Math Problem Statement

Which of the following statements about correlation are true? Check all that apply.

Correlation measures and describes the behavior of a single variable.
If the correlation between two variables is –1, it indicates a perfect negative linear relationship between the two variables.
If the correlation between two variables is 0, there is no clear linear relationship between the two variables.
If the correlation between two variables is close to 1, there is a positive linear relationship between the two variables.
The correlation indicates the strength and the direction of the linear relationship between two variables.
A positive correlation means that two variables tend to change in opposite directions.
A negative correlation means that two variables tend to change in opposite directions.

Solution

Here is an analysis of each statement regarding correlation:

  1. Correlation measures and describes the behavior of a single variable.
    False. Correlation measures the relationship between two variables, not just one.

  2. If the correlation between two variables is –1, it indicates a perfect negative linear relationship between the two variables.
    True. A correlation of –1 means the data points lie exactly on a straight line with a negative slope, showing a perfect negative linear relationship.

  3. If the correlation between two variables is 0, there is no clear linear relationship between the two variables.
    True. A correlation of 0 indicates no linear relationship. However, there could still be a non-linear relationship.

  4. If the correlation between two variables is close to 1, there is a positive linear relationship between the two variables.
    True. A correlation close to 1 indicates a strong positive linear relationship where as one variable increases, the other tends to increase.

  5. The correlation indicates the strength and the direction of the linear relationship between two variables.
    True. The value of the correlation coefficient shows the strength (how close to ±1) and direction (positive or negative) of the relationship.

  6. A positive correlation means that two variables tend to change in opposite directions.
    False. A positive correlation means the two variables tend to change in the same direction.

  7. A negative correlation means that two variables tend to change in opposite directions.
    True. A negative correlation indicates that as one variable increases, the other decreases, and vice versa.

Correct statements:

2, 3, 4, 5, and 7.


Would you like further clarification on any of these points?

Related Questions:

  1. How does correlation differ from causation?
  2. What are some limitations of using correlation as a measure of relationship?
  3. How can you determine whether a relationship is linear or non-linear?
  4. What are some real-world examples of positive and negative correlations?
  5. How does the strength of correlation (e.g., 0.2 vs. 0.8) impact its interpretation?

Tip:

When interpreting correlation, always visualize the data to check for potential non-linear relationships that a correlation coefficient might miss.

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Math Problem Analysis

Mathematical Concepts

Correlation
Linear Relationships
Statistical Analysis

Formulas

Correlation coefficient (r) formula

Theorems

Properties of correlation coefficients

Suitable Grade Level

Grades 10-12