If the mean and the variance of 10 observations is 15 and 4. If each observation is increased by 6, Find the mean and variance of the resulting observations. What happens if each observation is multiplied by 3. ( Need the answer for cross-checking).

Dear student
Given : Mean of 10 observations is 15 We know that,mean=sum of observationstotal number of observationssum of 10 observations=15×10=150when each observation is increased by 6,thenSum of new observations=sum of orginal 10 observations+6×10=150+60=210New mean=21010=21Now we will find Variance:Given : variance of 10 observations is 4Let x1,x2,...x10 be the given observationsVariance is 4If X is the mean of given observations ,then we getVariance X=110i=110xi-X2=4Let u1,u2,...u10 be the  new observations such thatui=xi+6 for i=1,2,...10  ...1U=1ni=110ui=110i=110xi+6=110i=110xi+6×1010  as i=1106=6×10=X+6  ...2ui-U=xi+6-6+X  from 1 and 2=xi-X110×ui-U2=110xi-X2squaring both sides and then dividing by 10110×i=110ui-U2=110i=110xi-X2110×i=110ui-U2=4Varince U=4 
When each observation is multiplied by 3The mean of 10 observations is 15So,sum of observations=mean×number of observations=15×10=150If each mean is multiplied by 3 then sum 150 will become=150×3=450So new mean=45010=45
Similarly try variance on your own.

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