A total of 100 randomly selected bottles from this machine are to be examined. The random variable $X$: ``Number of underfilled bottles'' in a sample is assumed to be binomially distributed with $p = 0.3$. (1) Determine the probability of the event ``Fewer than 30 underfilled bottles occur.'' (2) Give an event in the given context for which the probability can be calculated using the term $0.3 ^ { 4 } \cdot \binom { 96 } { 25 } \cdot 0.3 ^ { 25 } \cdot 0.7 ^ { 71 }$.
Qb
6 marksHypothesis test of binomial distributionsView
The machine operator responsible for the filling machine suspects that the machine actually works better than stated. By choosing $H _ { 0 } : p \geq 0.3$ as the null hypothesis, he wants to test his suspicion with a sample of 100 bottles. The number of underfilled bottles in the sample is again assumed to be binomially distributed. (1) Determine a decision rule appropriate to the null hypothesis at a significance level of $\alpha = 0.05$. (2) Describe the Type II error in the given context.
The beverage manufacturer acquires another machine. It is assumed that the filling quantities of all bottles are independent of each other. The continuous random variable $Y$: ``Filling quantity of a randomly selected bottle filled by this machine'' is assumed to be normally distributed with expected value $\mu = 331$ [ml] and standard deviation $\sigma = 1.34 [ \mathrm { ml } ]$. A filling with at most 327 ml is referred to in the following as a severe underfilling. (1) Determine the probability that a randomly selected bottle is a severe underfilling. Give your result rounded to five decimal places. [Check solution with four decimal places: 0.0014.] (2) Determine the expected number of severe underfilled bottles in a sample of 1500 bottles. (3) Determine the probability that a sample of 750 bottles contains more than two severe underfilled bottles. (4) The beverage manufacturer changes the machine parameters to $\mu _ { \text {new } } = 330 [ \mathrm { ml } ]$ and $\sigma _ { \text {new } } = 1.00 [ \mathrm { ml } ]$. Interpret the changed parameters in the given context. Assess how the probability that a randomly selected bottle is a severe underfilling changes due to the change in parameters.