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The article 鈥淩eadability 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 fourdigit 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 \(\left.1.34^{\circ}\right)\) \(x_{3}=\) viewing angle \(\left(\right.\) from \(0^{\circ}\) to \(\left.60^{\circ}\right)\) \(x_{4}=\) 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

Calculating the Mean Value of y

First, substitute the given values of \(x_{1}, x_{2}, x_{3}, x_{4}\) into the model equation. Therefore: \[y= 1.52 + (.02 \cdot 10) - (1.4 \cdot .5) + (.02 \cdot 50) - (.0006 \cdot 100) \] Calculate the value of y. This will give the mean error percentage knowing the levels of the variables in the regression equation.

02

Calculate the Mean Error Percentage for Specific Levels

Substitute the given values of backlight level, character subtense, viewing angle, and ambient light level into the model equation: \[y= 1.52 + (.02 \cdot 20) - (1.4 \cdot .5) + (.02 \cdot 10) - (.0006 \cdot 30) \] Then, calculate the value of y which will be the mean error percentage given these levels in the equation.

03

Interpret the Values of \(\beta_{2}\) and \(\beta_{3}\)

The coefficients \(\beta_{2}\) and \(\beta_{3}\) (-1.40 and .02, respectively), represent the change in error percentage (y) per unit change in character subtense and viewing angle respectively, holding other variables constant. In other words, \(\beta_{2}\) implies that for each degree increase in character subtense, the error percentage decreases by 1.4%, while \(\beta_{3}\) suggests that for each degree increase in viewing angle, the error percentage increases by .02%.

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Key Concepts

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

Independent Variables

In multiple regression analysis, independent variables (sometimes called predictors or input variables) are crucial. They stand for the factors we believe will influence or predict the dependent variable. In this exercise, the independent variables are:

• Backlight level (\(x_{1}\))
• Character subtense (\(x_{2}\))
• Viewing angle (\(x_{3}\))
• Ambient light (\(x_{4}\))

These variables are chosen based on theory or prior research as having potential impact on the dependent variable, which in this case is the error percentage in reading the LCD display.
Each independent variable needs to be measurable and are assigned specific numerical values. These numbers are then used for calculating the error percentage in this model.

Regression Equation

A regression equation is a mathematical formula used to describe the relationship between the dependent variable and one or more independent variables. The equation provides a model that predicts the dependent variable's values based on the values of the independent variables.
In the exercise, the regression equation is given as:\[y = 1.52 + 0.02x_{1} - 1.40x_{2} + 0.02x_{3} - 0.0006x_{4} + e\]Here, \(y\) represents the dependent variable (error percentage), and \(x_{1}, x_{2}, x_{3}, x_{4}\) are the independent variables detailed earlier. The model includes a constant term (1.52) and error term (\(e\)), representing noise or variability not captured by the independent variables. Calculating this equation with given \(x\) values provides the predicted mean value of \(y\).

Error Percentage

The error percentage signifies the amount of mistake or variance in reading the display correctly. It is the primary outcome we are trying to predict using this multiple regression model.
To find the error percentage using our model, substitute the given values for each of the independent variables into the regression equation. For instance, using the first scenario:\[ y = 1.52 + (0.02 \times 10) - (1.40 \times 0.5) + (0.02 \times 50) - (0.0006 \times 100) \]
By solving, you will derive the mean error percentage. This process is repeated for any other given variable scenarios, like changing backlight levels or viewing angles, to see how error percentages vary.

Coefficients

Coefficients in a regression model indicate the amount of change we'd expect in the dependent variable for a one-unit change in an independent variable, assuming all other variables stay constant.
The coefficients in our equation: 0.02 (for \(x_{1}\)), -1.40 (for \(x_{2}\)), 0.02 (for \(x_{3}\)), and -0.0006 (for \(x_{4}\)) each tell us about the sensitivity of the error percentage.

• \(\beta_{1} = 0.02\): Small increase in error with more backlight.
• \(\beta_{2} = -1.40\): Significant decrease as character subtense increases, meaning clearer or larger text dramatically reduces error.
• \(\beta_{3} = 0.02\): Marginal increase in error with a wider viewing angle.
• \(\beta_{4} = -0.0006\): More ambient light slightly reduces error.

Understanding these coefficients helps assess the predictors' impact on the error percentage and guides optimizing conditions for the best screen readability.