2.04d Normal approximation to binomial

6 A dressmaker makes dresses for Easifit Fashions. Each dress requires \(2.5 \mathrm {~m} ^ { 2 }\) of material. Faults occur randomly in the material at an average rate of 4.8 per \(20 \mathrm {~m} ^ { 2 }\).

1. Find the probability that a randomly chosen dress contains at least 2 faults. Each dress has a belt attached to it to make an outfit. Independently of faults in the material, the probability that a belt is faulty is 0.03 . Find the probability that, in an outfit,
2. neither the dress nor its belt is faulty,
3. the dress has at least one fault and its belt is faulty. The dressmaker attaches 300 randomly chosen belts to 300 randomly chosen dresses. An outfit in which the dress has at least one fault and its belt is faulty is rejected.
4. Use a suitable approximation to find the probability that fewer than 3 outfits are rejected.

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.

5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.

1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
2. Find the probability that exactly 8 people call in any half hour.
3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.

3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.

1. Find the probability that more than 3 major avalanches occur in a 3-month period.
2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.

6 In restaurant \(A\) an average of 2.2\% of tablecloths are stained and, independently, in restaurant \(B\) an average of 5.8\% of tablecloths are stained.

1. Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson approximation to find the probability that a total of more than 2 tablecloths are stained.
2. Random samples of \(n\) tablecloths are taken from each restaurant. The probability that at least one tablecloth is stained is greater than 0.99 . Find the least possible value of \(n\).

7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).

1. State two assumptions required for the Poisson model to be valid.
2. Find the probability that
   1. at least 1 patient will arrive in a 15-minute period,
   2. fewer than 4 patients will arrive in a 10-minute period.
   3. The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.

4 On average, 1 in 2500 people have a particular gene.

1. Use a suitable approximation to find the probability that, in a random sample of 10000 people, more than 3 people have this gene.
2. The probability that, in a random sample of \(n\) people, none of them has the gene is less than 0.01 . Find the smallest possible value of \(n\).

6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the \(5 \%\) significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.

1. Find the critical region for the test and, given that the number of cases is 2 , carry out the test.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. At another clinic the mean number of cases of asthma per month has the independent distribution \(\mathrm { Po } ( 13.1 )\). Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.

6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be

1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
2. fewer than 1000 calls in four weeks ( 672 hours).

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 \(\mathrm { P } ( X \leqslant 3 )\).

8

1. The following tables show the probability distributions for the random variables \(V\) and \(W\).

   \(v\)	- 1	0	1	\(> 1\)
   \(\mathrm { P } ( V = v )\)	0.368	0.368	0.184	0.080

   \(w\)	0	0.5	1	\(> 1\)
   \(\mathrm { P } ( W = w )\)	0.368	0.368	0.184	0.080

   For each of the variables \(V\) and \(W\) state how you can tell from its probability distribution that it does NOT have a Poisson distribution.
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 0 ) = p \quad \text { and } \quad \mathrm { P } ( X = 1 ) = 2.5 p$$ where \(p\) is a constant.
   1. Show that \(\lambda = 2.5\).
   2. Find \(\mathrm { P } ( X \geqslant 3 )\).
   3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu > 30\). Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y > 40 ) = 0.5793\) correct to 4 decimal places. Find \(\mu\).

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.

6 A machine is designed to generate random digits between 1 and 5 inclusive. Each digit is supposed to appear with the same probability as the others, but Max claims that the digit 5 is appearing less often than it should. In order to test this claim the manufacturer uses the machine to generate 25 digits and finds that exactly 1 of these digits is a 5 .

1. Carry out a test of Max's claim at the \(2.5 \%\) significance level.
2. Max carried out a similar hypothesis test by generating 1000 digits between 1 and 5 inclusive. The digit 5 appeared 180 times. Without carrying out the test, state the distribution that Max should use, including the values of any parameters.
3. State what is meant by a Type II error in this context.

7 In a certain lottery, 10500 tickets have been sold altogether and each ticket has a probability of 0.0002 of winning a prize. The random variable \(X\) denotes the number of prize-winning tickets that have been sold.

1. State, with a justification, an approximating distribution for \(X\).
2. Use your approximating distribution to find \(\mathrm { P } ( X < 4 )\).
3. Use your approximating distribution to find the conditional probability that \(X < 4\), given that \(X \geqslant 1\).

2 A six-sided die is suspected of bias. The die is thrown 100 times and it is found that the score is 2 on 20 throws. It is given that the probability of obtaining a score of 2 on any throw is \(p\).

1. Find an approximate \(94 \%\) confidence interval for \(p\).
2. Use your answer to part (i) to comment on whether the die may be biased.

3 The number of e-readers sold in a 10-day period in a shop is modelled by the distribution \(\operatorname { Po } ( 5.1 )\). Use an approximating distribution to find the probability that fewer than 140 e-readers are sold in a 300-day period.

6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.

1. State, in context, one condition for the number of accidents in a given period to be modelled by a Poisson distribution.
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that exactly 4 accidents will occur during a randomly chosen 12-week period.
3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).

7 All the seats on a certain daily flight are always sold. The number of passengers who have bought seats but fail to arrive for this flight on a particular day is modelled by the distribution \(\mathbf { B } ( 320,0.005 )\).

1. Explain what the number 320 represents in this context.
2. The total number of passengers who have bought seats but fail to arrive for this flight on 2 randomly chosen days is denoted by \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( 2 < X < 6 )\).
3. Justify the use of your approximating distribution.
   After some changes, the airline wishes to test whether the mean number of passengers per day who fail to arrive for this flight has decreased.
4. During 5 randomly chosen days, a total of 2 passengers failed to arrive. Carry out the test at the 2.5\% significance level.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.

1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
   (b) Justify the approximating distribution used in part (ii)(a).
3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .

7 The number of planes arriving at an airport every hour during daytime is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 5.2 )\).

1. State two assumptions required for the Poisson model to be valid in this context.
2. (a) Find the probability that the number of planes arriving in a 15 -minute period is greater than 1 and less than 4,
   (b) Find the probability that more than 3 planes will arrive in a 40-minute period.
3. The airport has enough staff to deal with a maximum of 60 planes landing during a 10-hour day. Use a suitable approximation to find the probability that, on a randomly chosen 10-hour day, staff will be able to deal with all the planes that land.

3 In a certain lottery, on average 1 in every 10000 tickets is a prize-winning ticket. An agent sells 6000 tickets.

1. Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.
2. Justify the use of your approximating distribution in this context.

1 It is known that, on average, 1 in 300 flowers of a certain kind are white. A random sample of 200 flowers of this kind is selected.

1. Use an appropriate approximating distribution to find the probability that more than 1 flower in the sample is white.
2. Justify the approximating distribution used in part (a).
   The probability that a randomly chosen flower of another kind is white is 0.02 . A random sample of 150 of these flowers is selected.
3. Use an appropriate approximating distribution to find the probability that the total number of white flowers in the two samples is less than 4 .

1 On average, 1 in 50000 people have a certain gene.
Use a suitable approximating distribution to find the probability that more than 2 people in a random sample of 150000 have the gene.

1 A random variable, \(X\), has the distribution \(\operatorname { Po } ( 31 )\). Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 40 )\).