Question
Download Solution PDFLet a, b ∈ R. Let the mean and the variance of 6 observations –3, 4, 7, –6, a, b be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is :
- \(\frac{13}{3}\)
- \(\frac{16}{3}\)
- \(\frac{11}{3}\)
- \(\frac{14}{3}\)
Answer (Detailed Solution Below)
Option 1 : \(\frac{13}{3}\)
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Detailed Solution
Download Solution PDFExplanation:
Mean and variance of 6 observations –3, 4, 7, –6, a, b be 2 and 23.
mean (μ) = 2
Then
\( \frac{\sum x_i}{6}=2\)
\( \frac{2+a+b}{6}=2\)
⇒ a + b = 10
⇒ a = 10 - b ...(i)
and
\(\frac{\sum x_i^2}{\mathrm{~N}}-μ^2=23 \)
⇒ \(\frac{9+16+49+36+a^2+b^2}{6}-4=23 \)
⇒ 110 + a2 + b2 = 162
⇒ a2 + b2 = 52
⇒ (10 - b)2 + b2 = 52
⇒ 100 - 20b + b2 + b2 = 52
⇒ 2b2 - 20b + 48 = 0
⇒ b2 - 10b + 24 = 0
⇒ (b - 6)(b - 4) = 0
⇒ b = 6, 4
when b = 6 then (i) implies a = 4
when b = 4 then (i) implies a = 6
Then the mean deviation about the mean of these 6 observations is :
\(\frac{\sum\left|x_{\mathrm{i}}-\mu\right|}{6}=\frac{5+2+5+8+2+4}{6}=\frac{13}{3} \)
Option (1) is true.
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