2.04d Normal approximation to binomial

1 In a certain country, 20540 adults out of a population of 6012300 have a degree in medicine.
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1. Use an approximating distribution to calculate the probability that, in a random sample of 1000 adults in this country, there will be fewer than 4 adults who have a degree in medicine. [4]
2. Justify the approximating distribution used in part (a).

2

1. The random variable \(W\) has a Poisson distribution.
   State the relationship between \(\mathrm { E } ( W )\) and \(\operatorname { Var } ( W )\).
2. The random variable \(X\) has the distribution \(\mathrm { B } ( n , p )\). Jyothi wishes to use a Poisson distribution as an approximate distribution for \(X\). Use the formulae for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\) to explain why it is necessary for \(p\) to be close to 0 for this to be a reasonable approximation.
3. Given that \(Y\) has the distribution \(\mathrm { B } ( 20000,0.00007 )\), use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( Y > 2 )\).

4 The number, \(X\), of books received at a charity shop has a constant mean of 5.1 per day.

1. State, in context, one condition for \(X\) to be modelled by a Poisson distribution.
   Assume now that \(X\) can be modelled by a Poisson distribution.
2. Find the probability that exactly 10 books are received in a 3-day period.
3. Use a suitable approximating distribution to find the probability that more than 180 books are received in a 30-day period.
   The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean 2.5 per day.
4. Find the probability that the total number of books and DVDs that are received at the shop in 1 day is more than 3 .

6 It is known that 1 in 5000 people in Atalia have a certain condition. A random sample of 12500 people from Atalia is chosen for a medical trial. The number having the condition is denoted by \(X\).

1. Use an appropriate approximating distribution to find \(\mathrm { P } ( X \leqslant 3 )\).
2. Find the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), and explain how your answers suggest that the approximating distribution used in (a) is likely to be appropriate.

2 Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following hypotheses. $$\begin{aligned} & \mathrm { H } _ { 0 } : \mathrm { P } ( \text { the spinner lands on blue } ) = \frac { 1 } { 5 } \\ & \mathrm { H } _ { 1 } : \mathrm { P } ( \text { the spinner lands on blue } ) \neq \frac { 1 } { 5 } \end{aligned}$$ Harry spins the spinner 300 times. It lands on blue on 45 spins.
Use a suitable approximation to carry out Harry's test at the \(5 \%\) significance level.

7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.

1. Find the standard deviation of the times spent by people visiting this dentist.
2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes.

1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.

5 A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.

1. Show that the probability that the spinner lands on the blue side is \(\frac { 1 } { 8 }\).
2. The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time.
3. The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times.

6 Human blood groups are identified by two parts. The first part is \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) or O and the second part (the Rhesus part) is + or - . In the UK, \(35 \%\) of the population are group \(\mathrm { A } + , 8 \%\) are \(\mathrm { B } + , 3 \%\) are \(\mathrm { AB } +\), \(37 \%\) are \(\mathrm { O } + , 7 \%\) are \(\mathrm { A } - , 2 \%\) are \(\mathrm { B } - , 1 \%\) are \(\mathrm { AB } -\) and \(7 \%\) are \(\mathrm { O } -\).

1. A random sample of 9 people in the UK who are Rhesus + is taken. Find the probability that fewer than 3 are group \(\mathrm { O } +\).
2. A random sample of 150 people in the UK is taken. Find the probability that more than 60 people are group A+.

1 On average, 1 clover plant in 10000 has four leaves instead of three.

1. Use an approximating distribution to calculate the probability that, in a random sample of 2000 clover plants, more than 2 will have four leaves.
2. Justify your approximating distribution.

2 Javier writes an article containing 52460 words. He plans to upload the article to his website, but he knows that this process sometimes introduces errors. He assumes that for each word in the uploaded version of his article, the probability that it contains an error is 0.00008 . The number of words containing an error is denoted by \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), giving your answers correct to three decimal places.
   Javier wants to use the Poisson distribution as an approximating distribution to calculate the probability that there will be fewer than 5 words containing an error in his uploaded article.
2. Explain how your answers to part (i) are consistent with the use of the Poisson distribution as an approximating distribution.
3. Use the Poisson distribution to calculate \(\mathrm { P } ( X < 5 )\).

6 Old televisions arrive randomly and independently at a recycling centre at an average rate of 1.2 per day.

1. Find the probability that exactly 2 televisions arrive in a 2-day period.
2. Use an appropriate approximating distribution to find the probability that at least 55 televisions arrive in a 50-day period.
   Independently of televisions, old computers arrive randomly and independently at the same recycling centre at an average rate of 4 per 7-day week.
3. Find the probability that the total number of televisions and computers that arrive at the recycling centre in a 3-day period is less than 4.

5

1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 42 )\).
   1. Use an appropriate approximating distribution to find \(\mathrm { P } ( X \geqslant 40 )\).
   2. Justify your use of the approximating distribution.
   3. A random variable \(Y\) has the distribution \(\mathrm { B } ( 60,0.02 )\).
      (a) Use an appropriate approximating distribution to find \(\mathrm { P } ( Y > 2 )\).
      (b) Justify your use of the approximating distribution.

1 A random variable \(X\) has the distribution \(\mathrm { B } ( 75,0.03 )\).

1. Use the Poisson approximation to the binomial distribution to calculate \(\mathrm { P } ( X < 3 )\).
2. Justify the use of the Poisson approximation.

1 On average, 2 people in every 10000 in the UK have a particular gene. A random sample of 6000 people in the UK is chosen. The random variable \(X\) denotes the number of people in the sample who have the gene. Use an approximating distribution to calculate the probability that there will be more than 2 people in the sample who have the gene.

6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the \(2 \%\) significance level, whether there are now fewer injuries than before, on average.

1. Find the critical region for the test.
2. Find the probability of a Type I error.
3. During the next 3 months there are a total of 3 injuries. Carry out the test.
4. Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.

3 When the council published a plan for a new road, only \(15 \%\) of local residents approved the plan. The council then published a revised plan and, out of a random sample of 300 local residents, 60 approved the revised plan. Is there evidence, at the \(2.5 \%\) significance level, that the proportion of local residents who approve the revised plan is greater than for the original plan?

7 At work Jerry receives emails randomly at a constant average rate of 15 emails per hour.

1. Find the probability that Jerry receives more than 2 emails during a 20 -minute period at work.
2. Jerry's working day is 8 hours long. Find the probability that Jerry receives fewer than 110 emails per day on each of 2 working days.
3. At work Jerry also receives texts randomly and independently at a constant average rate of 1 text every 10 minutes. Find the probability that the total number of emails and texts that Jerry receives during a 5 -minute period at work is more than 2 and less than 6 .

2 The number of enquiries received per day at a customer service desk has a Poisson distribution with mean 45.2. If more than 60 enquiries are received in a day, the customer service desk cannot deal with them all. Use a suitable approximating distribution to find the probability that, on a randomly chosen day, the customer service desk cannot deal with all the enquiries that are received.

5

1. The proportion of people having a particular medical condition is 1 in 100000 . A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by \(X\).
   1. State, with a justification, a suitable approximating distribution for \(X\), giving the values of any parameters.
   2. Use the approximating distribution to calculate \(\mathrm { P } ( X > 0 )\).
2. The percentage of people having a different medical condition is thought to be \(30 \%\). A researcher suspects that the true percentage is less than \(30 \%\). In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition. Use a binomial distribution to test the researcher's suspicion at the \(2 \%\) significance level.

1

1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 25 )\).
   Use the normal approximation to the Poisson distribution to find \(\mathrm { P } ( X > 30 )\).
2. A random variable \(Y\) has the distribution \(\mathrm { B } ( 100 , p )\) where \(p < 0.05\). Use the Poisson approximation to the binomial distribution to write down an expression, in terms of \(p\), for \(\mathrm { P } ( Y < 3 )\).

7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.

1. Calculate the probability of a Type I error.
2. Given that \(X = 2\), carry out the test. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-10_2716_40_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-11_2716_29_107_22}
3. The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years. Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
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1 A random variable \(X\) has the distribution \(\mathrm { B } \left( 4500000 , \frac { 1 } { 1000000 } \right)\).
Use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( X \geqslant 4 )\). \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-03_2716_29_107_22}

5 A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions \(\operatorname { Po } ( 3.7 )\) and \(\operatorname { Po } ( 2.6 )\) respectively. A bag of sweets is chosen at random.

1. Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5 .
2. Find the probability that the total number of lemon and orange sweets in the bag is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-06_2725_47_107_2002} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-07_2716_29_107_22} 10 bags of sweets are chosen at random.
3. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.

3 Random samples of size 120 are taken from the distribution \(\mathrm { B } ( 15,0.4 )\).

1. Describe fully the distribution of the sample mean.
2. Find the probability that the mean of a random sample of size 120 is greater than 6.1.