Approximating the Poisson to the Normal distribution

1 A random variable \(X\) has the distribution \(\mathrm { Po } ( 145 )\).

1. Use a suitable approximating distribution to calculate \(\mathrm { P } ( X \leqslant 150 )\).
2. Justify the use of your approximating distribution in this case.

3

1. The random variable \(X\) has a \(\mathrm { B } ( 60,0.02 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( X \leqslant 2 )\).
2. The random variable \(Y\) has a \(\operatorname { Po } ( 30 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( Y \leqslant 38 )\).

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

3 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean has changed.

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.

3 Flies stick to wet paint at random points. The average number of flies is 2 per square metre. A wall with area \(22 \mathrm {~m} ^ { 2 }\) is painted with a new type of paint which the manufacturer claims is fly-repellent. It is found that 27 flies stick to this wall. Use a suitable approximation to test the manufacturer's claim at the \(1 \%\) significance level. Take the null hypothesis to be \(\mu = 44\), where \(\mu\) is the population mean.

4 Small drops of two liquids, \(A\) and \(B\), are randomly and independently distributed in the air. The average numbers of drops of \(A\) and \(B\) per cubic centimetre of air are 0.25 and 0.36 respectively.

1. A sample of \(10 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Find the probability that the total number of drops of \(A\) and \(B\) in this sample is at least 4 .
2. A sample of \(100 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Use an approximating distribution to find the probability that the total number of drops of \(A\) and \(B\) in this sample is less than 60 .

5. Cars stop at a service station randomly at a rate of 3 every 5 minutes.

1. Calculate the probability that in a randomly selected 10 minute period,
   1. exactly 7 cars will stop at the service station,
   2. more than 7 cars will stop at the service station. Using a normal approximation, the probability that more than 40 cars will stop at the service station during a randomly selected \(n\) minute period is 0.2266 correct to 4 significant figures.
2. Find the value of \(n\).

1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.

1. Calculate an approximate \(98 \%\) confidence interval for the mean number of calls received per weekday night shift.
2. The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of \(\mathbf { 1 4 }\) hours' duration.
   (3 marks)

1. Explain briefly what you understand by
   1. a statistic,
   2. a sampling distribution.
   3. (a) Write down the condition needed to approximate a Poisson distribution by a Normal distribution.
   The random variable \(Y \sim \operatorname { Po } ( 30 )\).
2. Estimate \(\mathrm { P } ( Y > 28 )\).

2 The random variable \(D\) has the distribution \(\operatorname { Po } ( 20 )\). Using an appropriate approximation, which should be justified, calculate \(\mathrm { P } ( D \geqslant 25 )\).

1 A random variable has the distribution \(\mathrm { Po } ( 31 )\). Name an appropriate approximating distribution and state the mean and standard deviation of this approximating distribution.

4. (a) State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.
(b) Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.
During the winter the mean number of yachts hired per week is 5 .
(c) Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.
(d) Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer. In the summer there are 16 Saturdays on which a yacht can be hired.
(e) Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.