$x$ 1 2 3 4 $y$ 2 6 7 10
Least Squares Fitting
The problem requires fitting the given data to the linear equation $y = mx$ using the method of least squares to find the value of the slope, $m$.
Data Table Analysis
The provided data points $(x_i, y_i)$ are:
| $x$ | $y$ |
|---|---|
| 1 | 2 |
| 2 | 6 |
| 3 | 7 |
| 4 | 10 |
Summation Calculations
To apply the least squares method for the equation $y = mx$, we need the following sums:
- Sum of $x$ values: $\sum x_i = 1 + 2 + 3 + 4 = 10$
- Sum of $x^2$ values: $\sum x_i^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
- Sum of $x \cdot y$ values: $\sum x_i y_i = (1 \times 2) + (2 \times 6) + (3 \times 7) + (4 \times 10) = 2 + 12 + 21 + 40 = 75$
Slope Calculation Formula
The formula for the slope $m$ in the least squares fit for $y = mx$ is:
$m = \frac{\sum x_i y_i}{\sum x_i^2}$Calculating Slope 'm'
Substitute the calculated sums into the formula:
$m = \frac{75}{30}$ $m = 2.5$Final Result Presentation
The calculated value of $m$ is $2.5$. The question asks for the value rounded off to one decimal place. The value $2.5$ is already precise to one decimal place.
Therefore, the value of $m$ is 2.5.
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:
