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

1. A fair 5 -sided spinner has sides numbered \(1,2,3,4\) and 5

The spinner is spun once and the score of the side it lands on is recorded.

1. Write down the name of the distribution that can be used to model the score of the side it lands on. The spinner is spun 28 times.
   The random variable \(X\) represents the number of times the spinner lands on 2
2. 1. Find the probability that the spinner lands on 2 at least 7 times.
   2. Find \(\mathrm { P } ( 4 \leqslant X < 8 )\)

1. Afrika works in a call centre.

She assumes that calls are independent and knows, from past experience, that on each sales call that she makes there is a probability of \(\frac { 1 } { 6 }\) that it is successful. Afrika makes 9 sales calls.

1. Calculate the probability that at least 3 of these sales calls will be successful. The probability of Afrika making a successful sales call is the same each day.
   Afrika makes 9 sales calls on each of 5 different days.
2. Calculate the probability that at least 3 of the sales calls will be successful on exactly 1 of these days. Rowan works in the same call centre as Afrika and believes he is a more successful salesperson. To check Rowan's belief, Afrika monitors the next 35 sales calls Rowan makes and finds that 11 of the sales calls are successful.
3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence to support Rowan's belief.

A manufacturer of sweets knows that \(8 \%\) of the bags of sugar delivered from supplier \(A\) will be damp.
A random sample of 35 bags of sugar is taken from supplier \(A\).

1. Using a suitable model, find the probability that the number of bags of sugar that are damp is
   1. exactly 2
   2. more than 3 Supplier \(B\) claims that when it supplies bags of sugar, the proportion of bags that are damp is less than \(8 \%\) The manufacturer takes a random sample of 70 bags of sugar from supplier \(B\) and finds that only 2 of the bags are damp.
2. Carry out a suitable test to assess supplier B's claim. You should state your hypotheses clearly and use a \(10 \%\) level of significance.

1. Julia selects 3 letters at random, one at a time without replacement, from the word

\section*{VARIANCE} The discrete random variable \(X\) represents the number of times she selects a letter A.

1. Find the complete probability distribution of \(X\). Yuki selects 10 letters at random, one at a time with replacement, from the word \section*{DEVIATION}
2. Find the probability that he selects the letter E at least 4 times.

4. The random variable \(X \sim \mathrm {~B} ( 27,0.35 )\)

1. Find
   1. \(\mathrm { P } ( X = 10 )\)
   2. \(\mathrm { P } ( 12 \leqslant X < 15 )\) Historical records show that the proportion of defective items produced by a machine is 0.12 Following a maintenance service of the machine, a random sample of 60 items is taken and 3 defective items are found.
2. Carry out a suitable test to determine whether the proportion of defective items produced by the machine has decreased following the maintenance service. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
3. Write down the \(p\)-value for your test in part (b)

1. A nursery has a sack containing a large number of coloured beads of which \(14 \%\) are coloured red.

Aliya takes a random sample of 18 beads from the sack to make a bracelet.

1. State a suitable binomial distribution to model the number of red beads in Aliya's bracelet.
2. Use this binomial distribution to find the probability that
   1. Aliya has just 1 red bead in her bracelet,
   2. there are at least 4 red beads in Aliya's bracelet.
3. Comment on the suitability of a binomial distribution to model this situation. After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads.
4. Stating your hypotheses clearly, use a 5\% significance level to carry out a suitable test for the teacher.
5. Find the \(p\)-value in this case.

1. Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below.

One of the 184 days is selected at random.

1. Find the probability that it has a daily mean total cloud cover of 6 or greater. Magali is investigating whether the daily mean total cloud cover can be modelled using a binomial distribution. She uses the random variable \(X\) to denote the daily mean total cloud cover and believes that \(X \sim \mathrm {~B} ( 8,0.76 )\) Using Magali's model,
2. 1. find \(\mathrm { P } ( X \geqslant 6 )\)
   2. find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7
3. Explain whether or not your answers to part (b) support the use of Magali's model. There were 28 days that had a daily mean total cloud cover of 8 For these 28 days the daily mean total cloud cover for the following day is shown in the table below.

   Daily mean total cloud cover (oktas)	0	1	2	3	4	5	6	7	8
   Frequency (number of days)	0	0	1	1	2	1	5	9	9

4. Find the proportion of these days when the daily mean total cloud cover was 6 or greater.
5. Comment on Magali's model in light of your answer to part (d).

1. George throws a ball at a target 15 times.

Each time George throws the ball, the probability of the ball hitting the target is 0.48
The random variable \(X\) represents the number of times George hits the target in 15 throws.

1. Find
   1. \(\mathrm { P } ( X = 3 )\)
   2. \(\mathrm { P } ( X \geqslant 5 )\) George now throws the ball at the target 250 times.
2. Use a normal approximation to calculate the probability that he will hit the target more than 110 times.

1. A machine fills packets with sweets and \(\frac { 1 } { 7 }\) of the packets also contain a prize.

The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable \(T\) represents the number of packets of sweets that contain a prize in each box.

1. State a condition needed for \(T\) to be modelled by \(\mathrm { B } \left( 40 , \frac { 1 } { 7 } \right)\) A box is selected at random.
2. Using \(T \sim \mathrm {~B} \left( 40 , \frac { 1 } { 7 } \right)\) find
   1. the probability that the box has exactly 6 packets containing a prize,
   2. the probability that the box has fewer than 3 packets containing a prize. Kamil's sweet shop buys 5 boxes of these sweets.
3. Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than \(\frac { 1 } { 7 }\) A random sample of 110 packets is taken and 9 packets contain a prize.
4. Use a suitable test to assess Kamil's claim. You should

1. Xian rolls a fair die 10 times.

The random variable \(X\) represents the number of times the die lands on a six.

1. Using a suitable distribution for \(X\), find
   1. \(\mathrm { P } ( X = 3 )\)
   2. \(\mathrm { P } ( X < 3 )\) Xian repeats this experiment each day for 60 days and records the number of days when \(X = 3\)
2. Find the probability that there were at least 12 days when \(X = 3\)
3. Find an estimate for the total number of sixes that Xian will roll during these 60 days.
4. Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.

10 Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.

1. Use a binomial model to test the manufacturer's claim, at the \(2.5 \%\) significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
2. Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.

12 The variable \(X\) has the distribution \(\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)\). The probabilities \(\mathrm { P } ( X = r )\) for \(r = 0\) to 50 are given by the terms of the expansion of \(( a + b ) ^ { n }\) for specific values of \(a , b\) and \(n\).

1. State the values of \(a\), \(b\) and \(n\). A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the \(5 \%\) significance level.
2. Write down suitable hypotheses for the test.
3. Determine the rejection region for the test, showing the values of any relevant probabilities.

9 Last year, market research showed that \(8 \%\) of adults living in a certain town used a particular local coffee shop. Following an advertising campaign, it was expected that this proportion would increase. In order to test whether this had happened, a random sample of 150 adults in the town was chosen. The random variable \(X\) denotes the number of these 150 adults who said that they used the local coffee shop.

1. 1. Assuming that the proportion of adults using the local coffee shop is unchanged from the previous year, state a suitable binomial distribution with which to model the variable \(X\).
   2. The probabilities given by this model are the terms of the binomial expansion of an expression of the form \(( a + b ) ^ { n }\). Write down this expression, using appropriate values of \(a , b\) and \(n\). It was found that 18 of these 150 adults said that they use the local coffee shop.
2. Test, at the 5\% significance level, whether the proportion of adults in the town who use the local coffee shop has increased. It was later discovered by a statistician that the random sample of 150 adults had been chosen from shoppers in the town on a Friday and a Saturday.
3. Explain why this suggests that the assumptions made when using a binomial model for \(X\) may not be valid in this context.

11 Alex models the number of goals that a local team will score in any match as follows. The number of goals scored in any match is independent of the number of goals scored in any other match.

1. Alex chooses 3 matches at random. Use the model to determine the probability of each of the following.
   1. The team will score a total of exactly 1 goal in the 3 matches.
   2. The numbers of goals scored in the first 2 of the 3 matches will be equal, but the number of goals scored in the 3rd match will be different. During the first 10 matches this season, the team scores a total of 31 goals.
2. Without carrying out a formal test, explain briefly whether this casts doubt on the validity of Alex's model. \section*{END OF QUESTION PAPER}

5 Each day John either cycles to work or goes on the bus.

• If it is raining when John is ready to set off for work, the probability that he cycles to work is 0.4.
• If it is not raining when John is ready to set off for work, the probability that he cycles to work is 0.9 .
• The probability that it is raining when he is ready to set off for work is 0.2 .

You should assume that days on which it rains occur randomly and independently.

1. Draw a tree diagram to show the possible outcomes and their associated probabilities.
2. Calculate the probability that, on a randomly chosen day, John cycles to work. John works 5 days each week.
3. Calculate the probability that he cycles to work every day in a randomly chosen working week.

8 According to the latest research there are 19.8 million male drivers and 16.2 million female drivers on the roads in the UK.

1. A driver in the UK is selected at random. Find the probability that the driver is male.
2. Calculate the probability that there are 7 female drivers in a random sample of 25 UK drivers. When driving in a built-up area, Rebecca exceeded the speed limit and was obliged to attend a speed awareness course. Her husband said "It's nearly always male drivers who are speeding." When Rebecca attends the course, she finds that there are 25 drivers, 7 of whom are female. You should assume that the drivers on the speed awareness course constitute a random sample of drivers caught speeding.
3. In this question you must show detailed reasoning. Conduct a hypothesis test to determine whether there is any evidence at the \(5 \%\) level to suggest that male drivers are more likely to exceed the speed limit than female drivers.
4. State a modelling assumption that is necessary in order to conduct the hypothesis test in part (c).

15 Ali and Sam are playing a game in which Ali tosses a coin 5 times. If there are 4 or 5 heads, Ali wins the game. Otherwise Sam wins the game. They decide to play the game 50 times.

1. Initially Sam models the situation by assuming the coin is fair. Determine the number of games Ali is expected to win according to this model. Ali thinks the coin may be biased, with probability \(p\) of obtaining heads when the coin is tossed. Before playing the game, Ali and Sam decide to collect some data to estimate the value of \(p\). Sam tosses the coin 15 times and records the number of heads obtained. Ali tosses the coin 25 times and records the number of heads obtained.
2. Explain why it is better to use the combined data rather than just Sam's data or just Ali's data to estimate the value of \(p\). Ali records 20 heads and Sam records 8 heads.
3. Use the combined data to estimate the value of \(p\). Ali now models the situation using the value of \(p\) found in part (c) as the probability of obtaining heads when the coin is tossed.
4. Determine how many games Ali expects to win using this value of \(p\) to model the situation.
5. Ali wins 25 of the 50 games. Explain whether Sam's model or Ali's model is a better fit for the data. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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4 In a certain country it is known that 11\% of people are left-handed.

1. Calculate the probability that, in a random sample of 98 people from this country, 5 or fewer are found to be left-handed, giving your answer correct to 3 significant figures. An anthropologist believes that the proportion of left-handed people is lower in a particular ethnic group. The anthropologist collects a random sample of 98 people from this particular ethnic group in order to test the hypothesis that the proportion of left-handed people is less than \(11 \%\). The anthropologist carries out the test at the \(1 \%\) level.
2. Determine the critical region for this test.

12 A manufacturer of steel rods checks the length of each rod in randomly selected batches of 10 rods. 100 batches of 10 rods are checked and \(x\), the number of rods in each batch which are too long, is recorded. Summary statistics are as follows. \(n = 100\) $$\sum x = 210 \quad \sum x ^ { 2 } = 604$$

1. Calculate
   Layla decides to use a binomial distribution to model the number of rods which are too long in a batch of 10 .
2. Write down the parameters that Layla should use in her model.
3. Use Layla's model to determine the expected number of batches out of 100 in which there are exactly 2 rods which are too long.

10 A company operates trains. The company claims that \(92 \%\) of its trains arrive on time. You should assume that in a random sample of trains, they arrive on time independently of each other.

1. Assuming that \(92 \%\) of the company's trains arrive on time, find the probability that in a random sample of 30 trains operated by this company
   1. exactly 28 trains arrive on time,
   2. more than 27 trains arrive on time. A journalist believes that the percentage of trains operated by this company which arrive on time is lower than \(92 \%\).
2. To investigate the journalist's belief a hypothesis test will be carried out at the \(1 \%\) significance level. A random sample of 18 trains is selected. For this hypothesis test,

8 Every morning before breakfast Laura and Mike play a game of chess. The probability that Laura wins is 0.7 . The outcome of any particular game is independent of the outcome of other games. Calculate the probability that, in the next 20 games,

1. Laura wins exactly 14 games,
2. Laura wins at least 14 games.

8 A team called "The Educated Guess" enter a weekly quiz. If they win the quiz in a particular week, the probability that they will win the following week is 0.4 , but if they do not win, the probability that they will win the following week is 0.2 . In week 4 The Educated Guess won the quiz.

1. Calculate the probability that The Educated Guess will win the quiz in week 6. Every week the same 20 quiz teams, each with 6 members, take part in a quiz. Every member of every team buys a raffle ticket. Five winning tickets are drawn randomly, without replacement. Alf, who is a member of one of the teams, takes part every week.
2. Calculate the probability that, in a randomly chosen week, Alf wins a raffle prize.
3. Find the smallest number of weeks after which it will be \(95 \%\) certain that Alf has won at least one raffle prize.

4 A biased octagonal dice has faces numbered from 1 to 8 . The discrete random variable \(X\) is the score obtained when the dice is rolled once. The probability distribution of \(X\) is shown in the table below.

1. Determine the value of \(p\).
2. Find the probability that a score of at least 4 is obtained when the dice is rolled once. The dice is rolled 30 times.
3. Determine the probability that a score of 8 occurs exactly twice.

5 It is known that 40\% of people in Britain carry a certain gene.
A random sample of 32 people is collected.

1. Calculate the probability that exactly 12 people carry the gene.
2. Calculate the probability that at least 8 people carry the gene, giving your answer correct to \(\mathbf { 3 }\) decimal places.