2.04c Calculate binomial probabilities

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.

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.

5 Customers arrive at a shop at a constant average rate of 2.3 per minute.

1. State another condition for the number of customers arriving per minute to have a Poisson distribution.
   It is now given that the number of customers arriving per minute has the distribution \(\mathrm { Po } ( 2.3 )\).
2. Find the probability that exactly 3 customers arrive during a 1 -minute period.
3. Find the probability that more than 3 customers arrive during a 2 -minute period.
4. Five 1-minute periods are chosen at random. Find the probability that no customers arrive during exactly 2 of these 5 periods.

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.
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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.

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.

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.

2 A die has six faces numbered \(1,2,3,4,5,6\). Manjit suspects that the die is biased so that it shows a six on fewer throws than it would if it were fair. In order to test her suspicion, she throws the die a certain number of times and counts the number of sixes.

1. State suitable null and alternative hypotheses for Manjit's test.
2. There are no sixes in the first 15 throws. Show that this result is not significant at the \(5 \%\) level.
3. Find the smallest value of \(n\) such that, if there are no sixes in the first \(n\) throws, this result is significant at the 5\% level.

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.

1 The table shows the probability distribution of a random variable \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three values of \(X\) are chosen at random. Find the probability that \(X\) takes the value 2 at least twice.

4 In this question the product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).

1. The scatter diagram in Fig. 1 shows the results of an experiment involving some bivariate data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_597_595_434_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Write down the value of \(r _ { s }\) for these data.
2. On the diagram in the Answer Booklet, draw five points such that \(r _ { s } = 1\) and \(r \neq 1\).
3. The scatter diagram in Fig. 2 shows the results of another experiment involving 5 items of bivariate data. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b5ce3230-7528-439c-9e85-ef159a49cba3-4_604_608_1484_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Calculate the value of \(r _ { s }\).
4. A random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.6 )\). Find
   1. \(\mathrm { P } ( X \leqslant 14 )\),
   2. \(\mathrm { P } ( X = 14 )\),
   3. \(\quad \operatorname { Var } ( X )\).
   4. A random variable \(Y\) has the distribution \(\mathrm { B } ( 24,0.3 )\). Write down an expression for \(\mathrm { P } ( Y = y )\) and evaluate this probability in the case where \(y = 8\).
   5. A random variable \(Z\) has the distribution \(\mathrm { B } ( 2,0.2 )\). Find the probability that two randomly chosen values of \(Z\) are equal.
      (a) Find the number of ways in which 12 people can be divided into three groups containing 5 people, 4 people and 3 people, without regard to order.
      (b) The diagram shows 7 cards, each with a letter on it. $$\mathrm { A } \mathrm {~A} \mathrm {~A} \mathrm {~B} \text { } \mathrm { B } \text { } \mathrm { R } \text { } \mathrm { R }$$ The 7 cards are arranged in a random order in a straight line.
      1. Find the number of possible arrangements of the 7 letters.
      2. Find the probability that the 7 letters form the name BARBARA. The 7 cards are shuffled. Now 4 of the 7 cards are chosen at random and arranged in a random order in a straight line.
      3. Find the probability that the letters form the word ABBA .

7 Items from a production line are examined for any defects. The probability that any item will be found to be defective is 0.15 , independently of all other items.

1. A batch of 16 items is inspected. Using tables of cumulative binomial probabilities, or otherwise, find the probability that
   1. at least 4 items in the batch are defective,
   2. exactly 4 items in the batch are defective.
   3. Five batches, each containing 16 items, are taken.
      (a) Find the probability that at most 2 of these 5 batches contain at least 4 defective items.
      (b) Find the expected number of batches that contain at least 4 defective items.

8 At a doctor's surgery, records show that \(20 \%\) of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients. There are 16 appointments to see the doctor one afternoon.

1. Find the probability that all 16 patients turn up.
2. Find the probability that more than 3 patients do not turn up. To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.
3. The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.
4. Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up. A computerised appointment system is introduced at the surgery. It is decided to test, at the 5\% level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.
5. Write down suitable hypotheses and carry out the test.

3 Over a long period of time, 20\% of all bowls made by a particular manufacturer are imperfect and cannot be sold.

1. Find the probability that fewer than 4 bowls from a random sample of 10 made by the manufacturer are imperfect. The manufacturer introduces a new process for producing bowls. To test whether there has been an improvement, each of a random sample of 20 bowls made by the new process is examined. From this sample, 2 bowls are found to be imperfect.
2. Show that this does not provide evidence, at the \(5 \%\) level of significance, of a reduction in the proportion of imperfect bowls. You should show your hypotheses and calculations clearly.

8 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.

1. All 3 doughnuts eaten contain jam.
2. All 3 doughnuts are of the same kind.
3. The 3 doughnuts are all of a different kind.
4. The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
5. exactly 2 Saturdays out of the 5 ,
6. at least 1 Saturday out of the 5 .