A manufacturer produces plates. The proportion of plates that are flawed is $$\(45 \%\)$$, with flawed plates occurring independently. A random sample of 10 of these plates is selected. (a) Find the probability that the sample contains (i) fewer than 2 flawed plates, (ii) at least 6 flawed plates. (4) George believes that the proportion of flawed plates is not $$\(45 \%\)$$. To assess his belief George takes a random sample of 120 plates. The random variable $$\(F\)$$ represents the number of flawed plates found in the sample. (b) Using a normal approximation, find the maximum number of plates, $$\(c\)$$, and the minimum number of plates, $$\(d\)$$, such that $$\[ \mathrm{P}(F \leqslant c) \leqslant 0.05 \text { and } \mathrm{P}(F \geqslant d) \leqslant 0.05 \]$$ where $$\(F \sim \mathrm{B}(120,0.45)\)$$ (7) 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. (c) Use a suitable hypothesis test, at the 5\% level of significance, to assess the manufacturer's claim. State your hypotheses clearly. (4)

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