- Analyze the trend for each dataset by calculating the Pearson correlation coefficient \(r\).
- Table 1 shows strong negative correlation (\(r \approx -0.85\)).
- Table 2 shows strong positive correlation (\(r \approx 0.90\)).
- Table 3 shows no clear linear trend, with \(r \approx -0.10\).
- Table 4 shows very strong negative correlation (\(r \approx -0.97\)).
- The table representing no correlation is the third one: \(\boxed{\text{Table 3}}\).
Explanation
- Problem Analysis and Data Overview
To determine which table shows no linear correlation, we need to analyze the relationship between the independent variable \(x\) and the dependent variable \(y\) for each dataset. Linear correlation, measured by the Pearson correlation coefficient \(r\), indicates the strength and direction of a linear relationship between two variables. A value of \(r\) close to \(1\) indicates a strong positive linear correlation, a value close to \(-1\) indicates a strong negative linear correlation, and a value close to \(0\) indicates little to no linear correlation.
The given data points for the four tables are:
- Table 1: (3, -1), (5, -2), (6, -3), (8, -2), (10, -5), (14, -4), (15, -8)
- Table 2: (3, -6), (5, -7), (6, -4), (8, -2), (10, 0), (14, -1), (15, 3)
- Table 3: (3, -2), (5, -4), (6, 6), (8, 8), (10, 12), (14, 10), (15, -16)
- Table 4: (3, -3), (5, -5), (6, -9), (8, -11), (10, -13), (14, -15), (15, -17)
- Evaluating the Correlation Coefficients
We evaluate the correlation coefficient \(r\) for each table by examining the trend of \(y\) as \(x\) increases:
- Table 1: As \(x\) increases from 3 to 15, \(y\) generally decreases. Calculation shows \(r \approx -0.85\), which is a strong negative linear correlation.
- Table 2: As \(x\) increases, \(y\) generally increases. Calculation shows \(r \approx 0.90\), which is a strong positive linear correlation.
- Table 3: The values of \(y\) fluctuate significantly (increasing then dropping sharply at the end), showing no clear linear pattern. Calculation shows \(r \approx -0.10\), which is very close to 0.
- Table 4: There is a very consistent decrease in \(y\) as \(x\) increases. Calculation shows \(r \approx -0.97\), which is a very strong negative linear correlation.
- Conclusion and Final Answer
Based on the calculated correlation coefficients, Table 3 has a value of approximately \(-0.10\). Since this value is closest to 0 among the four datasets, it indicates the absence of a meaningful linear relationship. Therefore, Table 3 is the one that shows no correlation. The final answer is the third table.
Examples
The concept of correlation is vital in fields like data science and statistics to understand how variables interact. For example, a shopkeeper might analyze the correlation between the number of hours a store is open and daily sales to optimize operational hours. If they find a strong positive correlation, they can confidently increase hours to boost revenue. Conversely, if they find no correlation between the color of the shopping bags and customer satisfaction, they know changing the bag color won't affect customer loyalty. Understanding these relationships helps in making data-driven decisions in everyday business scenarios.