91Ó°ÊÓ

The AFL-CIO has undertaken a study of 30 secretaries' yearly salaries (in thousands of dollars). The organization wants to predict salaries from several other variables. The variables considered to be potential predictors of salary are The variables considered to be potential predictors of salary are \(\mathrm{X} 1=\) months of service \(\mathrm{X} 2=\) years of education \(\mathrm{X} 3=\) score on standardized test \(\mathrm{X} 4=\) words per minute (wpm) typing speed \(\mathrm{X} 5=\) ability to take dictation in words per minute A multiple regression model with all five variables was run on a computer package, resulting in the following output: \(\begin{array}{lccc}\text { Variable } & \text { Coefficient } & \text { Std. Error } & \text { t-Value } \\ \text { Intercept } & 9.788 & 0.377 & 25.960 \\\ \text { X1 } & 0.110 & 0.019 & 5.178 \\ \text { X2 } & 0.053 & 0.038 & 1.369 \\ \text { X3 } & 0.071 & 0.064 & 1.119 \\ \text { X4 } & 0.004 & 0.307 & 0.013 \\ \text { X5 } & 0.065 & 0.038 & 1.734\end{array}\) \(s=0.430 \quad R^{2}=0.863\) Assume that the residual plots show no violations of the conditions for using a linear regression model. a) What is the regression equation? b) From this model, what is the predicted Salary (in thousands of dollars) of a secretary with 10 years (120 months) of experience, 9th grade education (9 years of education), a 50 on the standardized test, 60 wpm typing speed, and the ability to take 30 wpm dictation? c) Test whether the coefficient for words per minute of typing speed \((X 4)\) is significantly different from zero at \(\alpha=0.\) d) How might this model be improved? e) A correlation of Age with Salary finds \(r=0.682,\) and the scatterplot shows a moderately strong positive linear association. However, if \(X 6=A g e\) is added to the multiple regression, the estimated coefficient of \(A g e\) turns out to be \(b_{6}=-0.154 .\) Explain some possible causes for this apparent change of direction in the relationship between age and salary.

Step by step solution

01

Identify Given Data

The multiple regression output includes the coefficients for the variables and gives predictions regarding secretaries' salaries in thousands of dollars. The regression equation involves these variables: Intercept, \(X_1\) through \(X_5\), along with their coefficients.

02

Formulate the Regression Equation

The regression equation can be formed using the coefficients provided in the output: \[ \text{Salary} = 9.788 + 0.110X_1 + 0.053X_2 + 0.071X_3 + 0.004X_4 + 0.065X_5 \]

03

Calculate Predicted Salary

Substitute the given values into the regression equation: 120 months for \(X_1\), 9 years for \(X_2\), 50 for \(X_3\), 60 for \(X_4\), and 30 for \(X_5\).\[ \text{Salary} = 9.788 + 0.110(120) + 0.053(9) + 0.071(50) + 0.004(60) + 0.065(30) \] Calculate each term and sum them to find the predicted salary.

04

Evaluate the Significance of the Coefficient for \(X_4\)

To test if \(X_4\), the typing speed, is significantly different from zero, examine the t-value provided (\(t = 0.013\)). With a high p-value, this coefficient is not significantly different from zero because it suggests typing speed doesn't contribute to variance in salary prediction based on a chosen \(\alpha = 0.05\).

05

Suggest Improvements for the Model

The model may be improved by removing non-significant predictors such as \(X_4\), or by adding new variables that are more predictive of salary. Checking multicollinearity issues among predictors could also be helpful.

06

Analyze the Change with Age Variable

Adding age (\(X_6\)) shows a negative coefficient due to collinearity. Age may be confounded with other variables like experience or education, causing distortions when other factors are kept constant.

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Key Concepts

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

Linear Regression

Linear regression is a statistical approach used to model the relationship between a dependent variable and one or more independent variables. It helps in predicting the outcome, which in this case is the yearly salary of secretaries. The process results in a regression equation that matches the relationship between variables. The equation formed is:\[ \text{Salary} = 9.788 + 0.110X_1 + 0.053X_2 + 0.071X_3 + 0.004X_4 + 0.065X_5 \]This formula allows us to forecast salary based on input values for the given variables. Each coefficient in the equation represents the expected change in the dependent variable when the corresponding independent variable increases by one unit, keeping other variables constant. By using linear regression, we can interpret and rely on large datasets more effectively. It is especially beneficial when assessing the linear relationship across multiple factors at once.

Predictor Variables

Predictor variables, also known as independent variables, are the factors that might influence the outcome or dependent variable, such as salary in this scenario. In our exercise, the predictor variables are:

• \(X_1\) : Months of service
• \(X_2\) : Years of education
• \(X_3\) : Score on standardized test
• \(X_4\) : Words per minute typing speed
• \(X_5\) : Ability to take dictation in words per minute

Each of these variables is expected to have some level of influence on salary. By analyzing these variables within a regression model, we gain insights into how changes in these predictors could potentially increase or decrease salaries. Incorporating multiple variables allows for a comprehensive view of influence while addressing the complexity of real-world scenarios.

Coefficient Significance

In multiple regression analysis, each predictor variable is assigned a coefficient that quantifies its contribution to the model. Coefficient significance determines whether a variable meaningfully predicts the dependent variable. The t-value calculated for each coefficient provides insight into its significance.For instance, the exercise highlights the absence of significance for the typing speed variable \(X_4\) as it had a very low t-value of 0.013, indicating it does not meaningfully contribute to the salary prediction in this scenario. To establish if a coefficient is significant, a t-test is performed, leading us to compare the t-value against a critical value derived from a statistical distribution. A high p-value, typically greater than 0.05, confirms the nonsignificance of a predictor. Understanding coefficient significance can help in refining models by identifying which predictors to retain or eliminate.

Model Improvement

Improving a regression model involves several strategies, primarily aimed at enhancing the accuracy and relevance of predictions. One critical step is to assess the significance of each predictor, as nonsignificant variables may dilute model accuracy. For example, based on the given results, removing \(X_4\) (typing speed) could improve the model as it does not significantly impact salary.Additionally, exploring potential interactions between existing predictors can reveal more complex relationships. Checking for multicollinearity, a situation where predictor variables are highly correlated, is essential because it can distort variable effects. Incorporating new, relevant variables that better explain variations in the outcome, such as age when properly adjusted for multicollinearity, might also enhance model predictions. Regularly evaluating and iterating on the model ensures it remains robust and reflective of real-world data and relationships.