2.04d Normal approximation to binomial

5. 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,
2. more than three. A customer bought three boxes.
3. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. The farmer delivered 10 boxes to a local shop.
4. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
5. Find the probability that a randomly chosen egg weighs more than 68 g .

6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that \(40 \%\) of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine.
2. Suggest a suitable sampling frame for the survey.
3. Identify the sampling units.
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
5. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
6. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
7. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END}

4. In an experiment, there are 250 trials and each trial results in a success or a failure.

1. Write down two other conditions needed to make this into a binomial experiment. It is claimed that \(10 \%\) of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
2. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
3. Comment on the acceptability of the assumptions you needed to carry out the test.

5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.

1. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
2. Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
3. Using a suitable approximation, find the probability that more than 42 are defective.
   (6)

7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.

1. 1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
   2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
3. Comment on your results for the tests in part (a) and part (b).

3. For a particular type of plant \(45 \%\) have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability that this batch contains

1. exactly 5 plants with white flowers,
2. more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batches of plants.
3. Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50 .
4. Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers.

4.

1. State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.
2. Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.
   During the winter the mean number of yachts hired per week is 5 .
3. Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.
4. Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer. In the summer there are 16 Saturdays on which a yacht can be hired.
5. Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.

6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.

1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
   1. a Poisson approximation,
   2. a Normal approximation.
2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.

1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.

Find the probability that

1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
   

4. A website receives hits at a rate of 300 per hour.

1. State a distribution that is suitable to model the number of hits obtained during a 1 minute interval.
2. State two reasons for your answer to part (a). Find the probability of
3. 10 hits in a given minute,
4. at least 15 hits in 2 minutes. The website will go down if there are more than 70 hits in 10 minutes.
5. Using a suitable approximation, find the probability that the website will go down in a particular 10 minute interval.

The probability of an electrical component being defective is 0.075 The component is supplied in boxes of 120

1. Using a suitable approximation, estimate the probability that there are more than 3 defective components in a box. A retailer buys 2 boxes of components.
2. Estimate the probability that there are at least 4 defective components in each box.
   

2. In a village, power cuts occur randomly at a rate of 3 per year.

1. Find the probability that in any given year there will be
   1. exactly 7 power cuts,
   2. at least 4 power cuts.
2. Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20

4. 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. fewer than 2 . One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,
3. use a suitable approximation to find the probability that there are enough first class stamps.
4. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid.

5. The time taken for a randomly selected person to complete a test is \(M\) minutes, where \(M \sim \mathrm {~N} \left( 14 , \sigma ^ { 2 } \right)\) Given that \(10 \%\) of people take less than 12 minutes to complete the test,

1. find the value of \(\sigma\) Graham selects 15 people at random.
2. Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test. Jovanna takes a random sample of \(n\) people. Using a normal approximation, the probability that fewer than 9 of these \(n\) people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
3. Find the value of \(n\).

In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9

1. Find
   1. \(\mathrm { P } ( X > 5 )\)
   2. \(\mathrm { P } ( 4 \leqslant X < 10 )\) The length of a working day is 7 hours.
2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

2. The continuous random variable \(X\) represents the error, in mm, made when a machine cuts piping to a target length. The distribution of \(X\) is rectangular over the interval \([ - 5.0,5.0 ]\). Find

1. \(\mathrm { P } ( X < - 4.2 )\),
2. \(\mathrm { P } ( | X | < 1.5 )\). A supervisor checks a random sample of 10 lengths of piping cut by the machine.
3. Find the probability that more than half of them are within 1.5 cm of the target length.
   (3 marks)
   If \(X < - 4.2\), the length of piping cannot be used. At the end of each day the supervisor checks a random sample of 60 lengths of piping.
4. Use a suitable approximation to estimate the probability that no more than 2 of these lengths of piping cannot be used.
   (5 marks)

3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable \(D\) represents the greatest distance thrown by each competitor and the random variable \(A\) represents the number of the attempt in which the competitor achieved their greatest distance.

1. State which of the two random variables \(D\) or \(A\) is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
2. Specify a suitable sampling frame and explain how such a sample could be taken.
   (2 marks)
   The coach assumes that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\), and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\),
3. find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
   (6 marks)
4. Comment on the assumption that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\).

6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that

1. at least 4 customers arrive in the next 10 minutes,
2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END

8. A six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6 A group of 50 students want to test whether or not the die is fair for the number six.
The 50 students each roll the die 30 times and record the number of sixes they each obtain.
Given that \(\bar { X }\) denotes the mean number of sixes obtained by the 50 students, and using $$\mathrm { H } _ { 0 } : p = \frac { 1 } { 6 } \text { and } \mathrm { H } _ { 1 } : p \neq \frac { 1 } { 6 }$$ where \(p\) is the probability of rolling a 6 ,

1. use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\), if \(\mathrm { H } _ { 0 }\) is true.
2. Hence find, in terms of \(\bar { X }\), the critical region for this test. Use a \(5 \%\) level of significance.
   

6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter \(p\) equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.

1. Determine the probability that a packet contains:
   1. less than or equal to 15 red pegs;
   2. exactly 10 red pegs;
   3. more than 5 but fewer than 15 red pegs.
2. Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet \(= 10.5\) Variance of number of red pegs per packet \(= 20.41\) Comment on the validity of Sly's claim.

7 The proportion of passengers who use senior citizen bus passes to travel into a particular town on 'Park \& Ride' buses between 9.30 am and 11.30 am on weekdays is 0.45 . It is proposed that, when there are \(n\) passengers on a bus, a suitable model for the number of passengers using senior citizen bus passes is the distribution \(\mathrm { B } ( n , 0.45 )\).

1. Assuming that this model applies to the 10.30 am weekday 'Park \& Ride' bus into the town:
   1. calculate the probability that, when there are \(\mathbf { 1 6 }\) passengers, exactly 3 of them are using senior citizen bus passes;
   2. determine the probability that, when there are \(\mathbf { 2 5 }\) passengers, fewer than 10 of them are using senior citizen bus passes;
   3. determine the probability that, when there are \(\mathbf { 4 0 }\) passengers, at least 15 but at most 20 of them are using senior citizen bus passes;
   4. calculate the mean and the variance for the number of passengers using senior citizen bus passes when there are \(\mathbf { 5 0 }\) passengers.
2. 1. Give a reason why the proposed model may not be suitable.
   2. Give a different reason why the proposed model would not be suitable for the number of passengers using senior citizen bus passes to travel into the town on the 7.15 am weekday 'Park \& Ride' bus.

4 The records at a passport office show that, on average, 15 per cent of photographs that accompany applications for passport renewals are unusable. Assume that exactly one photograph accompanies each application.

1. Determine the probability that in a random sample of 40 applications:
   1. exactly 6 photographs are unusable;
   2. at most 5 photographs are unusable;
   3. more than 5 but fewer than 10 photographs are unusable.
2. Calculate the mean and the standard deviation for the number of photographs that are unusable in a random sample of \(\mathbf { 3 2 }\) applications.
3. Mr Stickler processes 32 applications each day. His records for the previous 10 days show that the numbers of photographs that he deemed unusable were $$\begin{array} { l l l l l l l l l l } 8 & 6 & 10 & 7 & 9 & 7 & 8 & 9 & 6 & 7 \end{array}$$ By calculating the mean and the standard deviation of these values, comment, with reasons, on the suitability of the \(\mathrm { B } ( 32,0.15 )\) model for the number of photographs deemed unusable each day by Mr Stickler.

3 Stopoff owns a chain of hotels. Guests are presented with the bills for their stays when they check out.

1. Assume that the number of bills that contain errors may be modelled by a binomial distribution with parameters \(n\) and \(p\), where \(p = 0.30\). Determine the probability that, in a random sample of 40 bills:
   1. at most 10 bills contain errors;
   2. at least 15 bills contain errors;
   3. exactly 12 bills contain errors.
2. Calculate the mean and the variance for each of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\).
3. Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify for special customer care. However, he regularly finds errors in his bills when he checks out. Each month, during a 12-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recorded, each month, the number of occasions on which his bill contained errors. His recorded values were as follows. $$\begin{array} { l l l l l l l l l l l l } 2 & 1 & 4 & 3 & 1 & 3 & 0 & 3 & 1 & 0 & 5 & 1 \end{array}$$
   1. Calculate the mean and the variance of these 12 values.
   2. Hence state with reasons which, if either, of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\) is likely to provide a satisfactory model for these 12 values.

6 Each weekday, Monday to Friday, Trina catches a train from her local station. She claims that the probability that the train arrives on time at the station is 0.4 , and that the train's arrival time is independent from day to day.

1. Assuming her claims to be true, determine the probability that the train arrives on time at the station:
   1. on at most 3 days during a 2 -week period ( 10 days);
   2. on more than 10 days but fewer than 20 days during an 8-week period.
2. 1. Assuming Trina's claims to be true, determine the mean and standard deviation for the number of times during a week (5 days) that the train arrives on time at the station.
   2. Each week, for a period of 13 weeks, Trina's travelling colleague, Suzie, records the number of times that the train arrives on time at the station. Suzie's results are

      2

      2

      4

      1

      2

      3

      3

      2

      2

      0

      3

      2

      0

      Calculate the mean and standard deviation of these values.
   3. Hence comment on the likely validity of Trina's claims.