The linear relationship between two variables
Sample Solution
1. Understanding Correlation
Correlation is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is expressed as a correlation coefficient, typically represented by the symbol "r."
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+/- sign:
- Positive correlation (+): As one variable increases, the other variable also increases. For example, there is a positive correlation between height and weight in humans.
- Negative correlation (-): As one variable increases, the other variable decreases. For example, there is a negative correlation between hours of sleep and perceived stress levels.
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Numerical value:
- The numerical value of the correlation coefficient ranges from -1 to 1.
- A value of 1 indicates a perfect positive correlation, meaning the two variables are perfectly positively related.
- A value of -1 indicates a perfect negative correlation, meaning the two variables are perfectly negatively related.
- A value of 0 indicates no correlation between the variables.
Example of a Spurious Correlation:
A common example of a spurious correlation is the relationship between ice cream sales and drowning deaths. Both of these variables tend to increase in the summer months, leading to a positive correlation. However, there is no causal relationship between the two. The increase in both ice cream sales and drowning deaths is likely due to the warmer weather and increased outdoor activities.
Full Answer Section
2. Empirical Study and Correlation Analysis
Note: To provide a specific example, I would need to access recent research literature. However, I can illustrate the process and interpret a hypothetical correlation analysis.
Hypothetical Study:
Let's assume a study investigates the relationship between hours of sleep and academic performance in college students. The researchers might use Pearson correlation analysis to measure the linear relationship between these two continuous variables.
Hypothetical Findings:
- Correlation coefficient (r): 0.75
- Interpretation: The study finds a strong positive correlation (r = 0.75) between hours of sleep and academic performance in college students. This suggests that students who get more sleep tend to have higher academic performance. However, it's important to note that correlation does not imply causation. Other factors, such as study habits, motivation, and course difficulty, may also influence academic performance.