Calculating Linear Regression Slope
The question asks for the slope ($m$) of the linear regression plot of $Y$ on $X$ given three data points: $(2, 60)$, $(4, 70)$, and $(6, 90)$.
The formula for the slope ($m$) in linear regression of $Y$ on $X$ is: $m = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2}$ where $n$ is the number of data points.
Data Analysis
First, let's list the data points and calculate the necessary sums:
- Number of data points, $n = 3$.
- Data pairs $(X, Y)$: $(2, 60)$, $(4, 70)$, $(6, 90)$.
| X | Y | $X^2$ | XY | |
|---|---|---|---|---|
| 2 | 60 | 4 | 120 | |
| 4 | 70 | 16 | 280 | |
| 6 | 90 | 36 | 540 | |
| Sum | $\sum X = 12$ | $\sum Y = 220$ | $\sum X^2 = 56$ | $\sum XY = 940$ |
Slope Calculation
Now, substitute the calculated sums into the slope formula:
$m = \frac{3(940) - (12)(220)}{3(56) - (12)^2}$ $m = \frac{2820 - 2640}{168 - 144}$ $m = \frac{180}{24}$ $m = 7.5$Result
The slope ($m$) of the linear regression plot is $7.5$.
Important Questions from Regression Analysis
If a constant 60 is subtracted from each of the values of X and Y, then the regression coefficient is
Given the regression lines X + 2Y - 5 = 0, 2X + 3Y - 8 = 0 and Var(X) = 12, the value of Var(Y) is
The standard deviation of Y is double of standard deviation of x. The correlation coefficient between X and Y is 0.5.
The acute angle between lines of regression is
For the variables X, Y and Z, r xy = 0.80, r xz = 0.64, and r yz = 0.79, the square of multiple correlation coefficient \(\rm \mathop R\nolimits_{xyz}^2 \) is:
Dimension reduction methods have the goal of using the correlation structure among the predictor variables to accomplish which of the following:
A. To reduce the number of predictor components
B. To help ensure that these components are dependent
C. To provide a framework for interpretability of the results
D. To help ensure that these components are independent
E. To increase the number of predictor components
Choose the correct answer from the options given below:
