Test: Correlation & Regression - UGC NET MCQ

Correlation assesses the strength and direction of the relationship between two variables. It provides a numerical value, known as the correlation coefficient, which ranges from -1 to 1. A value close to 1 indicates a strong positive relationship, while a value close to -1 indicates a strong negative relationship. Understanding correlation is crucial in fields like economics, psychology, and healthcare, as it helps identify patterns and inform decisions based on observed relationships.

The primary purpose of correlation is to measure the strength and direction of a relationship between two quantitative variables. While it indicates how closely related the variables are, it does not imply causation. A high correlation does not mean that one variable causes changes in another; it merely shows that they move together in some way. This is crucial in data analysis, as misunderstanding correlation for causation can lead to faulty conclusions. Interestingly, the Pearson correlation coefficient is a widely used measure of correlation, ranging from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

- The assertion is true because correlation analysis indeed measures the strength and direction of a relationship between two variables.

- The reason is false because the correlation coefficient can be both positive and negative, indicating both direct and inverse relationships.

- Therefore, while both statements are true, the reason does not correctly explain the assertion.

- The Assertion is true as linear regression is favored for its simplicity and ease of interpretation.

- The Reason is false because while linear regression assumes a linear relationship, it can still be useful for approximating relationships that are not perfectly linear.

- Since the Reason does not accurately explain the Assertion, Option C is the correct answer.

The regression equation in linear regression provides an equation that predicts the value of a dependent variable based on one or more independent variables. This predictive model helps in understanding how the dependent variable changes as the independent variables vary. For instance, in a simple linear regression with one independent variable, the equation takes the form \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. This equation can be extremely useful in fields like economics and social sciences for forecasting outcomes and making informed decisions. An interesting fact is that regression analysis can be extended into multiple regression, where more than one independent variable is used, allowing for more complex and realistic modeling of relationships.

Statement 1 is correct because correlation indeed measures the strength and direction of a linear relationship between two variables, and regression establishes a predictive relationship through an equation. Statement 2 is incorrect; a negative correlation indicates that as one variable increases, the other variable decreases. Therefore, only Statement 1 is correct.

A positive correlation means that when one variable increases, the other variable also tends to increase. This relationship can be quantified using a correlation coefficient, which ranges from 0 to 1 for positive correlations. An interesting fact about correlation is that it does not imply causation; just because two variables are correlated does not mean that one causes the other to change.

- Assertion Analysis: The assertion is true as regression analysis is indeed used for predicting the value of a dependent variable based on independent variables.

- Reason Analysis: The reason is also true because the objective of regression analysis is to find the best-fit line that minimizes the residuals (the distance between observed and predicted values).

- Explanation of Relationship: The reason correctly explains the assertion, as the method of establishing a best-fit line is what enables the prediction of the dependent variable.

- The Assertion is true because regression analysis indeed quantifies the relationship between variables.

- The Reason is also true, as it highlights a critical aspect of statistical analysis: correlation may exist without a causal relationship.

- The Reason provides a correct explanation for the Assertion, as understanding the limitations of correlation is essential in interpreting regression results.

Both statements are correct.

- Statement 1 accurately describes Pearson's correlation coefficient, which quantifies the degree of linear relationship between two variables, producing a value between -1 and 1. A value of 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.

- Statement 2 correctly explains the method of least squares, which is a standard approach in regression analysis that aims to find the line of best fit by minimizing the sum of the squared differences (residuals) between observed values and those predicted by the model.

Since both statements are true, the correct answer is Option C: Both 1 and 2.