3. A manufacturer produces plates. The proportion of plates that are flawed is \(45 \%\), with flawed plates occurring independently.
A random sample of 10 of these plates is selected.
- Find the probability that the sample contains
- fewer than 2 flawed plates,
- at least 6 flawed plates.
(4)
George believes that the proportion of flawed plates is not \(45 \%\). To assess his belief George takes a random sample of 120 plates. The random variable \(F\) represents the number of flawed plates found in the sample.
- Using a normal approximation, find the maximum number of plates, \(c\), and the minimum number of plates, \(d\), such that
$$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$
where \(F \sim \mathrm {~B} ( 120,0.45 )\)
The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
- Use a suitable hypothesis test, at the \(5 \%\) level of significance, to assess the manufacturer's claim. State your hypotheses clearly.
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