Estimating Regression Coefficient b
The relationship between the variables Y and X in a simple regression equation is given by $Y = a + bX$. The coefficient $b$, also known as the slope, represents the change in Y for a unit change in X. Its estimate can be calculated using the correlation coefficient ($r$) and the standard deviations of X ($SX$) and Y ($SY$).
Regression Coefficient Formula
The formula to estimate the regression coefficient $b$ is:
$ b = r \times \frac{S_Y}{S_X} $
Given Values
- Correlation coefficient, $r = 0.75$
- Standard deviation of X, $SX = 3$
- Standard deviation of Y, $SY = 2$
Calculation Steps
- Substitute the given values into the formula for $b$:
$ b = 0.75 \times \frac{2}{3} $
- Perform the multiplication:
$ b = \frac{3}{4} \times \frac{2}{3} $
$ b = \frac{3 \times 2}{4 \times 3} $
$ b = \frac{6}{12} $
- Simplify the fraction to find the final estimate for $b$:
$ b = 0.5 $
Result
The estimate of the regression coefficient $b$ is 0.5.
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 :
