Two-tailed test setup or execution

3 The random variable \(G\) has the distribution \(\operatorname { Po } ( \lambda )\). A test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 4.5\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda \neq 4.5\), based on a single observation of \(G\). The critical region for the test is \(G \leqslant 1\) and \(G \geqslant 9\).

1. Find the significance level of the test.
2. Given that \(\lambda = 5.5\), calculate the probability that the test results in a Type II error.

1. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\) This observation is to be used to test, at a \(5 \%\) level of significance,

$$\mathrm { H } _ { 0 } : \lambda = k \quad \mathrm { H } _ { 1 } : \lambda \neq k$$ where \(k\) is a positive integer.
Given that the critical region for this test is \(( X = 0 ) \cup ( X \geqslant 9 )\)

1. find the value of \(k\), justifying your answer.
2. Find the actual significance level of this test.

2. A single observation \(x\) is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test \(\mathrm { H } _ { 0 } : \lambda = 7\) against \(\mathrm { H } _ { 1 } : \lambda \neq 7\).

1. Using a \(5 \%\) significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to \(2.5 \%\).
2. Write down the significance level of this test. The actual value of \(x\) obtained was 5 .
3. State a conclusion that can be drawn based on this value.

3. It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 8\) against \(\mathrm { H } _ { 1 } : \lambda \neq 8\), using a \(10 \%\) level of significance.

1. Find the critical region for this test, such that the probability in each tail is as close as possible to \(5 \%\).
2. Given that \(\lambda = 10\), find
   1. the probability of a type II error,
   2. the power of the test.
      (4)

A single observation x is to be taken from a Poisson distribution with parameter \(\lambda\). This observation is to be used to test H₀: \(\lambda\) = 7 against H₁: \(\lambda\) ≠ 7.

1. Using a 5\% significance level, find the critical region for this test assuming that the probability of rejection in either tail is as close as possible to 2.5\%. [5]
2. Write down the significance level of this test. [1]

The actual value of x obtained was 5.

1. State a conclusion that can be drawn based on this value. [2]

\emph{Liftsforall} claims that the lift they maintain in a block of flats breaks down at random at a mean rate of 4 times per month. To test this, the number of times the lift breaks down in a month is recorded.

1. Using a 5\% level of significance, find the critical region for a two-tailed test of the null hypothesis that 'the mean rate at which the lift breaks down is 4 times per month'. The probability of rejection in each of the tails should be as close to 2.5\% as possible. [3]

Over a randomly selected 1 month period the lift broke down 3 times.

1. Test, at the 5\% level of significance, whether \emph{Liftsforall}'s claim is correct. State your hypotheses clearly. [2]
2. State the actual significance level of this test. [1]

The residents in the block of flats have a maintenance contract with \emph{Liftsforall}. The residents pay \emph{Liftsforall} £500 for every quarter (3 months) in which there are at most 3 breakdowns. If there are 4 or more breakdowns in a quarter then the residents do not pay for that quarter. \emph{Liftsforall} installs a new lift in the block of flats. Given that the new lift breaks down at a mean rate of 2 times per month,

1. find the probability that the residents do not pay more than £500 to \emph{Liftsforall} in the next year. [6]