A certain type of steel is produced in a foundry. It has flaws (small bubbles) randomly distributed, and these can be detected by X-ray analysis. On average, there are 0.1 bubbles per \(\mathrm { cm } ^ { 3 }\), and the number of bubbles per \(\mathrm { cm } ^ { 3 }\) has a Poisson distribution.
In an ingot of \(40 \mathrm {~cm} ^ { 3 }\), find
the probability that there are less than two bubbles,
the probability that there are more than 3 but less than 10 bubbles.
A new machine is being considered. Its manufacturer claims that it produces fewer bubbles per \(\mathrm { cm } ^ { 3 }\). In a sample ingot of \(60 \mathrm {~cm} ^ { 3 }\), there is just one bubble.
Carry out a hypothesis test at the \(1 \%\) significance level to decide whether the new machine is better. State your hypotheses and conclusion carefully.