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Chapter 12: Problem 42

Verify that if each \(x_{i}\) is multiplied by a positive constant \(c\) and each \(y_{i}\) is multiplied by another positive constant \(d\), the \(t\) statistic for testing \(H_{0}: \beta_{1}=0\) versus \(H_{\mathrm{a}}: \beta_{1} \neq 0\) is unchanged in value (the value of \(\hat{\beta}_{1}\) will change, which shows that the magnitude of \(\hat{\beta}_{1}\) is not by itself indicative of model utility).

Short Answer

Expert verified
The t-statistic remains unchanged after multiplying by constants.

Step by step solution

01

Define the t-statistic and relevant formulas

The t-statistic for testing \( H_0: \beta_1 = 0 \) is given by \( t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)} \). Here, \( \hat{\beta}_1 \) is the estimated slope of the regression line, and \( SE(\hat{\beta}_1) \) is the standard error of the slope estimate.
02

Analyze the effect of multiplying by constants

Let \(\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \) under the original data. When \( x_i \) is multiplied by \( c \) and \( y_i \) by \( d \), the new slope \( \hat{\beta}_1' \) is \( \frac{d}{c} \cdot \hat{\beta}_1 \). Therefore, the numeric value of \( \hat{\beta}_1 \) changes, but its relationship to \( t \) requires a closer look.
03

Compute the new standard error

The standard error \( SE(\hat{\beta}_1) = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}} \), where \( s = \sqrt{\frac{1}{n-2} \sum (y_i - \hat{y}_i)^2} \). Under the transformed data, the error \( s' \) becomes \( d \cdot s \), and the total sum of squares for \( x_i \) becomes \( c^2 \cdot \sum (x_i - \bar{x})^2 \). Thus, the new standard error \( SE(\hat{\beta}_1') = \frac{d \cdot s}{c \sqrt{\sum (x_i - \bar{x})^2}} \).
04

Simplify the new t-statistic

Substitute the transformed values into the t-statistic formula: \( t' = \frac{\hat{\beta}_1'}{SE(\hat{\beta}_1')} = \frac{\frac{d}{c} \cdot \hat{\beta}_1}{\frac{d \cdot s}{c \sqrt{\sum (x_i - \bar{x})^2}}} = \frac{\frac{d}{c} \cdot \hat{\beta}_1}{\frac{d}{c} \cdot \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}}} \). This simplifies to \( \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)} \), showing that the t-statistic remains unchanged.

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Key Concepts

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

Regression Analysis
Regression analysis is a statistical tool used to explore the relationship between a dependent variable and one or more independent variables. The main goal of regression is to create an equation that can predict values of the dependent variable based on the independent variables. In simple linear regression, there is typically one dependent and one independent variable. This analysis helps in understanding how the dependent variable changes as the independent variable changes. Key aspects of regression analysis include:
  • Dependent Variable (Y): This is the outcome or the variable we are trying to predict or explain.
  • Independent Variable (X): This is the predictor or variable we are using to make predictions about Y.
  • Slope (\(\beta_1\)): This is the coefficient indicating the rate of change in the dependent variable for every one-unit change in the independent variable.
Regression analysis provides insights into the strength and type of relationship, whether it is positive or negative, and how statistically significant that relationship may be. This makes it a powerful tool for making predictions and decisions based on data.
Hypothesis Testing
Hypothesis testing in the context of regression involves testing assumptions about parameters like the slope of the regression line. The goal is to determine if there is a statistically significant relationship between the independent and dependent variables. For our case, we're testing if the slope (\(\beta_1\)) equals zero.The hypothesis has two parts:
  • Null Hypothesis (\(H_0: \beta_1 = 0\)): Implies that there is no relationship between the variables, meaning the slope is zero.
  • Alternative Hypothesis (\(H_a: \beta_1 eq 0\)): Suggests that a relationship exists, and the slope is different from zero.
To test these hypotheses, we use the t-statistic, which measures how many standard errors the estimated coefficient is from zero. A large t-statistic typically indicates that you can reject the null hypothesis, implying that the relationship is statistically significant. Hypothesis testing provides a framework for making inferences about population parameters based on sample data.
Standard Error
The standard error is a statistical term that measures the average amount by which an estimate deviates from the actual value. It plays a crucial role in regression analysis, particularly in assessing the reliability of estimated coefficients like the slope (\(\hat{\beta}_1\)).In regression, the standard error of a slope coefficient determines the precision of the estimated relationship between variables. It is calculated as:\[SE = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}}\]Where:
  • s: Represents the standard deviation of the residuals, indicating the scatter of data points around the regression line.
  • Sum of Squares: Accounts for variability in the independent variable.
A smaller standard error indicates that the coefficient estimate is more precise. When multiplying data by constants, both \(\hat{\beta}_1\) and the standard error are affected proportionally, yet the overall t-statistic remains unchanged due to these proportional adjustments. Understanding the standard error helps in evaluating the accuracy of our estimates and confidence in the model's predictions.

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Most popular questions from this chapter

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