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Question Number 105670 by Ar Brandon last updated on 30/Jul/20

$$\mathrm{A}\mathrm{machine}\mathrm{manufactures}\mathrm{washers},\mathrm{and}20\mathrm{\%}\mathrm{of}\mathrm{the}\mathrm{production}\mathrm{is}$$$$\mathrm{substandard}.\mathrm{A}\mathrm{random}\mathrm{sample}\mathrm{o}\mathrm{f}10\mathrm{washers}\mathrm{is}\mathrm{selected}.$$$$\mathrm{Find}\mathrm{the}\mathrm{mean}\mathrm{and}\mathrm{standard}\mathrm{deviation}\mathrm{of}\mathrm{the}\mathrm{number}$$$$\mathrm{of}\mathrm{substandard}\mathrm{washers}\mathrm{in}\mathrm{the}\mathrm{sample}.$$

Answered by Ar Brandon last updated on 30/Jul/20

$$\mathrm{This}\mathrm{is}\mathrm{a}\mathrm{binomial}\mathrm{probability}\mathrm{distribution}.\mathrm{Therefore};$$$$\mathrm{mean},\mathrm{E}\left(\mathrm{X}\right)=\mathrm{np}.\mathrm{Where}\mathrm{n}\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{trials}\mathrm{and}$$$$\mathrm{p}\mathrm{is}\mathrm{the}\mathrm{probability}\mathrm{of}\mathrm{X}\mathrm{on}\mathrm{a}\mathrm{trial}.\mathrm{In}\mathrm{this}\mathrm{case}\mathrm{n}=10,\mathrm{p}=20\mathrm{\%}$$$$\Rightarrow \mathrm{E}\left(\mathrm{X}\right)=10\times 0.2=2$$$$\mathrm{For}\mathrm{a}\mathrm{binomial}\mathrm{probability}\mathrm{distribution},$$$$\mathrm{variance},\mathrm{var}\left(\mathrm{X}\right)=\mathrm{np}(1-\mathrm{p})=2(1-0.2)=1.6$$$$\mathrm{standard}\mathrm{deviation},\sigma =\sqrt{\mathrm{var}\left(\mathrm{X}\right)}=\sqrt{1.6}=1.26(\mathrm{to}3\mathrm{S}\cdot \mathrm{F})$$

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