2.04b Binomial distribution: as model B(n,p)

6 An amateur tennis club purchases tennis balls that have been used previously in professional tournaments. The probability that each such ball fails a standard bounce test is 0.15 . The club purchases boxes each containing 10 of these tennis balls. Assume that the 10 balls in any box represent a random sample.

1. Determine the probability that the number of balls in a box which fail the bounce test is:
   1. at most 2 ;
   2. at least 2;
   3. more than 1 but fewer than 5 .
2. Determine the probability that, in \(\mathbf { 5 }\) boxes, the total number of balls which fail the bounce test is:
   1. more than 5 ;
   2. at least 5 but at most 10 .

6 A bin contains a very large number of paper clips of different colours. The proportion of each colour is shown in the table.

1. Packets are filled from the bin. Each filled packet contains exactly 30 paper clips which may be considered to be a random sample. Use binomial distributions to determine the probability that a filled packet contains:
   1. exactly 2 purple paper clips;
   2. a total of more than 10 red or purple paper clips;
   3. at least 5 but at most 10 green paper clips.
2. Jumbo packets are also filled from the bin. Each filled jumbo packet contains exactly 100 paper clips.
   1. Assuming that the number of paper clips in a jumbo packet may be considered to be a random sample, calculate the mean and the variance of the number of red paper clips in a filled jumbo packet.
   2. It is claimed that the proportion of red paper clips in the bin is greater than 0.22 and that jumbo packets do not contain random samples of paper clips. An analysis of the number of red paper clips in each of a random sample of 50 filled jumbo packets resulted in a mean of 22.1 and a standard deviation of 4.17. Comment, with numerical justification, on each of the two claims.

6 The probability that an online order from a supermarket chain has at least one item missing when delivered is 0.06 . Online orders are 'incomplete' if they contain substitute items and/or have at least one item missing when delivered. The probability that an order is incomplete is 0.15 .

1. Calculate the probability that exactly 2 out of a random sample of 26 online orders have at least one item missing when delivered.
2. Determine the probability that the number of incomplete orders in a random sample of 50 online orders is:
   1. fewer than 10 ;
   2. more than 5;
   3. more than 6 but fewer than 12 .
3. Farokh, the manager of one of the supermarket's stores, examines 50 randomly selected online orders from each of a random sample of 100 of the store's customers. He records, for each of the 50 orders, the number, \(x\), that were incomplete. His summarised results, correct to three significant figures, for the 100 customers selected are $$\bar { x } = 4.33 \text { and } s ^ { 2 } = 3.94$$ Use this information to compare the performance of the store managed by Farokh with that of the supermarket chain as a whole.
   [0pt] [5 marks]

5 An analysis of the number of vehicles registered by each household within a city resulted in the following information.

1. A random sample of 30 households within the city is selected. Use a binomial distribution with \(n = 30\), together with relevant information from the table in each case, to find the probability that the sample contains:
   1. exactly 3 households with no registered vehicles;
   2. at most 5 households with three or more registered vehicles;
   3. more than 10 households with at least two registered vehicles;
   4. more than 5 households but fewer than 10 households with exactly two registered vehicles.
2. If a random sample of \(\mathbf { 1 5 0 }\) households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-16_1075_1707_1628_153}

3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.

1. Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
   1. aged between 41 and 65;
   2. aged 66 or over and charged at band 2;
   3. aged between 19 and 40 and charged at most at band 1;
   4. aged 41 or over, given that the patient was charged at band 2;
   5. charged at least at band 2, given that the patient was not aged 66 or over.
2. Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
   [0pt] [5 marks]

6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

1. A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
   1. exactly 4 red loom bands;
   2. at most 10 yellow loom bands;
   3. at least 30 blue or green loom bands;
   4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
2. The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box. Estimate values for the mean and the variance of \(R\).
   [0pt] [2 marks]

1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.

1. 1. Write down the distribution of \(X\), the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
   2. Determine \(\mathrm { P } ( X = 20 )\).
   3. Determine \(\mathrm { P } ( 6 \leqslant X \leqslant 18 )\).
2. Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
3. When \(n\) cars pass the speed camera, the number of cars, \(Y\), that exceed 60 mph may be modelled by the distribution \(\mathrm { B } ( n , 0.2 )\). Given that \(n = 20\), determine \(\mathrm { P } ( Y \geqslant 5 )\).
4. Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .

5

1. The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(X\) having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
   1. at least 9 minor accidents;
   2. more than 5 but fewer than 10 minor accidents.
2. The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(Y\) having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
3. The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(T\) having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots \\ 0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
   1. state the value of \(\lambda\);
   2. determine \(\mathrm { P } ( T > 16 )\);
   3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

Lupin seeds are sold in packets of 15 . On average, 9 seeds in a packet are green and 6 are red. Find, to 2 decimal places, the probability that in any particular packet there are

1. less than 2 red seeds,
2. more red than green seeds. The seeds from 10 packets are then combined together.
3. Use a suitable approximation to find the probability that the total number of green seeds is more than 100 .
   

4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).

1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
2. Evaluate this probability when \(p = 0.6\).

5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.

1. Find the probability that none are left-handed.
2. Find the probability that at least two are left-handed.
3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
   Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}

3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.

1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
3. Determine the expected number of answers that Sarah will get right.
4. Find the probability that Sarah gets more than 25 correct answers out of 30.

6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is \(5 \%\). If a tray is selected at random, find

1. the probability that at least two of the apples in it are blemished,
2. the probability that exactly two are blemished. Trays are now packed in boxes of 50 trays each. In one such box, find
3. the probability that at most one tray contains at least two blemished apples,
4. the expected number of trays containing at least two blemished apples.
5. Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.

3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).

1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).

4. A random variable \(X\) has the distribution \(\mathrm { B } ( 80,0.375 )\).

1. Write down the mean and variance of \(X\).
2. Use the Normal approximation to the binomial distribution to estimate \(\mathrm { P } ( X > 40 )\).

6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was \(35 \%\), and the percentage who voted for the Monster Raving Loony party was \(2 \%\).

1. Find the probability that a random sample of 10 electors includes at least two Conservative voters. Use suitable approximations to find
2. the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
3. the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
4. One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.

2. Suggest, with reasons, suitable distributions for modelling each of the following:

1. the number of times the letter J occurs on each page of a magazine,
2. the length of string left over after cutting as many 3 metre long pieces as possible from partly used balls of string,
3. the number of heads obtained when spinning a coin 15 times.

7

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
   2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find, in terms of \(n\) and \(p\), an expression for \(\operatorname { Var } ( X )\).
2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
   1. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
   2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution \(\mathrm { B } ( 10,0.65 )\) has one mode, and find the two values for the mode of the distribution \(B ( 35,0.5 )\).
3. The random variable \(Y\) has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate \(\mathrm { P } ( Y \leqslant k )\), where \(k\) denotes the mode of \(Y\).
   (3 marks)

7 It is claimed that the proportion, \(P\), of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .

1. In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the \(10 \%\) level of significance to investigate the claim.
   (6 marks)
2. It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
   1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, \(\widehat { P }\), and the \(10 \%\) level of significance to again investigate the claim.
   2. The critical value of \(\widehat { P }\) for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that \(P > 0.50\) based on a random sample of 500 people and using the \(10 \%\) level of significance.
      (5 marks)

5

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
   2. Given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find an expression for \(\operatorname { Var } ( X )\).
2. 1. The random variable \(Y\) has a binomial distribution with \(\mathrm { E } ( Y ) = 3\) and \(\operatorname { Var } ( Y ) = 2.985\). Find values for \(n\) and \(p\).
   2. The random variable \(U\) has \(\mathrm { E } ( U ) = 5\) and \(\operatorname { Var } ( U ) = 6.25\). Show that \(U\) does not have a binomial distribution.
3. The random variable \(V\) has the distribution \(\operatorname { Po } ( 5 )\) and \(W = 2 V + 10\). Show that \(\mathrm { E } ( W ) = \operatorname { Var } ( W )\) but that \(W\) does not have a Poisson distribution.
4. The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected. Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative.
   [0pt] [3 marks]

7.

1. Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
2. Write down the distribution of the number of sixes obtained when the ten dice are thrown.
3. Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
4. Find the probability that the mean number of sixes obtained is more than 1.8

5. (a) Explain briefly what you understand by

1. an unbiased estimator,
2. a consistent estimator.
   of an unknown population parameter \(\theta\). From a binomial population, in which the proportion of successes is \(p , 3\) samples of size \(n\) are taken. The number of successes \(X _ { 1 } , X _ { 2 }\), and \(X _ { 3 }\) are recorded and used to estimate \(p\).
   (b) Determine the bias, if any, of each of the following estimators of \(p\). $$\begin{aligned} & \hat { p } _ { 1 } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } } { 3 n } \\ & \hat { p } _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \\ & \hat { p } _ { 3 } = \frac { 2 X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \end{aligned}$$ (c) Find the variance of each of these estimators.
   (d) State, giving a reason, which of the three estimators for \(p\) is
3. the best estimator,
4. the worst estimator.

5. Define

1. a Type I error,
2. the size of a test. Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded. The probability of Jane stating the correct symbol is denoted by \(p\).
   To test the hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) against \(\mathrm { H } _ { 1 } : p > 0.25\), a critical region of \(X > 6\) is used.
3. Find the size of this test.
4. Show that the power function of this test is \(8 p ^ { 7 } - 7 p ^ { 8 }\). Given that \(p = 0.3\), calculate
5. the power of this test,
6. the probability of a Type II error.
7. Suggest two ways in which you might reduce the probability of a Type II error.

7. A bag contains marbles of which an unknown proportion \(p\) is red. A random sample of \(n\) marbles is drawn, with replacement, from the bag. The number \(X\) of red marbles drawn is noted. A second random sample of \(m\) marbles is drawn, with replacement. The number \(Y\) of red marbles drawn is noted. Given that \(p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }\) is an unbiased estimator of \(p\),

1. show that \(a + b = 1\). Given that \(p _ { 2 } = \frac { ( X + Y ) } { n + m }\),
2. show that \(p _ { 2 }\) is an unbiased estimator for \(p\).
3. Show that the variance of \(p _ { 1 }\) is \(p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)\).
4. Find the variance of \(p _ { 2 }\).
5. Given that \(a = 0.4 , m = 10\) and \(n = 20\), explain which estimator \(p _ { 1 }\) or \(p _ { 2 }\) you should use.

A drug is claimed to produce a cure to a certain disease in \(35 \%\) of people who have the disease. To test this claim a sample of 20 people having this disease is chosen at random and given the drug. If the number of people cured is between 4 and 10 inclusive the claim will be accepted. Otherwise the claim will not be accepted.

1. Write down suitable hypotheses to carry out this test.
2. Find the probability of making a Type I error. The table below gives the value of the probability of the Type II error, to 4 decimal places, for different values of \(p\) where \(p\) is the probability of the drug curing a person with the disease.

   P (cure)	0.2	0.3	0.4	0.5
   P (Type II error)	0.5880	\(r\)	0.8565	\(s\)

3. Calculate the value of \(r\) and the value of \(s\).
4. Calculate the power of the test for \(p = 0.2\) and \(p = 0.4\)
5. Comment, giving your reasons, on the suitability of this test procedure.