91Ó°ÊÓ

A number of investigations have focused on the problem of assessing loads that can be manually handled in a safe manner. The article "Anthropometric, Muscle Strength, and Spinal Mobility Characteristics as Predictors in the Rating of Acceptable Loads in Parcel Sorting" (Ergonomics [1992]: \(1033-1044\) ) proposed using a regression model to relate the dependent variable \(y=\) individual's rating of acceptable load \((\mathrm{kg})\) to \(k=3\) independent (predictor) variables: \(x_{1}=\) extent of left lateral bending \((\mathrm{cm})\) $$ \begin{aligned} &x_{2}=\text { dynamic hand grip endurance (sec) } \\ &x_{3}=\text { trunk extension ratio }(\mathrm{N} / \mathrm{kg}) \end{aligned} $$ Suppose that the model equation is $$ y=30+.90 x_{1}+.08 x_{2}-4.50 x_{3}+e $$ and that \(\sigma=5\). a. What is the population regression function? b. What are the values of the population regression \(\underline{\mathrm{co}}\) efficients? c. Interpret the value of \(\beta_{1}\). d. Interpret the value of \(\beta_{3}\). e. What is the mean value of rating of acceptable load when extent of left lateral bending is \(25 \mathrm{~cm}\), dynamic hand grip endurance is \(200 \mathrm{sec}\), and trunk extension ratio is \(10 \mathrm{~N} / \mathrm{kg}\) ? f. If repeated observations on rating are made on different individuals, all of whom have the values of \(x_{1}, x_{2}\), and \(x_{3}\) specified in Part (e), in the long run approximately what percentage of ratings will be between \(13.5 \mathrm{~kg}\) and \(33.5 \mathrm{~kg} ?\)

Step by step solution

01

Identify the Population Regression Function

The population regression function is given in the problem as \(y = 30 + 0.90x_{1} + 0.08x_{2} - 4.50x_{3} + e\).

02

Identify the Population Regression Coefficients

The coefficients of the population regression are the numerical factors of the predictor variables. Therefore, the coefficients of \(x_{1}, x_{2}\), and \(x_{3}\) are \(0.90, 0.08\), and \(-4.50\) respectively.

03

Interpret the Coefficient of \(x_{1}\)

The coefficient of \(x_{1} = 0.90\) means that for a 1-cm increase in the extent of left lateral bending, the individual's rating of acceptable load increases, on average, by 0.90 kg, holding all other predictors constant.

04

Interpret the Coefficient of \(x_{3}\)

The coefficient of \(x_{3} = -4.50\) means that for a 1-unit increase in the trunk extension ratio, the individual's rating of acceptable load decreases, on average, by 4.50 kg, holding all other predictors constant.

05

Compute the Mean Value of Rating of Acceptable Load

Substitute \(x_{1} = 25, x_{2} = 200\), and \(x_{3} = 10\) in the regression function to compute the mean value of rating of acceptable load: \( y = 30 + 0.90*(25) + 0.08*(200) - 4.50*(10) = 22.5 kg.\)

06

Compute the Percentage of Ratings Between 13.5 kg and 33.5 kg

Using the property of normally distributed residuals with standard deviation \(\sigma = 5\), we can convert the value of individual's rating to z-scores. The z-score is a measure of how many standard deviations an element is from the mean. For the lower limit, \(z1 = (13.5 - 22.5) / 5 = -1.80\). For the upper limit, \(z2 = (33.5 - 22.5) / 5 = 2.20\). The percentage of ratings between these two values is the area under a normal distribution curve between these z-scores. Using the standard normal distribution table, we know that proportions for \(z1\) and \(z2\) are 0.0359 and 0.9861 respectively. Therefore, the proportion of ratings between the two limits is \(0.9861 - 0.0359 = 0.9502\) or approximately 95%.

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Key Concepts

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

Regression Model

In the context of statistical analysis, a regression model is a powerful tool used to examine the relationship between a dependent variable and one or more independent variables, often referred to as predictor variables. The core purpose of this model is to explore how the dependent variable changes as the predictor variables are varied.

For example, in the given exercise, the regression model relates the variable y (individual's rating of acceptable load in kilograms) to three predictor variables. These variables (extent of left lateral bending, dynamic hand grip endurance, and trunk extension ratio) are thought to influence the amount of load an individual finds acceptable to handle safely.

The equation provided in the exercise, \( y = 30 + 0.90x_{1} + 0.08x_{2} - 4.50x_{3} + e \) where e represents the error term, is the representation of the regression model. This equation allows us to predict the average load an individual can handle based on the values of the predictor variables. The accuracy of this prediction depends on the regression coefficients and the variability captured by the error term.

Predictor Variables

In any regression model, the predictor variables are the independent variables that provide us with the means to predict or estimate the dependent variable. They are deliberately selected based on their presumed influence on the outcome variable.

In our exercise, the predictor variables are:

• Extent of left lateral bending (\(x_{1}\)) measured in centimeters
• Dynamic hand grip endurance (\(x_{2}\)) measured in seconds
• Trunk extension ratio (\(x_{3}\)) measured in Newtons per kilogram

These chosen measures are believed to have a direct impact on the perceived acceptable load that an individual can handle during parcel sorting. They form the core of the regression model's exploratory purpose and they determine the nature and strength of the relationship with the dependent variable.

The role of each predictor variable is quantified by its corresponding regression coefficient, which tells us the expected change in the dependent variable for a one-unit change in the predictor, holding all other predictors constant. Understanding the interplay between these variables is essential in assessing their individual contributions to the predictive power of the model.

Regression Coefficients

The regression coefficients in a regression model are the numerical values that represent the expected change in the dependent variable for each unit change in the predictor variables. Essentially, they tell us the strength and direction of the relationship between the individual predictor variables and the dependent variable.

In our scenario, the regression function \( y = 30 + 0.90x_{1} + 0.08x_{2} - 4.50x_{3} + e \) contains the coefficients: 0.90 for the extent of left lateral bending (\(x_{1}\)), 0.08 for dynamic hand grip endurance (\(x_{2}\)), and -4.50 for the trunk extension ratio (\(x_{3}\)). These coefficients reveal that while each centimeter of left lateral bending and each second of grip endurance contribute positively to the acceptable load, an increase in the trunk extension ratio actually reduces the acceptable load.

This negative coefficient for \(x_{3}\) is particularly interesting because it highlights that not all physical capacities increase the safe load limit. The regression coefficients provide valuable insights into the nature of the predictors' influence and are central to making informed predictions about the dependent variable.