The-set-X-and-Y-have-five-elementseach-Given-that-X-25-Y-55-X-2-165-and-Y-2-765-and-a-linear-Function-y-px-q-tranforms-the-set-X-into-the-set-Y-where-p-and-q-are-positive-constants-a-Fi

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Question Number 42299 by Rio Michael last updated on 22/Aug/18

$$ThesetXandYhavefiveelementseach.$$$$Giventhat\mathrm{\Sigma }X=25,\mathrm{\Sigma }Y=55,\mathrm{\Sigma }{X}^2=165and\mathrm{\Sigma }{Y}^2=765$$$$andalinearFunctiony=px+qtranformsthesetXintotheset$$$$Y,wherepandqarepositiveconstants.$$$$a)FindthemeanandVarianceofXandY$$$$hence,orotherwise,$$$$b)findthevaluesofpandq.$$

Answered by MJS last updated on 22/Aug/18

$$V=\frac{\mathrm{\Sigma }{x}^2}{n}-{\left(\frac{\mathrm{\Sigma }x}{n}\right)}^2$$$$\mathrm{mean}=\stackrel{-}{x}=\frac{\mathrm{\Sigma }x}{n}$$$$1.n=5\mathrm{\Sigma }X=25\mathrm{\Sigma }{X}^2=165$$$$V=\frac{165}{5}-{\left(\frac{25}{5}\right)}^2=33-25=8$$$$\stackrel{-}{X}=\frac{25}{5}=5$$$$2.n=5\mathrm{\Sigma }Y=55\mathrm{\Sigma }{Y}^2=765$$$$V=\frac{765}{5}-{\left(\frac{55}{5}\right)}^2=153-121=32$$$$\stackrel{-}{Y}=\frac{55}{5}=11$$$$$$$$\mathrm{\Sigma }Y=p\mathrm{\Sigma }X+q\Rightarrow 55=25p+q\Rightarrow q=55-25p$$$$\stackrel{-}{Y}=p\stackrel{-}{X}+q\Rightarrow 11=5p+q\Rightarrow 11=5p+55-25p\Rightarrow$$$$\Rightarrow 20p=44\Rightarrow p=\frac{11}{5}\Rightarrow q=55-25\times \frac{11}{5}=0$$$$$$$$y=\frac{11}{5}x$$

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