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
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): If the securities with less than perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly.
Reason (R): The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive.
In the light of the above statements, choose the most appropriate answer from the options given below:
The two-regression equation of variable \(\rm{x}\) and \(\rm{y}\) are
\(\rm{y = 0.8x + 9.8}\) and \(\rm{x = 10.2 + 0.6y}\)
The coefficient of correlation between \(\rm{x}\) and \(\rm{y}\) is
- 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?
$$ \begin{bmatrix} & P & Q & R & S \\ P & 0 & 0 & 0 & 0 \\ Q & 0 & 3.73 & 1.85 & 3.23 \\ R & 0 & 1.85 & 0.93 & 1.61 \\ S & 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} $$
