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Answer :
- The correlation coefficient $r = 0.23$ suggests a positive correlation.
- The p-value $p = 0.221$ indicates that the correlation is not statistically significant.
- Therefore, there is no statistically significant correlation between seating position and final grades.
- The most accurate statement is that there was not a statistically significant correlation: $\boxed{\text{There was not a statistically significant correlation between where students sat and the final grades.}}$
### Explanation
1. Understand the problem and provided data
We are given the correlation coefficient $r = 0.23$, the sample size $n = 30$, and the p-value $p = 0.221$. We need to determine the most accurate statement about the correlation between seating position and final grades.
2. Interpret the correlation coefficient
The correlation coefficient $r = 0.23$ indicates a positive correlation. This means that as students sit closer to the front of the class, their grades tend to be higher. However, we need to check if this correlation is statistically significant.
3. Assess the statistical significance
The p-value is $p = 0.221$. A correlation is considered statistically significant if the p-value is less than a predetermined significance level (alpha), which is commonly set at 0.05. In this case, $0.221 > 0.05$, so the correlation is not statistically significant.
4. Conclusion
Since the correlation is not statistically significant, we cannot definitively conclude that there is a correlation between seating position and final grades, even though the correlation coefficient is positive. Therefore, the most accurate statement is: There was not a statistically significant correlation between where students sat and the final grades.
### Examples
Understanding correlations is useful in many real-world scenarios. For example, businesses use correlation analysis to determine the relationship between advertising spending and sales revenue. Similarly, researchers use it to study the relationship between diet and health outcomes. In education, instructors can use correlation to explore the relationship between study habits and exam performance, although it's important to remember that correlation doesn't imply causation.
- The p-value $p = 0.221$ indicates that the correlation is not statistically significant.
- Therefore, there is no statistically significant correlation between seating position and final grades.
- The most accurate statement is that there was not a statistically significant correlation: $\boxed{\text{There was not a statistically significant correlation between where students sat and the final grades.}}$
### Explanation
1. Understand the problem and provided data
We are given the correlation coefficient $r = 0.23$, the sample size $n = 30$, and the p-value $p = 0.221$. We need to determine the most accurate statement about the correlation between seating position and final grades.
2. Interpret the correlation coefficient
The correlation coefficient $r = 0.23$ indicates a positive correlation. This means that as students sit closer to the front of the class, their grades tend to be higher. However, we need to check if this correlation is statistically significant.
3. Assess the statistical significance
The p-value is $p = 0.221$. A correlation is considered statistically significant if the p-value is less than a predetermined significance level (alpha), which is commonly set at 0.05. In this case, $0.221 > 0.05$, so the correlation is not statistically significant.
4. Conclusion
Since the correlation is not statistically significant, we cannot definitively conclude that there is a correlation between seating position and final grades, even though the correlation coefficient is positive. Therefore, the most accurate statement is: There was not a statistically significant correlation between where students sat and the final grades.
### Examples
Understanding correlations is useful in many real-world scenarios. For example, businesses use correlation analysis to determine the relationship between advertising spending and sales revenue. Similarly, researchers use it to study the relationship between diet and health outcomes. In education, instructors can use correlation to explore the relationship between study habits and exam performance, although it's important to remember that correlation doesn't imply causation.
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