.
4. 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.

3. 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. Assuming this model and a mean of 0.7 weeds per \(\mathrm { m } ^ { 2 }\), find
2. the probability that in a randomly chosen plot of size \(4 \mathrm {~m} ^ { 2 }\) there will be fewer than 3 of these weeds.
3. Using a suitable approximation, find the probability that in a plot of \(100 \mathrm {~m} ^ { 2 }\) there will be more than 66 of these weeds.
   (6)