LaTasha argued that there was no correlation between the variables and in a given data set. To evaluate her claim, we can examine the regression equation derived from the data. If the slope of the regression line is close to zero, it suggests that changes in have little to no effect on , supporting LaTasha’s argument of no correlation. However, if the slope is significantly positive or negative, it indicates a clear trend—either increasing or decreasing—meaning there is a correlation and LaTasha would be incorrect. Therefore, the regression equation provides a mathematical basis for confirming or refuting her claim.
Answer (1)
To understand LaTasha's claim about the correlation between two variables using a regression equation, let's break down the concept of correlation and regression analysis.
Correlation :
Correlation measures the strength and direction of a linear relationship between two variables. It is quantified by the correlation coefficient, usually represented by r . A value of r close to 0 suggests no linear relationship, while a value close to 1 or -1 indicates a strong linear relationship—positive or negative, respectively.
Regression Analysis :
Regression involves fitting a line through the data points to model the relationship between the independent variable ( x ) and the dependent variable ( y ). The equation of a simple linear regression line is given by:
y = m x + b
Where:
y is the predicted value.
m is the slope of the line.
x is the independent variable.
b is the y-intercept.
Evaluating LaTasha's Claim :
Slope ( m ) Analysis :
If m (the slope) is close to zero, it suggests that changes in x have little to no effect on y , supporting LaTasha's claim of no correlation.
Conversely, if m is significantly positive or negative, it indicates a linear relationship between x and y , implying that LaTasha's claim is incorrect.
Conclusion :
By examining the slope of the regression line derived from the given data, we can mathematically assess LaTasha's argument about the absence or presence of correlation between the variables. A slope close to zero would confirm her statement, whereas a strong positive or negative slope would show a correlation.
In summary, the slope of the regression line from the data set provides a clear mathematical basis to evaluate and confirm or refute LaTasha's claim regarding the correlation between the variables.
