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

It is known that on average 85% of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.

1. 1. Find the probability that exactly 12 germinate. [3]
   2. Find the probability that fewer than 12 germinate. [2]

The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the 1% significance level to investigate whether he is correct.

1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
2. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test. [4]
3. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35, complete the test. [3]
4. If \(n\) is small, there is no point in carrying out the test at the 1% significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer. [3]

25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.

1. Find the probability that there are no plants with red flowers in the sample. [2]
2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]

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]

On a production line, bags are filled with cement and weighed as they emerge. It is found that 20\% of the bags are underweight. In a random sample consisting of \(n\) bags, the variance of the number of underweight bags is found to be 2.4.

1. Show that \(n = 15\). [2 marks]
2. Use cumulative binomial probability tables to find the probability that, in a further random sample of 15 bags, the number that are underweight is
   1. less than 3, [3 marks]
   2. at least 5. [2 marks]

Ten samples of 15 bags each are tested. Find the probability that

1. all these batches contain less than 5 underweight bags, [3 marks]
2. the fourth batch tested is the first to contain less than 5 underweight bags. [3 marks]

An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.

1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
2. Find the probability that on one particular day he repairs
   1. no CD players,
   2. more than 6 CD players. [3 marks]
3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]

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]

Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that

1. none of the dice show a score of 6, [3 marks]
2. more than one of the dice shows a score of 6, [4 marks]
3. there are equal numbers of odd and even scores showing on the dice. [3 marks]

One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]

The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives

1. 3 complaints, [3 marks]
2. 10 or more complaints. [2 marks]

The supermarket is open on six days each week.

1. Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [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]

The stem-and-leaf diagram shows the heights, in centimetres, of 15 plants. $\begin{array}{l|l} 0 & 2
1 & 0
2 & 4
3 & 0\ 2\ 4\ 9
4 & 1\ 2\ 4\ 7\ 9
5 & 3\ 7
6 & 2 \end{array}$ Key: \(2 | 5\) means 25 cm.

1. Draw a box-and-whisker plot to illustrate the data. [4]

A statistician intends to analyse the data, but wants to ignore any outliers before doing so.

1. Discuss briefly whether there are any heights in the diagram which the statistician should ignore. [3]

Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.

1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times. [2 marks]
2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions. [2 marks]
3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid. [2 marks]

A mathematical puzzle is published every day in a newspaper. Over a long period of time Paula is able to solve the puzzle correctly 60% of the time.

1. For a randomly chosen 14-day period find the probability that:
   1. Paula correctly solves exactly 8 puzzles [1 mark]
   2. Paula correctly solves at least 7 but not more than 11 puzzles. [2 marks]
2. State one assumption that is necessary for the distribution used in part (a) to be valid. [1 mark]

An archer is training for the Olympics. Each of the archer's training sessions consists of 30 attempts to hit the centre of a target. The archer consistently hits the centre of the target with 79% of their attempts. It can be assumed that the number of times the centre of the target is hit in any training session can be modelled by a binomial distribution.

1. Find the mean of the number of times that the archer hits the centre of the target during a training session. [1 mark]
2. Find the probability that the archer hits the centre of the target exactly 22 times during a particular training session. [1 mark]
3. Find the probability that the archer hits the centre of the target 18 times or less during a particular training session. [1 mark]
4. Find the probability that the archer hits the centre of the target more than 26 times in a training session. [2 marks]

The discrete random variables \(X\) and \(Y\) can be modelled by the distributions $$X \sim \text{B}(40, p)$$ $$Y \sim \text{B}(25, 0.6)$$ It is given that the mean of \(X\) is equal to the variance of \(Y\)

1. Find the value of \(p\) [3 marks]
2. Find P(\(Y = 17\)) [1 mark]

Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution B(\(n\), 0.20).

1. There are 10 students in Ellie's statistics class. Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
   1. two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
   2. at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
2. Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
   1. Name the sampling method used by Declan. [1 mark]
   2. Describe one weakness of this sampling method. [1 mark]
   3. Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [6 marks]

Abu visits his local hardware store to buy six light bulbs. He knows that 15% of all bulbs at this store are faulty.

1. State a distribution which can be used to model the number of faulty bulbs he buys. [1 mark]
2. Find the probability that all of the bulbs he buys are faulty. [1 mark]
3. Find the probability that at least two of the bulbs he buys are faulty. [2 marks]
4. Find the mean of the distribution stated in part (a). [1 mark]
5. State two necessary assumptions in context so that the distribution stated in part (a) is valid. [2 marks]

Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

1. 1. Find the mean number of times he falls off in a day. [1 mark]
   2. Find the variance of the number of times he falls off in a day. [1 mark]
2. 1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
   2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
   1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
   2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts. Of the shirts with defects, the proportion of each type of defect is as shown in the table below.

Type of defect	Colour	Fabric	Sewing	Sizing
Probability	0.25	0.30	0.40	0.05

Shirts with defects are packed in boxes of 30 at random.

1. Find the probability that:
   1. a box contains exactly 5 shirts with a colour defect [2 marks]
   2. a box contains fewer than 15 shirts with a sewing defect [2 marks]
   3. a box contains at least 20 shirts which do not have a fabric defect. [3 marks]
2. Tiana wants to investigate the proportion, \(p\), of defective shirts with a fabric defect. She wishes to test the hypotheses H\(_0\): \(p = 0.3\) H\(_1\): \(p < 0.3\) She takes a random sample of 60 shirts with a defect and finds that \(x\) of them have a fabric defect.
   1. Using a 5\% level of significance, find the critical region for \(x\). [5 marks]
   2. In her sample she finds 13 shirts with a fabric defect. Complete the test stating her conclusion in context. [2 marks]

The random variable \(X\) is such that \(X \sim B(n, p)\) The mean value of \(X\) is 225 The variance of \(X\) is 144 Find \(p\). Circle your answer. [1 mark] 0.36 \quad 0.6 \quad 0.64 \quad 0.8

James is playing a mathematical game on his computer. The probability that he wins is 0.6 As part of an online tournament, James plays the game 10 times. Let \(Y\) be the number of games that James wins.

1. State two assumptions, in context, for \(Y\) to be modelled as \(B(10, 0.6)\) [2 marks]
2. Find \(P(Y = 4)\) [1 mark]
3. Find \(P(Y \geq 4)\) [2 marks]
4. After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them. Test at a 5% significance level whether James's claim is correct. [6 marks]

A customer service centre records every call they receive. It is found that 30% of all calls made to this centre are complaints. A sample of 20 calls is selected. The number of calls in the sample which are complaints is denoted by the random variable \(X\).

1. State two assumptions necessary for \(X\) to be modelled by a binomial distribution. [2 marks]
2. Assume that \(X\) can be modelled by a binomial distribution.
   1. Find P(\(X = 1\)) [1 mark]
   2. Find P(\(X < 4\)) [2 marks]
   3. Find P(\(X \geq 10\)) [2 marks]
3. In a random sample of 10 calls to a school, the number of calls which are complaints, \(Y\), may be modelled by a binomial distribution $$Y \sim \text{B}(10, p)$$ The standard deviation of \(Y\) is 1.5 Calculate the possible values of \(p\). [3 marks]

It is known that, on average, 40% of the drivers who take their driving test at a local test centre pass their driving test. Each day 32 drivers take their driving test at this centre. The number of drivers who pass their test on a particular day can be modelled by the distribution B \((32, 0.4)\)

1. State one assumption, in context, required for this distribution to be used. [1 mark]
2. Find the probability that exactly 7 of the drivers on a particular day pass their test. [1 mark]
3. Find the probability that, at most, 16 of the drivers on a particular day pass their test. [1 mark]
4. Find the probability that more than 12 of the drivers on a particular day pass their test. [2 marks]
5. Find the mean number of drivers per day who pass their test. [1 mark]
6. Find the standard deviation of the number of drivers per day who pass their test. [2 marks]