The sample mean and sample variance of five data values are, respectively 13.6 and 25.8. If three of the data values are 7, 13 and 20, what are the other two data values
Answers
Answer:
Let the remaining two observation be x and y
The observation are 2,4,10,12,14,x,y
Mean =
x
ˉ
=
7
2+4+10+12+14+x+y
=8
⇒56=42+x+y
⇒x+y=14 ......(1)
Variance =16=
n
1
i=l
∑
7
(X
i
−
X
ˉ
)
2
16=
7
1
[(−6)
2
+(−4)
2
+(2)
2
+(4)
2
+(6)
2
+x
2
+y
2
−2×8(x+y)+2×(8)
2
]
16=
7
1
[36+16+4+6+36+x
2
+y
2
−16(14)+2(64)] ...........[using (1)]
16=
7
1
[108+x
2
+y
2
−224+128]
16=
7
1
[12+x
2
+y
2
]
⇒x
2
+y
2
=112−12=100
x
2
+y
2
=100 ..........(2)
From (1), we obtain
x
2
+y
2
+2xy=196 ....(3)
From (2) and (3), we obtain
2xy=196−100
⇒2xy=96 ...........(4)
subtracting (4) from (2), we obtain
x
2
+y
2
−2xy=100−96
⇒(x−y)
2
=4
⇒x−y=±2 ............(5)
Therefore, from (1) and (5) we obtain
x=8 and y=6 when x−y=2
x=6 and y=8 when x−y=−2
Thus the remaining observation are 6 and 8