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NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Statistics

NDA 2015 · Mathematics · Statistics

The geometric mean of the observations x_1, x_2, x_3, . . x_ n is...

The geometric mean of the observations x1,x2,x3,…..xn\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \ldots . . \mathrm{x}_{\mathrm{n}} is G1\mathrm{G}_1, The geometric mean of the observations y1,y2,y3,…yn\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3, \ldots \mathrm{y}_{\mathrm{n}} is G2\mathrm{G}_2. The geometric mean of observations x1y1,x2y2,x3y3,….xnyn\frac{x_1}{y_1}, \frac{x_2}{y_2}, \frac{x_3}{y_3}, \ldots . \frac{x_n}{y_n} is

Options

  1. AG1G2\mathrm{G}_1 \mathrm{G}_2
  2. Bln⁡(G1G2)\ln \left(\mathrm{G}_{1} \mathrm{G}_{2}\right)
  3. CG1G2\frac{\mathrm{G}_1}{\mathrm{G}_2}
  4. Dln⁡(G1G2)\ln \left(\frac{G_1}{G_2}\right)

Correct answer

C. G1G2\frac{\mathrm{G}_1}{\mathrm{G}_2}

Step-by-step solution

G1=[x1×x2×x3×……×xn]1/nG2=[y1×y2×y3×……×yn]1/n⇒G1G2=[x1y1×x2y2×x3y3×………×xnyn]1/n∴G1G2 is the  G.M of x1y1,x2y2,x3y3,………….xnyn\begin{aligned} & \mathrm{G}_1=\left[\mathrm{x}_1 \times \mathrm{x}_2 \times \mathrm{x}_3 \times \ldots \ldots \times \mathrm{x}_{\mathrm{n}}\right]^{1 / n} \\ & \mathrm{G}_2=\left[\mathrm{y}_1 \times \mathrm{y}_2 \times \mathrm{y}_3 \times \ldots \ldots \times \mathrm{y}_{\mathrm{n}}\right]^{1 / n} \\ & \Rightarrow \frac{\mathrm{G}_1}{\mathrm{G}_2}=\left[\frac{\mathrm{x}_1}{\mathrm{y}_1} \times \frac{\mathrm{x}_2}{\mathrm{y}_2} \times \frac{\mathrm{x}_3}{\mathrm{y}_3} \times \ldots \ldots \ldots \times \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}\right]^{1 / n} \\ & \therefore \frac{\mathrm{G}_1}{\mathrm{G}_2} \text{ is\text{ the } G.M of } \frac{\mathrm{x}_1}{\mathrm{y}_1}, \frac{\mathrm{x}_2}{\mathrm{y}_2}, \frac{\mathrm{x}_3}{\mathrm{y}_3}, \ldots \ldots \ldots \ldots . \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}\end{aligned}
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