91Ó°ÊÓ

Given the linear regression equation $$ x_{1}=1.6+3.5 x_{2}-7.9 x_{3}+2.0 x_{4} $$ (a) Which variable is the response variable? Which variables are the explanatory variables? (b) Which number is the constant term? List the coefficients with their corresponding explanatory variables. (c) If \(x_{2}=2, x_{3}=1\), and \(x_{4}=5\), what is the predicted value for \(x_{1} ?\) (d) Explain how each coefficient can be thought of as a "slope" under certain conditions. Suppose \(x_{3}\) and \(x_{4}\) were held at fixed but arbitrary values and \(x_{2}\) was increased by 1 unit. What would be the corresponding change in \(x_{1}\) ? Suppose \(x_{2}\) increased by 2 units. What would be the expected change in \(x_{1} ?\) Suppose \(x_{2}\) decreased by 4 units. What would be the expected change in \(x_{1} ?\) (e) Suppose that \(n=12\) data points were used to construct the given regression equation and that the standard error for the coefficient of \(x_{2}\) is \(0.419\). Construct a \(90 \%\) confidence interval for the coefficient of \(x_{2}\). (f) Using the information of part (e) and level of significance \(5 \%\), test the claim that the coefficient of \(x_{2}\) is different from zero. Explain how the conclusion of this test would affect the regression equation.

Step by step solution

01

Identify the Response and Explanatory Variables

In a regression equation, the response variable is the variable we are trying to predict, and it is typically on the left-hand side of the equation. From the given regression equation \(x_{1}=1.6+3.5x_{2}-7.9x_{3}+2.0x_{4}\), the response variable is \(x_{1}\). The explanatory variables, which are the factors used to predict the response, are \(x_{2}\), \(x_{3}\), and \(x_{4}\).

02

Find the Constant Term and Coefficients

The constant term, also known as the intercept, is the standalone number in the regression equation that is not multiplied by any variable. In \(x_{1}=1.6+3.5x_{2}-7.9x_{3}+2.0x_{4}\), the constant term is \(1.6\). The coefficients are the numbers multiplying each explanatory variable: \(3.5\) for \(x_{2}\), \(-7.9\) for \(x_{3}\), and \(2.0\) for \(x_{4}\).

03

Calculate the Predicted Value of x_1

Substitute the given values of \(x_{2}=2\), \(x_{3}=1\), and \(x_{4}=5\) into the equation \(x_{1}=1.6+3.5x_{2}-7.9x_{3}+2.0x_{4}\). This gives:\[ x_{1} = 1.6 + 3.5(2) - 7.9(1) + 2.0(5) \]\[ x_{1} = 1.6 + 7.0 - 7.9 + 10.0 \]\[ x_{1} = 10.7 \]

04

Interpret Coefficients as Slopes

Each coefficient represents the change in the response variable given a one-unit change in the corresponding explanatory variable, assuming other variables remain constant. If \(x_{3}\) and \(x_{4}\) are held constant, increasing \(x_{2}\) by 1 unit increases \(x_{1}\) by \(3.5\) units. Similarly, increasing \(x_{2}\) by 2 units increases \(x_{1}\) by \(3.5 \times 2 = 7.0\) units. Decreasing \(x_{2}\) by 4 units decreases \(x_{1}\) by \(3.5 \times 4 = 14.0\) units.

05

Construct a 90% Confidence Interval for Coefficient of x_2

First calculate the critical value for \(t\) with \(n-2 = 10\) degrees of freedom at a 90% confidence level, which is approximately \(t_{0.95, 10} \approx 1.812\). The confidence interval for \(3.5\) is calculated by:\((3.5 - t \times SE, 3.5 + t \times SE) = (3.5 - 1.812 \times 0.419, 3.5 + 1.812 \times 0.419)\)This results in the confidence interval: \((2.24, 4.76)\).

06

Test the Hypothesis for the Coefficient of x_2

The null hypothesis \(H_0\) is that the coefficient of \(x_{2}\) is zero, while the alternative hypothesis \(H_1\) is that it is not zero. Calculate the test statistic:\[ t = \frac{3.5 - 0}{0.419} = 8.35 \]With a significance level of 5%, the critical value is \(t_{0.975, 10} \approx 2.228\). Since \(8.35 > 2.228\), we reject \(H_0\) and conclude the coefficient is significantly different from zero, indicating \(x_{2}\) has a meaningful impact on \(x_{1}\).

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Key Concepts

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

Response and Explanatory Variables

In a regression equation, understanding the role of response and explanatory variables is key. The response variable, often located on the left side of the equation, is what you want to predict or explain. In our example, the response variable is \(x_{1}\). It represents the outcome or the main focus of the study. On the other hand, the explanatory variables are used to explain variations in the response variable. Here, the explanatory variables are \(x_{2}, x_{3},\) and \(x_{4}\). They provide the necessary input data for the prediction model. Each one has a unique contribution to how the final outcome is calculated. Understanding these roles helps set the stage for more in-depth regression analysis.

Identifying these variables correctly is a fundamental skill when analyzing data with linear regression. It establishes a clear framework for how different factors impact your outcome.

Confidence Interval

A confidence interval gives us a range within which we expect the true value of a parameter, like a regression coefficient, to lie. It's derived using the sample data and provides valuable insight into precision and reliability. For instance, when we constructed the 90% confidence interval for the coefficient of \(x_{2}\), we calculated: - Critical \(t\)-value for \(n-2=10\) degrees of freedom
- Combined it with the standard error to find the range
This resulted in the interval \([2.24, 4.76]\), suggesting that with 90% certainty, the coefficient of \(x_{2}\) lies within this range.

Confidence intervals are essential for understanding the reliability of your estimates. A narrow range indicates high precision, whereas a wider range suggests less certainty. This concept is crucial when making predictions about data behavior.

Hypothesis Testing

Hypothesis testing allows us to make informed decisions based on data. In the context of linear regression, we often test whether regression coefficients are different from zero. For the coefficient of \(x_{2}\), we set up the null hypothesis \(H_{0}: \beta_{2} = 0\) (it has no effect), versus the alternative \(H_{1}: \beta_{2} eq 0\).
To test this, we calculate the \(t\)-statistic which is a ratio of the estimated coefficient to its standard error: \[ t = \frac{3.5}{0.419} = 8.35 \]
We then compare this statistic to a critical value determined by our chosen significance level (5% in this case). Here, since \(8.35\) is greater than the critical value \(2.228\), we reject the null hypothesis. This means \(x_{2}\) significantly contributes to the variability in \(x_{1}\).

Hypothesis testing offers a structured way to assess the impact of predictors, ensuring that conclusions are statistically valid.

Regression Coefficients

Regression coefficients are the numerical values that multiply the explanatory variables in a regression equation. They represent the change in the response variable with each one-unit change in the explanatory variable while holding others constant. For the model \(x_{1} = 1.6 + 3.5x_{2} - 7.9x_{3} + 2.0x_{4}\), each coefficient tells a distinct story:

• **\(3.5\)** for \(x_{2}\): Increasing \(x_{2}\) by one unit leads to a 3.5-unit increase in \(x_{1}\).
• **\(-7.9\)** for \(x_{3}\): Increasing \(x_{3}\) by one unit results in a 7.9-unit decrease in \(x_{1}\)
• **\(2.0\)** for \(x_{4}\): A one-unit increase in \(x_{4}\) raises \(x_{1}\) by 2.0 units

Each coefficient expresses a directional change, vital for interpreting how different predictors impact the outcome. Therefore, understanding these coefficients as 'slopes' enables us to assess how changes in predictors alter the response variable.