State Poisson approximation with justification

3 In the data-entry department of a certain firm, it is known that \(0.12 \%\) of data items are entered incorrectly, and that these errors occur randomly and independently.

1. A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by \(X\).
   1. State the distribution of \(X\), including the values of any parameters.
   2. State an appropriate approximating distribution for \(X\), including the values of any parameters. Justify your choice of approximating distribution.
   3. Use your approximating distribution to find \(\mathrm { P } ( X > 2 )\).
2. Another large random sample of \(n\) data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 . Use an approximating distribution to find the largest possible value of \(n\).

3 It is known that \(1.8 \%\) of children in a certain country have not been vaccinated against measles. A random sample of 200 children in this country is chosen.

1. Use a suitable approximating distribution to find the probability that there are fewer than 3 children in the sample who have not been vaccinated against measles.
2. Justify your approximating distribution.

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).

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 4000,0.001 )\).

1. Use a suitable approximating distribution to find \(\mathrm { P } ( 2 \leqslant X < 5 )\).
2. Justify your approximating distribution in this case.

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.

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.

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.

5 The probability that a new car of a certain type has faulty brakes is 0.008 . A random sample of 520 new cars of this type is chosen, and the number, \(X\), having faulty brakes is noted.

1. Describe fully the distribution of \(X\) and describe also a suitable approximating distribution. Justify this approximating distribution.
2. Use your approximating distribution to find
   (a) \(\mathrm { P } ( X > 3 )\),
   (b) the smallest value of \(n\) such that \(\mathrm { P } ( X = n ) > \mathrm { P } ( X = n + 1 )\).

1 It is known that \(1.2 \%\) of rods made by a certain machine are bent. The random variable \(X\) denotes the number of bent rods in a random sample of 400 rods.

1. State the distribution of \(X\).
2. State, with a reason, a suitable approximate distribution for \(X\).
3. Use your approximate distribution to find the probability that the sample will include more than 2 bent rods.

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

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 \(X \geqslant 1\).

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

1. 1. A random variable \(X\) has the distribution \(\mathrm { B } ( 2540,0.001 )\). Use the Poisson approximation to the binomial distribution to find \(\mathrm { P } ( X > 1 )\).
   2. Explain why the Poisson approximation is appropriate in this case.
2. Two independent random variables, \(S\) and \(T\), have distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively. Find the mean and standard deviation of \(S + T\).

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.

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.

2 The probability that a randomly chosen plant of a certain kind has a particular defect is 0.01 . A random sample of 150 plants is taken.

1. Use an appropriate approximating distribution to find the probability that at least 1 plant has the defect. Justify your approximating distribution. The probability that a randomly chosen plant of another kind has the defect is 0.02 . A random sample of 100 of these plants is taken.
2. Use an appropriate approximating distribution to find the probability that the total number of plants with the defect in the two samples together is more than 3 and less than 7 .

2 The random variable \(Y\) has the distribution \(\mathrm { B } ( 140,0.03 )\). Use a suitable approximation to find \(\mathrm { P } ( Y = 5 )\). Justify your approximation.

1 A newspaper article consists of 800 words. For each word, the probability that it is misprinted is 0.005 , independently of all other words. Use a suitable approximation to find the probability that the total number of misprinted words in the article is no more than 6 . Give a reason to justify your approximation.

3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.

8 A company has 3600 employees, of whom \(22.5 \%\) live more than 30 miles from their workplace. A random sample of 40 employees is obtained.

1. Use a suitable approximation, which should be justified, to find the probability that more than 5 of the employees in the sample live more than 30 miles from their workplace.
2. Describe how to use random numbers to select a sample of 40 from a population of 3600 employees.

2 Suppose that 3\% of the population of a large city have red hair.

1. A random sample of 10 people from the city is selected. Find the probability that there is at least one person with red hair in this sample. A random sample of 60 people from the city is selected. The random variable \(X\) represents the number of people in this sample who have red hair.
2. Explain why the distribution of \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Hence find
   (A) \(\mathrm { P } ( X = 2 )\),
   (B) \(\mathrm { P } ( X > 2 )\).
4. Discuss whether or not it would be appropriate to model \(X\) using a Normal approximating distribution. A random sample of 5000 people from the city is selected.
5. State the exact distribution of the number of people with red hair in the sample.
6. Use a suitable Normal approximating distribution to find the probability that there are at least 160 people with red hair in the sample.

5. In a manufacturing process, \(2 \%\) of the articles produced are defective. A batch of 200 articles is selected.

1. Giving a justification for your choice, use a suitable approximation to estimate the probability that there are exactly 5 defective articles.
2. Estimate the probability there are less than 5 defective articles.