Find Type I error probability

4 The number of accidents per 3-month period on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) has been 5.7. Following some changes to the road, the council carries out a hypothesis test to determine whether the value of \(\lambda\) has decreased. If there are fewer than 3 accidents in a randomly chosen 3 -month period, the council will conclude that the value of \(\lambda\) has decreased.

1. Find the probability of a Type I error.
2. Find the probability of a Type II error if the mean number of accidents per 3-month period is now actually 0.9 .

4 At a doctors' surgery, the number of missed appointments per day has a Poisson distribution. In the past the mean number of missed appointments per day has been 0.9 . Following some publicity, the manager carries out a hypothesis test to determine whether this mean has decreased. If there are fewer than 3 missed appointments in a randomly chosen 5-day period, she will conclude that the mean has decreased.

1. Find the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. Find the probability of a Type II error if the mean number of missed appointments per day is 0.2 .

The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per 10 m\(^2\). An adjustment is made to the machine. It is required to test at the 5% significance level whether the mean number of faults has decreased. A randomly selected 30 m\(^2\) of cloth is checked and the number of faults is found.

1. State suitable null and alternative hypotheses for the test. [1]
2. Find the probability of a Type I error. [3]

Exactly 3 faults are found in the randomly selected 30 m\(^2\) of cloth.

1. Carry out the test at the 5% significance level. [2]

Later a similar test was carried out at the 5% significance level, using another randomly selected 30 m\(^2\) of cloth.

1. Given that the number of faults actually has a Poisson distribution with mean 0.5 per 10 m\(^2\), find the probability of a Type II error. [2]

The number of accidents per year on a certain road has the distribution \(\text{Po}(\lambda)\). 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. [4]
2. Given that \(X = 2\), carry out the test. [3]
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. [3]
4. Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than \(10\) accidents in \(30\) years. [4]

1. Define
   1. a Type I error,
   2. a Type II error. [2]

A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.

1. [(b)] Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
2. Calculate the probability of the Type I error for this test. [3]
3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]

1. Define
   1. a Type I error,
   2. a Type II error. [2]

A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.

1. Assuming that the number of chicks reared per year follows a Poisson distribution test, at the 5\% significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly. [4]
2. Calculate the probability of the Type I error for this test. [3]
3. Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error. [2]