To assess whether a machine works well, the mean of the filling quantity and the spread of the filling quantity are considered. A machine works better the closer the filling is on average to the value 330 ml and the smaller the spread is. For the sample from Machine $A$, the mean of the filling quantity is 330 ml and the standard deviation is approximately $1.34 \mathrm { ml }$. Investigate based on the samples which of the two machines works better.
In this part of the task, Machine $A$ is examined more closely. 100 randomly selected bottles from this machine are to be examined. The random variable $X$: ``Number of underfills'' in a sample is assumed to be binomially distributed with $p = 0.3$. (1) Determine the probability of the event ``Fewer than 30 underfills occur''. (2) Determine the probability of the event ``At least 40 underfills occur''. (3) Give an event in the given context whose probability together with the probabilities from (2) and (3) sums to 1. (4) Give an event in the given context whose probability can be calculated using the term $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.
Qc
2 marksHypothesis test of binomial distributionsView
In this part of the task, Machine $B$ is examined more closely. In the manufacturer's specifications for Machine $B$, it states that 30\% underfills can be expected. The responsible machine operator has the suspicion that Machine $B$ works better than specified. With a sample of 200 bottles, he tests his suspicion. If at most 45 underfills occur in this sample, he assumes that the machine works better than specified. Determine the probability with which the machine operator assumes that the machine works better, even though the machine actually produces underfills with a probability of $p = 0.3$.