Statistics 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 4 7 b . Find
(a) The values of a and b,
(b) The variance of the new data.

Solution:
(a)
Mean  x ¯ =b 1+a+2a+8+9+15 6 =b 33+3a=6b 3a=6b33 a=2b11 ------(1) New median = 4b 7 ( 2a3 )+( 83 ) 2 = 4b 7 2a+2 2 = 4b 7 14a+14=8b 7a=4b7 ------(2) Substitute (1) into (2), 7( 2b11 )=4b7 14b77=4b7 10b=70 b=7 From (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,  σ 2 = x 2 N x ¯ 2 σ 2 = ( 2 ) 2 + ( 0 ) 2 + ( 3 ) 2 + ( 5 ) 2 + ( 6 ) 2 + ( 12 ) 2 6 ( 2+0+3+5+6+12 6 ) 2 σ 2 = 218 6 16=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 x 2 .
(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  x ¯ = x N 8= x 20 x=160 Standard deviation, σ= x 2 N x ¯ 2 3= x 2 N x ¯ 2 9= x 2 20 8 2 x 2 20 =73 x 2 =1460
(b)
Sum of certain numbers, M is 72 with mean of 9, 72 M =9 M=8 Mean of the remaining numbers= 16072 208 =7 1 3 Variance of the remaining numbers= 1460800 12 ( 7 1 3 ) 2                                                             =5553 7 9 =1 2 9