5.02d Binomial: mean np and variance np(1-p)

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.

1. A salesman sells insurance to people. Each day he chooses a number of people to contact. The probability that the salesman sells insurance to a person he contacts is 0.05

On Monday he chooses to contact 10 people.

1. Find the probability that on Monday the salesman sells insurance to
   1. exactly 1 person,
   2. at least 3 people.
2. Find the number of people he should contact each day in order to sell insurance, on average, to 3 people per day.
3. Calculate the least number of people he must choose to contact on Friday, so that the probability of selling insurance to at least 1 person on Friday exceeds 0.99

1. A large number of students sat an examination. All of the students answered the first question. The first question was answered correctly by \(40 \%\) of the students.

In a random sample of 20 students who sat the examination, \(X\) denotes the number of students who answered the first question correctly.

1. Write down the distribution of the random variable \(X\)
2. Find \(\mathrm { P } ( 4 \leqslant X < 9 )\) Students gain 7 points if they answer the first question correctly and they lose 3 points if they do not answer it correctly.
3. Find the probability that the total number of points scored on the first question by the 20 students is more than 0
4. Calculate the variance of the total number of points scored on the first question by a random sample of 20 students.

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.

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.

6. The owner of a small restaurant decides to change the menu. A trade magazine claims that \(40 \%\) of all diners choose organic foods when eating away from home. On a randomly chosen day there are 20 diners eating in the restaurant.

1. Assuming the claim made by the trade magazine to be correct, suggest a suitable model to describe the number of diners \(X\) who choose organic foods.
2. Find \(\mathrm { P } ( 5 < X < 15 )\).
3. Find the mean and standard deviation of \(X\). The owner decides to survey her customers before finalising the new menu. She surveys 10 randomly chosen diners and finds 8 who prefer eating organic foods.
4. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the proportion of diners in her restaurant who prefer to eat organic foods is higher than the trade magazine's claim. State your hypotheses clearly.
   (5)

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 .

7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).

1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
   (4) END

5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.

1. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
2. Estimate the probability there are less than 5 defective articles.

4. Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager questions whether or not there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.

1. Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's question. You should state the probability of rejection in each tail which should be less than 0.05 .
2. Write down the actual significance level of a test based on your critical region from part (a). The manager found that 11 customers from the sample of 20 had bought baked beans in single tins.
3. Comment on this finding in the light of your critical region found in part (a).

1. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].

A stick is selected at random from the box.

1. Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
2. Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
3. Find the probability of winning a soft toy.

5. Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm .

1. Find the probability that Jim's plank contains at most 3 defects. Shivani buys 6 planks each of length 100 cm .
2. Find the probability that fewer than 2 of Shivani's planks contain at most 3 defects.
3. Using a suitable approximation, estimate the probability that the total number of defects on Shivani's 6 planks is less than 18.

2. A test statistic has a distribution \(\mathrm { B } ( 25 , p )\). Given that $$\mathrm { H } _ { 0 } : p = 0.5 \quad \mathrm { H } _ { 1 } : p \neq 0.5$$

1. find the critical region for the test statistic such that the probability in each tail is as close as possible to \(2.5 \%\).
2. State the probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) using this critical region.

4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.

1. Find the probability that in the next 4 weeks the estate agent sells,
   1. exactly 3 houses,
   2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
2. Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
3. Use a suitable approximation to estimate the probability that the estate agent receives a bonus.

2. A fair coin is spun 6 times and the random variable \(T\) represents the number of tails obtained.

1. Give two reasons why a binomial model would be a suitable distribution for modelling \(T\).
2. Find \(\mathrm { P } ( T = 5 )\)
3. Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is \(\frac { 1 } { 4 }\) This second coin is spun 6 times.
4. Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.

6. A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.

1. Show that the mean number of defective items per sample is 2.91 A factory manager suggests that the data can be modelled by a binomial distribution with \(n = 6\). He uses the mean from the sample above and calculates expected frequencies as shown in the table below.

   Number of defective items	0	1	2	3	4	5	6
   Expected frequency	1.87	10.54	24.82	\(a\)	22.01	8.29	\(b\)

2. Calculate the value of \(a\) and the value of \(b\) giving your answers to 2 decimal places.
3. Test, at the \(5 \%\) level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly.

5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).

1. Given that $$\mathrm { E } ( X ) = n p \quad \text { and } \quad \mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$$ find an expression for \(\operatorname { Var } ( X )\).
2. Given that \(X\) has a mean of 36 and a standard deviation of 4.8:
   1. find values for \(n\) and \(p\);
   2. use a distributional approximation to estimate \(\mathrm { P } ( 30 < X < 40 )\).

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)

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]