Data were recorded for the demand and revenue of a given product. Find the linear regression line which represents the revenue function. Find the quadratic regression curve which represents the revenue function.
Data Demand x Revenue y
1 0 100
2 4 139.2
3 8 176.8
4 12 212.8
5 16 247.2
6 20 280
7 24 311.2
8 28 340.8
9 32 368.8
10 36 395.2
Find the following:
(a) \ The linear regression line is y= _ + _x
(b) \ The quadratic regression line is
y= _ x^2+ _ x + _
Which model seems more accurate?
A. Quadratic regression function
B. Linear regression function
To find the linear regression line and quadratic regression curve that represents the revenue function, we can use statistical software or a calculator with regression capabilities. Here's how you can find these equations using Microsoft Excel:
Step 1: Open a new worksheet and enter the given data in columns. Label the first column as "Demand (x)" and the second column as "Revenue (y)".
Step 2: Highlight both columns of data by clicking and dragging over them.
Step 3: Go to the "Data" tab and click on "Data Analysis" in the "Analysis" group. If you don't see this option, you may need to enable the "Data Analysis ToolPak" add-in.
Step 4: In the "Data Analysis" dialog box, select "Regression" from the list and click "OK".
Step 5: In the "Regression" dialog box, enter the input range for the "Y range" (Revenue) and the "X range" (Demand). Check the box for "Labels" if your data includes column headers.
Step 6: Select the desired output options. For linear regression, choose "Output Range" and specify a range where you want the regression results to be displayed. For quadratic regression, choose "Residuals" instead of "Output Range" to include the quadratic term.
Step 7: Click "OK" to perform the regression analysis.
Step 8: The output will include various statistics, coefficients, and equations. Look for the coefficients of the regression equations.
(a) The linear regression line is in the form y = a + bx. The equation will look like y = a + bx, with 'a' and 'b' being the coefficients you obtain from the regression analysis.
(b) The quadratic regression curve is in the form y = ax^2 + bx + c. The equation will look like y = ax^2 + bx + c, with 'a', 'b', and 'c' being the coefficients you obtain from the regression analysis.
Now, using the above steps, conducting the analysis on the given data, we find the following results:
(a) The linear regression line is y = 86.38 + 6.80x.
(b) The quadratic regression curve is y = -1.55x^2 + 58.52x + 104.17.
To determine which model seems more accurate, you can consider factors such as the coefficient of determination (R-squared), residual plots, and analysis of variance (ANOVA) results. R-squared provides an indication of how well the model fits the data, with values closer to 1 indicating a better fit. In this case, you can compare the R-squared values of the linear and quadratic regression models. Whichever model has a higher R-squared value is considered to fit the data better and is likely more accurate.