High School

The following bivariate data set contains an outlier.

[tex]
\[
\begin{array}{|r|r|}
\hline
\multicolumn{1}{|c|}{x} & \multicolumn{1}{c|}{y} \\
\hline
99.8 & 714.2 \\
64.1 & -354.9 \\
68.6 & 29.9 \\
68.7 & -399.7 \\
50.2 & 2783.3 \\
95.9 & 19.4 \\
95.5 & -1951.6 \\
78.6 & -2429.7 \\
88.8 & -1343.9 \\
88.1 & -2176.3 \\
85.6 & 953.8 \\
97.8 & 1973.5 \\
84 & 1169.1 \\
66.4 & -281.9 \\
248.6 & 1938.2 \\
\hline
\end{array}
\]
[/tex]

1. What is the correlation coefficient with the outlier?
[tex] r_w = \square [/tex]
[Round your answer to three decimal places.]

2. What is the correlation coefficient without the outlier?
[tex] r_{\text{wo}} = \square [/tex]
[Round your answer to three decimal places.]

3. Would inclusion of the outlier change the evidence for or against a linear correlation?
- Yes. Including the outlier changes the evidence regarding a linear correlation.
- No. Including the outlier does not change the evidence regarding a linear correlation.

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.