Correlation Strength and Common Variance
The strength of a relationship indicated by a correlation coefficient ($r$) is determined by its absolute value, $|r|$. Common variance, which represents the proportion of variance shared between two variables, is proportional to the square of the correlation coefficient, $r^2$. Therefore, a higher absolute value of $r$ signifies a stronger relationship and greater common variance.
Ranking Correlation Coefficients
To arrange the given correlation coefficients in descending order of strength, we compare their absolute values:
- The correlation coefficients are: $-0.8$, $0.7$, $-0.5$, $0.4$.
- Calculate the absolute value for each:
- $|-0.8| = 0.8$
- $|0.7| = 0.7$
- $|-0.5| = 0.5$
- $|0.4| = 0.4$
- Arrange these absolute values in descending order: $0.8$, $0.7$, $0.5$, $0.4$.
- The corresponding original correlation coefficients in descending order of strength are: $-0.8$, $0.7$, $-0.5$, $0.4$.
Final Ordering
The correlation coefficients arranged in descending order of their strength (based on common variance) are: $-0.8$, $0.7$, $-0.5$, $0.4$. This matches Option A.
Important Questions from Correlation Analysis
- Let $X$ be a random variable that follows $Uniform(-1, 1)$ distribution. The conditional distribution of the random variable $Y$ given $X = x$ is the $Uniform(x^2 - 0.1, x^2 + 0.1)$ distribution.
The value of $correlation(X, Y)$ is __________ . (Answer in integer) - In the variance-covariance matrix given below, which one of the following pairs of elements has the highest correlation?
$$ \begin{bmatrix} & P & Q & R & S \\ P & 0 & 0 & 0 & 0 \\ Q & 0 & 3.73 & 1.85 & 3.23 \\ R & 0 & 1.85 & 0.93 & 1.61 \\ S & 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} $$ - Consider 62 pair of observations $(x_i, y_i)$, i = 1, …., 62, obtained from the distribution of $(X, Y)$. The Pearson correlation coefficient ($R$) of these observations is 0.27.
Which one of the following statements is CORRECT for a significance level $\alpha = 0.05$ and $t_{0.025,60} = 2.0$?
Hint: Test statistic $t = \frac{R\sqrt{n - 2}}{\sqrt{1 - R^2}}$ - Let $X = \cos \theta$ and $Y = \sin \theta$, where $\theta$ is a random variable uniformly distributed over $(0, 2\pi)$. Which of the following options is/are CORRECT?
Hint: $\text{cov}(X, Y) = E[(X - E(X)) (Y - E(Y))]$ - The multiple correlation ranges between :
