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Let nge3. A list of numbers x_(1),x_(2),....,x_(n) has mean mu and standard deviation sigma . A new list of numbers y_(1),(y_(2),....,y_(n) is made as follows : y_(1)=(x_(1)+ x_(2))/(2),y_(2)=(x_(1)+x_(2))/(2) and y_(j)"for"j=3,4,....,n. The mean and the standard deviation of the new list are hatmuand hatsigma . Then whcih of the following is necessarily true? |
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Answer» `mu=hatmu and sigmalehat SIGMA` `hatmu=(y_(1)+y_(2)+....+y_(n))/(n)=((x_(1)+x_(2))/(2)+(x_(1)+x_(2))/(2)+x_(3)+....+x_(n))/(n)` `hatmu=(x_(1)+x_(2)+....+x_(n))/(n)=murArrhatmu=mu` `sigma^(2)=(sumx_(i)^(2))/(n)-mu^(2)` `sigma^(2)=(x_(1)^(2)+x_(2)^(2)+....+x_(n)^(2))/(n)-mu^(2)""....(1)` `hatsigma^(2)=(x_(1)^(2)+y_(2)^(2)+....+y_(n)^(2))/(n)-mu^(2) (((x_(1)+x_(2))/(2))^(2)+((x_(1)+x_(2))/(2))^(2)+x_(3)^(2)+....x+_(n)^(2))/(n)-mu^(2)` `hatsigma^(2)=((x_(1)^(2)+x_(2)^(2))/(2)+x_(1)x_(2)+x_(3)^(2)+....+x_(n)^(2))/(n)-mu^(2)"" ....(2)` `sigma^(2)-hatsigma^(2)=(x_(1)^(2)+x_(2)^(2))/(n)-((x_(1)^(2)+x_(2)^(2)+2x_(1)x_(2))/(2N))=(x_(1)^(2)+x_(2)^(3)-2x_(1)x_(2))/(2n)` `(x _(1)-x_(2))^(2)/(2n)ge0rArrgehat sigma&mu=hatmu` |
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