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

3 A factory owner models the number of employees who use the factory canteen on any day by the distribution \(\mathrm { B } ( 25 , p )\). In the past the value of \(p\) was 0.8 . A new menu is introduced in the canteen and the owner wants to test whether the value of \(p\) has increased. On a randomly chosen day he notes that the number of employees who use the canteen is 23 .

1. Use the binomial distribution to carry out the test at the \(10 \%\) significance level.
2. Given that there are 30 employees at the factory comment on the suitability of the owner's model. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-04_2713_33_111_2016} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-05_2716_29_107_22}

4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.

1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.

1 At the 2009 election, \(\frac { 1 } { 3 }\) of the voters in Chington voted for the Citizens Party. One year later, a researcher questioned 20 randomly selected voters in Chington. Exactly 3 of these 20 voters said that if there were an election next week they would vote for the Citizens Party. Test at the \(2.5 \%\) significance level whether there is evidence of a decrease in support for the Citizens Party in Chington, since the 2009 election.

3 At an election in 2010, 15\% of voters in Bratfield voted for the Renewal Party. One year later, a researcher asked 30 randomly selected voters in Bratfield whether they would vote for the Renewal Party if there were an election next week. 2 of these 30 voters said that they would.

1. Use a binomial distribution to test, at the \(4 \%\) significance level, the null hypothesis that there has been no change in the support for the Renewal Party in Bratfield against the alternative hypothesis that there has been a decrease in support since the 2010 election.
2. (a) Explain why the conclusion in part (i) cannot involve a Type I error.
   (b) State the circumstances in which the conclusion in part (i) would involve a Type II error.

4 The proportion of people who have a particular gene, on average, is 1 in 1000. A random sample of 3500 people in a certain country is chosen and the number of people, \(X\), having the gene is found.

1. State the distribution of \(X\) and state also an appropriate approximating distribution. Give the values of any parameters in each case. Justify your choice of the approximating distribution.
2. Use the approximating distribution to find \(\mathrm { P } ( X \leqslant 3 )\).

1 On average 1 in 25000 people have a rare blood condition. Use a suitable approximating distribution to find the probability that fewer than 2 people in a random sample of 100000 have the condition.

7 Each day at a certain doctor's surgery there are 70 appointments available in the morning and 60 in the afternoon. All the appointments are filled every day. The probability that any patient misses a particular morning appointment is 0.04 , and the probability that any patient misses a particular afternoon appointment is 0.05 . All missed appointments are independent of each other. Use suitable approximating distributions to answer the following.

1. Find the probability that on a randomly chosen morning there are at least 3 missed appointments.
2. Find the probability that on a randomly chosen day there are a total of exactly 6 missed appointments.
3. Find the probability that in a randomly chosen 10-day period there are more than 50 missed appointments.

1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).

1 On average, 1 in 150 components made by a certain machine are faulty. The random variable \(X\) denotes the number of faulty components in a random sample of 500 components.

1. Describe fully the distribution of \(X\).
2. State a suitable approximating distribution for \(X\), giving a justification for your choice.
3. Use your approximating distribution to find the probability that the sample will include at least 3 faulty components.

4 A train company claims that \(92 \%\) of trains on a particular line arrive on time. Sanjeep suspects that the true percentage is less than \(92 \%\). He chooses a random sample of 20 trains on this line and finds that exactly 16 of them arrive on time. Making an assumption that should be stated, test at the 5\% significance level whether Sanjeep's suspicion is justified.
[0pt] [6]

7 At a certain hospital it was found that the probability that a patient did not arrive for an appointment was 0.2 . The hospital carries out some publicity in the hope that this probability will be reduced. They wish to test whether the publicity has worked.

1. It is suggested that the first 30 appointments on a Monday should be used for the test. Give a reason why this is not an appropriate sample.
   A suitable sample of 30 appointments is selected and the number of patients that do not arrive is noted. This figure is used to carry out a test at the 5\% significance level.
2. Explain why the test is one-tail and state suitable null and alternative hypotheses.
3. State what is meant by a Type I error in this context.
4. Use the binomial distribution to find the critical region, and find the probability of a Type I error.
5. In fact 3 patients out of the 30 do not arrive. State the conclusion of the test, explaining your answer.

5 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \sqrt { } 2\).
2. Find \(\mathrm { P } ( X > 0.4 )\).
3. Find the upper quartile of \(X\).
4. Find the probability that exactly 3 out of 5 random observations of \(X\) have values greater than the upper quartile.

3 A book contains 40000 words. For each word, the probability that it is printed wrongly is 0.0001 and these errors occur independently. The number of words printed wrongly in the book is represented by the random variable \(X\).

1. State the exact distribution of \(X\), including the values of any parameters.
2. State an approximate distribution for \(X\), including the values of any parameters, and explain why this approximate distribution is appropriate.
3. Use this approximate distribution to find the probability that there are more than 3 words printed wrongly in the book.

7 The number of workers, \(X\), absent from a factory on a particular day has the distribution \(\mathrm { B } ( 80,0.01 )\).

1. Explain why it is appropriate to use a Poisson distribution as an approximating distribution for \(X\).
2. Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 . Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.
3. During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the \(2 \%\) significance level.

6 At the last election, 70\% of people in Apoli supported the president. Luigi believes that the same proportion support the president now. Maria believes that the proportion who support the president now is \(35 \%\). In order to test who is right, they agree on a hypothesis test, taking Luigi's belief as the null hypothesis. They will ask 6 people from Apoli, chosen at random, and if more than 3 support the president they will accept Luigi's belief.

1. Calculate the probability of a Type I error.
2. If Maria's belief is true, calculate the probability of a Type II error.
3. In fact 2 of the 6 people say that they support the president. State which error, Type I or Type II, might be made. Explain your answer.

5 It is known that when seeds of a certain type are planted, on average \(10 \%\) of the resulting plants reach a height of 1 metre. A gardener wishes to investigate whether a new fertiliser will increase this proportion. He plants a random sample of 18 seeds of this type, using the fertiliser, and notes how many of the resulting plants reach a height of 1 metre.

1. In fact 4 of the 18 plants reach a height of 1 metre. Carry out a hypothesis test at the \(8 \%\) significance level.
2. Explain which of the errors, Type I or Type II, might have been made in part (i). Later, the gardener plants another random sample of 18 seeds of this type, using the fertiliser, and again carries out a hypothesis test at the \(8 \%\) significance level.
3. Find the probability of a Type I error.

5

1. Narika has a die which is known to be biased so that the probability of throwing a 6 on any throw is \(\frac { 1 } { 100 }\). She uses an approximating distribution to calculate the probability of obtaining no 6s in 450 throws. Find the percentage error in using the approximating distribution for this calculation.
2. Johan claims that a certain six-sided die is biased so that it shows a 6 less often than it would if the die were fair. In order to test this claim, the die is thrown 25 times and it shows a 6 on only 2 throws. Test at the \(10 \%\) significance level whether Johan's claim is justified.

6 Louise and Marie play a series of tennis matches. It is given that, in any match, the probability that Louise wins the first two sets is \(\frac { 3 } { 8 }\).

1. Find the probability that, in 5 randomly chosen matches, Louise wins the first two sets in exactly 2 of the matches. It is also given that Louise and Marie are equally likely to win the first set.
2. Show that P (Louise wins the second set, given that she won the first set) \(= \frac { 3 } { 4 }\).
3. The probability that Marie wins the first two sets is \(\frac { 1 } { 3 }\). Find P(Marie wins the second set, given that she won the first set).

7 It is known that, on average, one match box in 10 contains fewer than 42 matches. Eight boxes are selected, and the number of boxes that contain fewer than 42 matches is denoted by \(Y\).

1. State two conditions needed to model \(Y\) by a binomial distribution. Assume now that a binomial model is valid.
2. Find
   1. \(\mathrm { P } ( Y = 0 )\),
   2. \(\mathrm { P } ( Y \geqslant 2 )\).
   3. On Wednesday 8 boxes are selected, and on Thursday another 8 boxes are selected. Find the probability that on one of these days the number of boxes containing fewer than 42 matches is 0 , and that on the other day the number is 2 or more.

9 A variable \(X\) has the distribution \(\mathrm { B } ( 11 , p )\).

1. Given that \(p = \frac { 3 } { 4 }\), find \(\mathrm { P } ( X = 5 )\).
2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find \(p\).
3. Given that \(\operatorname { Var } ( X ) = 1.76\), find the two possible values of \(p\).

5

1. \(20 \%\) of people in the large town of Carnley support the Residents' Party. 12 people from Carnley are selected at random. Out of these 12 people, the number who support the Residents' Party is denoted by \(U\). Find
   1. \(\mathrm { P } ( U \leqslant 5 )\),
   2. \(\quad \mathrm { P } ( U \geqslant 3 )\).
   3. \(30 \%\) of people in Carnley support the Commerce Party. 15 people from Carnley are selected at random. Out of these 15 people, the number who support the Commerce Party is denoted by \(V\). Find \(\mathrm { P } ( V = 4 )\).

7

1. Andrew plays 10 tennis matches. In each match he either wins or loses.
   1. State, in this context, two conditions needed for a binomial distribution to arise.
   2. Assuming these conditions are satisfied, define a variable in this context which has a binomial distribution.
   3. The random variable \(X\) has the distribution \(\mathrm { B } ( 21 , p )\), where \(0 < p < 1\). Given that \(\mathrm { P } ( X = 10 ) = \mathrm { P } ( X = 9 )\), find the value of \(p\).

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p .16\) shoppers are selected at random.

1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
   1. at least 8,
   2. between 4 and 9 inclusive.
   3. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 .

4

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
3. The random variable \(Z\) has the distribution \(\mathrm { B } ( n , 0.27 )\). Find the smallest value of \(n\) such that \(\mathrm { P } ( Z \geqslant 1 ) > 0.95\).

7 On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by X .

1. State two conditions needed for X to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
2. Find \(\mathrm { P } ( \mathrm { X } \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.