Final answer:
A scatterplot can exhibit positive, negative, or zero correlation, with correlation coefficient values indicating the strength and direction of the relationship. Data patterns must be closely examined to decide if linear regression is suitable. If the correlation coefficient is near zero and the scatterplot shows a nonlinear pattern, linear regression is not recommended.
Step-by-step explanation:
When analyzing a scatterplot, identifying the correlation between variables is crucial for understanding their relationship. A scatterplot with data points closely forming a line with a positive slope indicates a positive correlation, where an increase in one variable tends to be associated with an increase in the other (Figure 12.10 (a)). Conversely, a scatterplot with a negative slope suggests a negative correlation, where an increase in one variable is associated with a decrease in the other (Figure 12.10 (b)). A scatterplot where the data points do not follow any discernible trend or pattern signifies a zero correlation, and here the correlation coefficient (r) is around 0 (Figure 12.10 (c)).
When interpreting a specific scatterplot with a correlation coefficient (r), such as r=.55 (Figure 1-16), it's essential to comprehend the strength of the relationship. A correlation coefficient of .55 indicates a moderately strong relationship, which might be linear, but one should always consider the pattern of the scatterplot before concluding if a linear regression model is appropriate.
The pattern of the scatterplot should be inspected thoroughly. If the points suggest a nonlinear relationship, alternative models, such as a curve, may be more suitable. Thus, when deciding on using linear regression, the pattern must indicate that the X and Y variables have a linear relationship (Figure 2.12).
In summary, if the scatterplot does not exhibit a linear pattern and the correlation coefficient is not significantly different from zero, there is insufficient evidence to assert that a significant linear relationship exists between X and Y variables. As a result, linear regression would not be a reliable method for modeling these variables.