4 The number of faults in cloth made on a certain machine has a Poisson distribution with mean 2.4 per \(10 \mathrm {~m} ^ { 2 }\). 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 \(30 \mathrm {~m} ^ { 2 }\) of cloth is checked and the number of faults is found.

1. State suitable null and alternative hypotheses for the test.
2. Find the probability of a Type I error.
   Exactly 3 faults are found in the randomly selected \(30 \mathrm {~m} ^ { 2 }\) of cloth.
3. Carry out the test at the \(5 \%\) significance level.
   Later a similar test was carried out at the \(5 \%\) significance level, using another randomly selected \(30 \mathrm {~m} ^ { 2 }\) of cloth.
4. Given that the number of faults actually has a Poisson distribution with mean 0.5 per \(10 \mathrm {~m} ^ { 2 }\), find the probability of a Type II error.
   \(5 X\) is a random variable with distribution \(\mathrm { B } ( 10,0.2 )\). A random sample of 160 values of \(X\) is taken.
5. Find the approximate distribution of the sample mean, including the values of the parameters.
6. Hence find the probability that the sample mean is less than 1.8 .