Long Questions (Question 3 & 4)


Question 3:
The mean of the data 1, a, 2a, 8, 9 and 15 which has been arranged in ascending order is b. If each number of the data is subtracted by 3, the new median is 47b . Find
(a) The values of a and b,
(b) The variance of the new data.

Solution:
(a)
Mean ˉx=b1+a+2a+8+9+156=b33+3a=6b3a=6b33a=2b11 ------(1)New median =4b7(2a3)+(83)2=4b72a+22=4b714a+14=8b7a=4b7 ------(2)Substitute (1) into (2),7(2b11)=4b714b77=4b710b=70b=7From (1), a=2(7)11=3


(b)

New data is (1 – 3), (3 – 3), (6 – 3), (8 – 3), (9 – 3), (15 – 3)
New data is  – 2, 0, 3, 5, 6, 12

Variance, σ2=x2Nˉx2σ2=(2)2+(0)2+(3)2+(5)2+(6)2+(12)26(2+0+3+5+6+126)2σ2=218616=20.333



Question 4:
A set of data consists of 20 numbers. The mean of the numbers is 8 and the standard deviation is 3.

(a) Calculate   x and x2 .

(b) A sum of certain numbers is 72 with mean of 9 and the sum of the squares of these numbers of 800, is taken out from the set of 20 numbers. Calculate the mean and variance of the remaining numbers.

Solution:
(a)
Mean ˉx=xN8=x20x=160Standard deviation, σ=x2Nˉx23=x2Nˉx29=x22082x220=73x2=1460


(b)
Sum of certain numbers, M is 72 with mean of 9,72M=9M=8Mean of the remaining numbers=16072208=713Variance of the remaining numbers=146080012(713)2=555379=129