The sample mean and sample variance of five data values are, respectively, x = 104 and s2 = 4. If three of the data values are 102, 100, 105, what are the other two data values
Answers
Given : The sample mean and sample variance of five data values are, respectively, x = 104 and s²= 4
three of the data values are 102, 100, 105
To Find : what are the other two data values
Solution:
let say remaining two values are 104+a & 104+b
Mean = ( 102 + 100 + 105 + 104+a +104+ b) / 5
=> 104 = ( 515 + a + b)/6
=> 520 = 515 + a + b
=> a + b = 5
xi - mean (xi - mean)²
102 -2 4
100 -4 16
105 1 1
104+a a a²
b b b²
21 + a² + b²
S² = ∑ (xi- Mean)² /(n-1)
4 = (21 + a² + b²)/(5 - 1)
=> 16 = 21 + a² + b²
=> a² + b² = - 5
a² + b² can not be negative
Hence mistake in Data
Taking s = 4
=> s² = 16
=> 16 = (21 + a² + b²)/(5 - 1)
=> a² + b² = 43
a + b = 5
Squaring both sides
=> a² + b² + 2ab = 25
=> 2ab = -18
=> ab = - 9
x² - 5x - 9 = 0
=> x = (5 ± √61)/2
=> x = 6.4 , - 1.4
a = 6.4 b = - 1.4
b = -1.4 a = 6.4
104 + a = 110.4
104 + b = 102.6
the other two data values are 110.4 and 102.6
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