Identify distribution and parameters

5 Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The random variable \(X\) is the number of these 12 integers which are multiples of 5 .

1. State the distribution of \(X\) and give its parameters.
2. Calculate the probability that \(X\) is between 3 and 5 inclusive. Fiona now produces \(n\) random integers between 7 and 21 inclusive.
3. Find the least possible value of \(n\) if the probability that none of these integers is a multiple of 5 is less than 0.01.

7 At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
2. Find
   (a) \(\mathrm { P } ( X = 3 )\),
   (b) \(\mathrm { P } ( X \geqslant 1 )\).
3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .

6. Past information at a computer shop shows that \(40 \%\) of customers buy insurance when they purchase a product. In a random sample of 30 customers, \(X\) buy insurance.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a) to be suitable. The probability that fewer than \(r\) customers buy insurance is less than 0.05
3. Find the largest possible value of \(r\). A second random sample, of 100 customers, is taken.
   The probability that at least \(t\) of these customers buy insurance is 0.938 , correct to 3 decimal places.
4. Using a suitable approximation, find the value of \(t\). The shop now offers an extended warranty on all products. Following this, a random sample of 25 customers is taken and 6 of them buy insurance.
5. Test, at the \(10 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy insurance has decreased. State your hypotheses clearly.

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 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)

7. As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable \(X\).

1. Suggest a suitable distribution for \(X\).
   (2) Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable \(S\) represents the final score, in points, for an applicant who chooses answers to this test at random.
2. Show that \(S = 5 X - 20\)
3. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\). An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.
4. Find \(\mathrm { P } ( S \geqslant 20 )\)
   (4) Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4
5. Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.

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. A bag contains 40 beads of the same shape and size. The ratio of red to green to blue beads is \(1 : 3 : 4\) and there are no beads of any other colour. In an experiment, a bead is picked at random, its colour noted and the bead replaced in the bag. This is done ten times.

1. Suggest a suitable distribution for modelling the number of times a blue bead is picked out and give the value of any parameters needed.
2. Explain why this distribution would not be suitable if the beads were not replaced in the bag.
3. Find the probability that of the ten beads picked out
   1. five are blue,
   2. at least one is red. The experiment is repeated, but this time a bead is picked out and replaced \(n\) times.
4. Find in the form \(a ^ { n } < b\), where \(a\) and \(b\) are exact fractions, the condition which \(n\) must satisfy in order to have at least a \(99 \%\) chance of picking out at least one red bead.

6 A bank monitors the amounts of cash withdrawn from a cash machine. It categorises any withdrawal of an amount of \(\pounds 50\) or less as 'small' and any withdrawal of an amount greater than \(\pounds 50\) as 'large'. Over a long period of time the bank finds that the proportion of withdrawals that are small is 0.43 .
The bank wishes to model a sample of 10 withdrawals to examine the number of small withdrawals.

1. 1. State a suitable probability distribution for such a model, justifying your answer.
   2. State one assumption needed for the model to be valid.
2. 1. Find the probability that exactly 4 of the 10 withdrawals are small.
   2. Find the probability that exactly 4 of the 10 withdrawals are large.
   3. Find the probability that no more than 4 of the 10 withdrawals are large.
3. Find the probability that, in the 10 withdrawals, the 7th withdrawal is large and there are exactly 3 that are small.

7. Emma throws a fair coin 15 times and records the number of times it shows a head.
a State the appropriate distribution to model the number of times the coin shows a head giving any relevant parameter values.
b Find the probability that Emma records:
i exactly 8 heads
ii at least 4 heads.

6. A retail bakery makes cherry muffins where, due to the production process, \(15 \%\) of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.

1. State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid.
2. Using your model from part (a), find the probability that a randomly selected box contains:
   1. exactly 3 muffins with a lower than expected quantity of cherries,
   2. at least 5 muffins with a lower than expected quantity of cherries.
3. The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries.
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5. At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
2. Find
   (a) \(\mathrm { P } ( X = 3 )\),
3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .
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9 A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents' evening. The number of students in the sample who are from years 12 and 13 is denoted by \(X\).

1. State a suitable binomial model for \(X\). Use your model to answer the following.
2. 1. Write down an expression for \(\mathrm { P } ( X = x )\).
   2. State, in set notation, the values of \(x\) for which your expression is valid.
3. Find \(\mathrm { P } ( 5 \leqslant X \leqslant 9 )\).
4. State one disadvantage of using a random sample in this context.