A certain list of 100 data has a mean of 6 and a standard deviation of D, where D is positive. Which of the following pairs of data, when added to the list, must result in a list of 102 data with standard deviation less than D?
a) -6 and 0
b) 0 and 0
c) 0 and 6
d) 0 and 12
e) 6 and 6
Please explain your answer.
I think E 6,6 should be the answer.
sum of 100 numbers = 100 * 6 = 600
D = SQRT( 1/100 * summation of [(x[i] - 6) ^2 ]
with addition of 2 terms the total number of terms will be 102, so the bracketed part of SQRT above is already smaller,
My approach involves not increasing the summation part, and 6, 6 is ideal option for this
adding 6 + 6 to 600 = 612
mean will be 612/102 = 6..unchanged
since the added terms are 6,6 , the difference (6-6) = 0 and wont add to the summation part of SQRT above
thus SD will be smaller.