Question

If $r=0.5,\sum xy=120,\sum x^2=90,{\sigma }_y=8$, then $n=$

• A. $100$
• B. $10$
• C. $15$
• D. $50$

Answer 1

The correct option is B

$10$

Explanation for the correct option :

Step 1: Equation using standard deviation :

From the question, we know that $r=0.5,\sum xy=120,\sum x^2=90,\text{and}{\sigma }_y=8$

The formula for the standard deviation is given by:

${\mathit{\sigma }}_{\mathbf{y}}\mathbf{=}\sqrt{\frac{\mathbf{\sum }{\mathbf{y}}^{\mathbf{2}}}{\mathbf{n}}}$

From this, we have

$8=\sqrt{\frac{\sum y^2}{n}}$

Squaring both sides,

$64n=\sum y^2$

Step 2: Determination of the value of $n$:

Now the formula for the coefficient of correlation $\left(r\right)$ is given by:

$\mathit{r}\mathbf{=}\frac{\mathbf{\sum }\mathbf{x}\mathbf{y}}{\sqrt{\left[\sum x^2\sum y^2\right]}}$

Substituting the given values we get,

$0.5=\frac{120}{\sqrt{90\times 64n}}$

Squaring both sides,

$\begin{array}{rcl}\therefore {\left(0.5\right)}^2& =& \frac{{\left(120\right)}^2}{90\times 64n}\\ & \Rightarrow & 0.25=\frac{14400}{5760n}\\ & \Rightarrow & n=\frac{14400}{5760(0.25)}\\ & \Rightarrow & n=\frac{14400}{1440}\\ & \Rightarrow & n=10\end{array}$

Hence ,option B is correct.