91Ó°ÊÓ

The article "Readability of Liquid Crystal Displays: A Response Surface" (Human Factors [1983]: \(185-190\) ) used a multiple regression model with four independent variables, where \(y=\) error percentage for subjects reading a four-digit liquid crystal display \(x_{1}=\) level of backlight (from 0 to \(\left.122 \mathrm{~cd} / \mathrm{m}\right)\) \(x_{2}=\) character subtense (from \(.025^{\circ}\) to \(1.34^{\circ}\) ) \(x_{3}=\) viewing angle (from \(0^{\circ}\) to \(60^{\circ}\) ) \(x_{4}=\) level of ambient light (from 20 to \(1500 \mathrm{~lx}\) ) The model equation suggested in the article is $$ y=1.52+.02 x_{1}-1.40 x_{2}+.02 x_{3}-.0006 x_{4}+e $$ a. Assume that this is the correct equation. What is the mean value of \(y\) when \(x_{1}=10, x_{2}=.5, x_{3}=50\), and \(x_{4}=100 ?\) b. What mean error percentage is associated with a backlight level of 20 , character subtense of \(.5\), viewing angle of 10 , and ambient light level of 30 ? c. Interpret the values of \(\beta_{2}\) and \(\beta_{3}\).

Step by step solution

01

Calculate Mean Value of y for Given Values of x1, x2, x3 and x4 - Part (a)

To find the mean value of y when \(x_{1} = 10\), \(x_{2} = 0.5\), \(x_{3} = 50\), and \(x_{4} = 100\), substitute these given values in the multiple regression equation. According to the regression equation, \(y = 1.52 + 0.02x_{1} - 1.40x_{2} + 0.02x_{3} - 0.0006x_{4} + e\) . Note that 'e' represents the random error component, but we are looking at the mean value of y, so 'e' equals 0.

02

Calculate Mean Error Percentage for Given values of x1, x2, x3 and x4 - Part (b)

Repetitive process of step 1, substituting the provided backlight, character subtense, viewing angle, and ambient light level values into the regression equation, and then solve for y. The new values to be used are \(x_{1} = 20\), \(x_{2} = 0.5\), \(x_{3} = 10\), and \(x_{4} = 30\). The result will be mean error percentage associated with the provided conditions.

03

Interpret the Values of Beta Coefficients - Part (c)

To interpret given coefficients \(\beta_{2}\) and \(\beta_{3}\), we need to locate these in the given equation. They are the coefficients of the independent variables \(x_{2}\) and \(x_{3}\) which are -1.40 and 0.02 respectively. The coefficient of a variable in a multiple regression equation represents the average change in the dependent variable for each 1-unit change in the respective independent variable, holding all other variables constant. Thus, the coefficient \(\beta_{2} = -1.40\) implies that for each 1-degree increase in the character subtense, the error percentage \(y\) decreases by 1.40, all else being equal. Similarly, the coefficient \(\beta_{3} = 0.02\) implies that for each 1-degree increase in viewing angle, the error percentage \(y\) increases by 0.02, all else being equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Variables

In the context of multiple regression, independent variables are the input factors that influence the outcome variable, often denoted as the dependent variable. In this exercise, the dependent variable is the error percentage of reading a liquid crystal display, denoted by \(y\). The independent variables are:

• \(x_1\) - Level of backlight, ranging from 0 to 122 cd/m\(^{2}\).
• \(x_2\) - Character subtense, ranging from 0.025\(^{\circ}\) to 1.34\(^{\circ}\).
• \(x_3\) - Viewing angle, ranging from 0\(^{\circ}\) to 60\(^{\circ}\).
• \(x_4\) - Level of ambient light, ranging from 20 to 1500 lx.

These variables serve as predictors in the regression model, explaining variations in the error percentage \(y\). By understanding these factors, we can determine their impact individually and collectively on the readability of displays.

Error Percentage

The error percentage, denoted as \(y\), is the dependent variable in the regression equation. It represents the percentage of errors made by subjects while reading a four-digit display. Understanding this concept is crucial because it allows us to assess the effectiveness of different display settings.The regression model provides a tool to estimate this error percentage based on the given values of the independent variables. The model equation includes a component for random error \(e\), which accounts for variability not explained by the independent variables. However, when calculating mean values, this error term is typically assumed to be zero, simplifying our calculations.In practical terms, reducing the error percentage is vital for improving display readability, and tweaking the independent variables can help achieve this goal.

Regression Coefficients

In multiple regression analysis, regression coefficients are evaluated to understand the relationship between independent variables and the dependent variable. These coefficients indicate the extent of change in the dependent variable resulting from a one-unit change in an independent variable, holding all other variables constant.For this exercise, the given equation is\[ y = 1.52 + 0.02x_1 - 1.40x_2 + 0.02x_3 - 0.0006x_4 + e \]The regression coefficients are as follows:

• \(\beta_1 = 0.02\): Suggests a slight increase in error percentage with increasing backlight level.
• \(\beta_2 = -1.40\): Indicates a significant decrease in error percentage with an increase in character subtense.
• \(\beta_3 = 0.02\): Implies a small increase in error percentage with an increasing viewing angle.
• \(\beta_4 = -0.0006\): Shows a negligible decrease in error percentage with higher ambient light levels.

By interpreting these coefficients, we understand the relative impact of each independent variable on the error percentage, helping us refine display conditions for better readability.

Mean Value Calculation

Calculating the mean value of a dependent variable in a regression analysis requires substituting known values of independent variables into the regression equation. This process helps determine expected outcomes under specific conditions.The provided exercise explored two sets of independent variable values:
1. \(x_1 = 10\), \(x_2 = 0.5\), \(x_3 = 50\), \(x_4 = 100\)2. \(x_1 = 20\), \(x_2 = 0.5\), \(x_3 = 10\), \(x_4 = 30\)To compute the mean error percentage for each scenario, we plug these values into the regression model:

• For the first set: \(y = 1.52 + 0.02(10) - 1.40(0.5) + 0.02(50) - 0.0006(100)\)
• For the second set: \(y = 1.52 + 0.02(20) - 1.40(0.5) + 0.02(10) - 0.0006(30)\)

By following this method, you can find the average error percentages associated with given settings, aiding in decision-making about optimal display configurations.