Question

HELP PLS FOR 100 POINTS
7. a. Use Desmos to compute a line of regression for the data to the right
b. Open a new window and use the same data and find a quadratic curve of regression. View the windows side by side.
c. Which seems to fit the data better? How do you know?
d. In Desmos, with the data and linear regression graphed, click the "Plot" button under the word "residuals" for both graphs. Does this give you a clearer answer to #9?________. What do you think "residuals" are?
e. What is the correlation coefficient for each of the residuals? (look for "r") _____(linear); _______(quadratic). What do you think the correlation coefficient measures?

Answer 1

Answer:

(a) Attachment 1

(b) Attachment 2

(c) The quadratic regression curve provides a better fit for the data as the data points are closer to this curve than they are to the line of regression.

(d) Yes.
Residuals represent the vertical distances between observed data points and the predicted values of the regression line or curve.

(e) r = 0.9761 (linear)
     R² = 0.9996 (quadratic)
The correlation coefficient measures the degree of the relationship between two variables, with values ranging from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation.

Step-by-step explanation:

Part (a)

Linear regression is a statistical method that models the relationship between a dependent variable (x) and independent variable (y) by fitting a linear equation to the observed data.

See attachment 1 for the graph of the line of regression for the given data.

Part (b)

Quadratic regression is a statistical method that models the relationship between a dependent variable (x) and independent variable (y) by  fitting a quadratic equation (second-degree polynomial) to the observed data.

See attachment 2 for the graph of the quadratic curve of regression for the given data.

Part (c)

From observation of the graphs, it appears that the quadratic regression curve provides a better fit for the data as the data points are closer to this curve than they are to the line of regression.

Part (d)

Residuals represent the vertical distances between observed data points and the predicted values of the regression line or curve, offering insights into which type of regression curve better fits the data by revealing patterns in prediction errors.

If residuals are randomly scattered around the x-axis, it suggests a good fit of the regression curve, indicating that the model captures the data's variability. However, the presence of patterns, such as residuals forming a specific shape or consistently deviating from zero, signals potential inadequacies in the chosen regression model.

In the case of the quadratic curve of regression, residuals are randomly scattered around the x-axis (attachment 3), while the residuals of the linear regression curve form the shape of a downward opening parabola (attachment 4). Therefore, the residuals support the conclusion that the quadratic regression curve provides a better fit.

Part (e)

Correlation coefficients are statistical measures that quantify the degree and direction of the relationship between two variables, with values ranging from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation.

From observation of the attached graphs, the correlation coefficients:

• Linear: r = 0.9761
• Quadratic: R² = 0.9996