Final answer:
To find the best linear model for a data set, we draw a line that best captures the data trend, check for outliers, and ensure that the linear model remains valid for the range of predictions. We derive the least-squares line using regression analysis, with the slope indicating the relationship between the variables.
Step-by-step explanation:
To determine which linear model best fits a set of data, we need to consider several factors. First, we should analyze whether the data seems to follow a linear trend. We can do this by plotting the data and attempting to draw a line that represents the general direction of the data points. If a line seems to capture the pattern of the data well, then a linear model may be appropriate.
Next, we need to consider the presence of outliers, which are data points that do not fit the overall pattern. Outliers can significantly affect the slope of the best-fit line and the predictions made by the linear model. If outliers exist, we might need to investigate their causes and consider removing them from the analysis or using a different model that is less sensitive to outliers.
If we are using the linear model to make predictions, such as estimating the cost of a 300 oz. size of laundry detergent based on the existing data, we must also verify that the model remains valid for such extrapolations. The least-squares line or best-fit line is typically found using regression analysis, which minimizes the sum of the squared differences (residuals) between the observed values and the values predicted by the line. The slope of this line indicates the rate of change between the two variables being plotted. A positive slope suggests a positive relationship, whereas a negative slope suggests a negative relationship.