Question

(a) Describe the nature of the association between price and quality rating for the sewing machines. (b) One of the 14 sewing machines substantially affects the appropriateness of using a linear regression model to predict quality rating based on price. Report the approximate price and quality rating of that machine and explain your choice. (c) Chris is interested in buying one of the 14 sewing machines. He will consider buying only those machines for which there is no other machine that has both higher quality and lower price. On the scatterplot reproduced below, circle all data points corresponding to machines that Chris will consider buying.

Answer 1

Scatter plots visually display the relationship between independent and dependent variables. The least-squares line fits the data for predictive insights, while significance of the relationship can be validated by the correlation coefficient's value.

To analyze the relationship between two variables, we begin by identifying the independent and dependent variables. Upon deciding this, a scatter plot is created to visually inspect any apparent association between the variables. The perceived relationship can be quantified through the calculation of a least-squares line, commonly denoted as ý = a + bx. This line attempts to best fit the data points and predict the dependent variable's values based on the independent variable.

To further assess the strength and direction of the relationship, the correlation coefficient is computed. A significant correlation coefficient suggests a strong association between the variables, validating the appropriateness of using linear regression for predictions. Outliers are detected by examining deviations from the trend, which can influence the regression model's accuracy. Lastly, the least-squares line's equation enables estimation of values for specific data points, thereby aiding in decision-making processes or predictions.

An interpretation of the slope of this regression line offers insight into the relationship dynamics between the variables, indicating how a unit change in the independent variable impacts the dependent variable. This step is critical for the sound analysis and eventual application of regression models.