Final answer:
A scatter plot shows a linear relationship if the data points form a straight line, indicating a direct relationship. A significant correlation coefficient and a linear pattern together suggest that linear regression is appropriate for modeling the data and making predictions.
Step-by-step explanation:
To determine if a scatter plot shows a linear relationship, you should first look at the pattern the points make on the graph. When data is graphed and the result is a straight line, this is indicative of a direct relationship between the variables. A linear relationship suggests that as one variable increases, the other variable tends to increase (positive relationship) or decrease (negative relationship) at a constant rate.
When considering if the variables would be good candidates for linear regression, it's important to assess if there is a clear, consistent pattern that resembles a line. Moreover, the significance of the correlation coefficient, often denoted as 'r', plays a crucial role. If r is significant, and the scatter plot displays a linear trend, the data can be modeled with a linear equation and used for prediction within the domain of observed x values. It is essential to note that if the pattern seems more curved than straight, other forms of regression might be more appropriate.
However, if the scatter plot shows a significant linear trend and the correlation coefficient is significantly different from zero, this suggests that there is a significant linear relationship. Under such circumstances, a statistician might conclude that the linear regression model is appropriate, and the regression line can be used to predict values.