The correlation between two variables x and y is 0.78. Which of these statements is true? O Larger values of x tend to go with smaller values of y. O A larger value of x is the cause of a larger value of y. O Larger values of x tend to go with larger values of y. O A larger value of x is the cause of a smaller value of y.

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### Understanding Correlation

The correlation between two variables \(x\) and \(y\) is \(0.78\). Which of these statements is true?

- \( \odot \) Larger values of \(x\) tend to go with smaller values of \(y\).
- \( \odot \) A larger value of \(x\) is the cause of a larger value of \(y\).
- \( \odot \) Larger values of \(x\) tend to go with larger values of \(y\).
- \( \odot \) A larger value of \(x\) is the cause of a smaller value of \(y\).

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### Explanation:

In this context, we are dealing with the concept of **correlation** in statistics, which measures the strength and direction of a linear relationship between two variables.

- A **positive correlation** close to 1 implies that as one variable increases, the other variable also tends to increase.
- A **negative correlation** close to -1 implies that as one variable increases, the other variable tends to decrease.
- A correlation value of \(0.78\) indicates a **strong positive correlation**, meaning that larger values of \(x\) are associated with larger values of \(y\).

It is important to note that correlation does not imply causation. Therefore, statements about causality, such as "a larger value of \(x\) is the cause of a larger (or smaller) value of \(y\)," cannot be determined solely based on the correlation coefficient.

Based on these principles, the correct statement is:

- \( \odot \) Larger values of \(x\) tend to go with larger values of \(y\).

By understanding these concepts, you can make more informed interpretations of the relationships between different variables in your data analyses.
Transcribed Image Text:### Understanding Correlation The correlation between two variables \(x\) and \(y\) is \(0.78\). Which of these statements is true? - \( \odot \) Larger values of \(x\) tend to go with smaller values of \(y\). - \( \odot \) A larger value of \(x\) is the cause of a larger value of \(y\). - \( \odot \) Larger values of \(x\) tend to go with larger values of \(y\). - \( \odot \) A larger value of \(x\) is the cause of a smaller value of \(y\). **Submit Question** --- ### Explanation: In this context, we are dealing with the concept of **correlation** in statistics, which measures the strength and direction of a linear relationship between two variables. - A **positive correlation** close to 1 implies that as one variable increases, the other variable also tends to increase. - A **negative correlation** close to -1 implies that as one variable increases, the other variable tends to decrease. - A correlation value of \(0.78\) indicates a **strong positive correlation**, meaning that larger values of \(x\) are associated with larger values of \(y\). It is important to note that correlation does not imply causation. Therefore, statements about causality, such as "a larger value of \(x\) is the cause of a larger (or smaller) value of \(y\)," cannot be determined solely based on the correlation coefficient. Based on these principles, the correct statement is: - \( \odot \) Larger values of \(x\) tend to go with larger values of \(y\). By understanding these concepts, you can make more informed interpretations of the relationships between different variables in your data analyses.
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