To determine if LaTasha was correct in arguing that there was no correlation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to analyze the relationship between the variables using the regression equation and the correlation coefficient.
Given the data set:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
y & 4 & 5 & 4 & 5 & 4 & 5 & 4 \\
\hline
\end{array}
\][/tex]
### Step 1: Determine the Regression Equation
The regression equation is represented as:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept.
From the provided results, we have:
[tex]\[ m \approx 2.172 \times 10^{-16} \][/tex]
[tex]\[ b \approx 4.429 \][/tex]
Thus, the regression equation is:
[tex]\[ y = (2.172 \times 10^{-16})x + 4.429 \][/tex]
Given that [tex]\( 2.172 \times 10^{-16} \)[/tex] is effectively zero, the regression line can be approximated as:
[tex]\[ y \approx 4.429 \][/tex]
### Step 2: Calculate the Correlation Coefficient
The correlation coefficient, denoted as [tex]\( r \)[/tex], quantifies the degree to which [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are linearly related. It ranges from -1 to 1, where:
- [tex]\( r = 1 \)[/tex] indicates perfect positive correlation,
- [tex]\( r = -1 \)[/tex] indicates perfect negative correlation,
- [tex]\( r = 0 \)[/tex] indicates no linear correlation.
From the results, the correlation coefficient is:
[tex]\[ r = 0.0 \][/tex]
### Analysis
The correlation coefficient [tex]\( r = 0.0 \)[/tex] indicates that there is no linear correlation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This means changes in [tex]\( x \)[/tex] do not systematically result in changes in [tex]\( y \)[/tex].
Furthermore, analyzing the regression equation [tex]\( y \approx 4.429 \)[/tex] also supports this conclusion. Since the slope [tex]\( m \)[/tex] is essentially zero, the line is nearly horizontal, suggesting that [tex]\( y \)[/tex] values do not change as [tex]\( x \)[/tex] changes.
### Conclusion
LaTasha's argument that there is no correlation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is indeed correct. The correlation coefficient [tex]\( r = 0.0 \)[/tex] and the regression equation [tex]\( y \approx 4.429 \)[/tex] both suggest that there is no linear relationship between the variables. The [tex]\( y \)[/tex] values are fluctuating around a constant value without any dependence on [tex]\( x \)[/tex].
