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 + 2 a + 8 + 9 + 15 6 = b 33 + 3 a = 6 b 3 a = 6 b 33 a = 2 b 11  ——(1) New median  = 4 b 7 ( 2 a 3 ) + ( 8 3 ) 2 = 4 b 7 2 a + 2 2 = 4 b 7 14 a + 14 = 8 b 7 a = 4 b 7  ——(2) Substitute (1) into (2), 7 ( 2 b 11 ) = 4 b 7 14 b 77 = 4 b 7 10 b = 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 = 160 72 20 8 = 7 1 3 Variance of the remaining numbers = 1460 800 12 ( 7 1 3 ) 2                                                              = 55 53 7 9 = 1 2 9

Leave a Reply

Your email address will not be published. Required fields are marked *