Answer :
To solve this problem, we are analyzing a bivariate data set to determine the effect of an outlier on the correlation coefficient.
### Step 1: Identify the Outlier
Upon examining the provided data set, we find that the point [tex]\((248.6, 1938.2)\)[/tex] is much different from the other data values. This makes it a potential outlier that can affect the correlation.
### Step 2: Calculate the Correlation Coefficient with the Outlier
The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables, [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Including the outlier, we calculate the correlation coefficient for the entire data set:
- Correlation with the outlier: [tex]\( r_w = 0.251 \)[/tex]
### Step 3: Remove the Outlier and Calculate the Correlation Coefficient Again
To see how the outlier affects the correlation, we remove the point [tex]\((248.6, 1938.2)\)[/tex] from our data set. We then recalculate the correlation coefficient for the remaining data:
- Correlation without the outlier: [tex]\( r_{\text{wo}} = -0.214 \)[/tex]
### Step 4: Analyze the Change in Correlation
Now, we need to evaluate whether the outlier significantly changes our understanding of the relationship between the two variables:
- With the outlier: The correlation coefficient is 0.251, suggesting a weak positive linear relationship.
- Without the outlier: The correlation coefficient is -0.214, indicating a weak negative linear relationship.
### Conclusion
The inclusion of the outlier changes the evidence regarding a linear correlation. It shifts the correlation from weakly positive to weakly negative, indicating the outlier has a notable impact on the perceived relationship between the variables.
Therefore, the answer to whether including the outlier changes the evidence for or against a linear correlation is:
Yes. Including the outlier changes the evidence regarding a linear correlation.
### Step 1: Identify the Outlier
Upon examining the provided data set, we find that the point [tex]\((248.6, 1938.2)\)[/tex] is much different from the other data values. This makes it a potential outlier that can affect the correlation.
### Step 2: Calculate the Correlation Coefficient with the Outlier
The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables, [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Including the outlier, we calculate the correlation coefficient for the entire data set:
- Correlation with the outlier: [tex]\( r_w = 0.251 \)[/tex]
### Step 3: Remove the Outlier and Calculate the Correlation Coefficient Again
To see how the outlier affects the correlation, we remove the point [tex]\((248.6, 1938.2)\)[/tex] from our data set. We then recalculate the correlation coefficient for the remaining data:
- Correlation without the outlier: [tex]\( r_{\text{wo}} = -0.214 \)[/tex]
### Step 4: Analyze the Change in Correlation
Now, we need to evaluate whether the outlier significantly changes our understanding of the relationship between the two variables:
- With the outlier: The correlation coefficient is 0.251, suggesting a weak positive linear relationship.
- Without the outlier: The correlation coefficient is -0.214, indicating a weak negative linear relationship.
### Conclusion
The inclusion of the outlier changes the evidence regarding a linear correlation. It shifts the correlation from weakly positive to weakly negative, indicating the outlier has a notable impact on the perceived relationship between the variables.
Therefore, the answer to whether including the outlier changes the evidence for or against a linear correlation is:
Yes. Including the outlier changes the evidence regarding a linear correlation.
