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Determine which plot shows the strongest Linear Correlation.

This question aims to find the linear correlation between different indicator points on the XY-axis. The coefficient of linear correlation indicators analyzes the strength of the linear relationship between different variables.

The correlation is called positive if the linear coefficient is greater than zero, and it is called negative if the linear coefficient is greater than zero. The value of zero indicates that there is no correlation between the indicators.

Expert Answer:

Pearson-product moment correlation is the most commonly used correlation in finding the linear relationship between two variables $x$ and $y$. This correlation coefficient tells us the degree of movement of different variables, and it is represented by (\rho) since this coefficient is used to find the linear correlation, so it is not used to find the non-linear correlation.

Formula:

\[\rho = \frac{cov(X , Y)}{\sigma_X \sigma_Y}\]

To find the correlation coefficient, we need to divide the product of the standard deviations of two variables. The dispersion of data from its average is called standard deviation, and the change of two variables is measured by covariance.

The two variables move in such a way that the increment or decrease in the first variable causes the same results in the other variables. If one variable is increasing, then the other variable must increase. Similarly, if one variable is decreasing, then the other variable must decrease and the inverse relation between two variables in negative correlation is observed.

The value of the Pearson coefficient ranges from $-1$ to $+1$. It means the value $-1$ indicates the minimum value of correlation while the value $+1$ indicates the maximum value of correlation.

The positive correlation has a value greater than $0$ and less than $+1$. This type of correlation indicates that when one variable moves higher, the other variable must follow its movement to create a positive result altogether.

The negative correlation describes the inverse relationship between two variables. If the value of the coefficient is less than $0$ and its minimum value is $-1$, then it indicates a negative correlation. The increase in one variable causes a decrease in the other variable and vice versa in a negative correlation.

Example:

The calculation of the correlation between two variables like the heating bills and the outside temperature gives a value of $-0.95$. This value indicates that with the increase in the outside temperature, the prices of heating bills decrease is an example of negative correlation.

If the price of oil per liter and the fare of railway tickets per seat are equal, then it means they can be represented on the graph as strong indicators having a positive correlation.

Numerical Solution:

The plot having a value of $+0.75$ shows that it is a positive correlation.

Determine which plot shows the strongest Linear Correlation.

Figure 1

In this plot, the value of $x$ is increasing, and the value of $y$ is increasing as well, and $+0.75$ is greater than $+1$. It means it is showing a positive correlation.

Image/Mathematical drawings are created in Geogebra.

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