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
There is no value of x that can simultaneously satisfy both the given equations. Therefore, find the ‘least squares error’ solution to the two equations, i.e., find the value of x that minimizes the sum of squares of the errors in the two equations. __________
2x = 3
4x = 1
- Using linear regression (least squares), the best-fit line for the given dataset is $y = 2.2x + 2.3$. Given the residual sum of squares is 25.8, the coefficient of determination ($r^2$), is ______________ . (rounded off to two decimal places)
$x$ 2 3 4 5 $y$ 8 9 7 16 - The data given in the table are fitted to the equation $y = mx$ using the method of least squares. The value of $m$ is ______ (rounded off to one decimal place).
$x$ 1 2 3 4 $y$ 2 6 7 10 The graph below shows a plot of annual rainfall on the X axis and number of plant species on the Y axis. Based on the pattern in the graph, which one of the following statistical methods would be most appropriate to model the relationship between annual rainfall and the number of plant species?
- Arrange the following steps of formulating research proposal in a logical sequential order :
(i) Defining research problem.
(ii) Formulation of hypothesis.
(iii) Review of literature.
(iv) Collection of data.
(v) Research Design.
Choose the correct answer from the options given below :
