91Ó°ÊÓ

In regression analysis, understanding degrees of freedom (df) is crucial. Simply put, degrees of freedom refers to the number of independent values or quantities that we can assign to a statistical analysis.

In the context of simple linear regression, the degrees of freedom specifically related to inference about the slope \( \beta \) is calculated using the formula:

• \( df = n - 2 \)

where \( n \) stands for the total number of observations. For example, if we have 32 observations, like in our exercise, the degrees of freedom is \( df = 32 - 2 = 30 \).

This means out of 32 observations, 30 are free to vary independently. The subtraction by two accounts for the two estimated parameters: the slope and the intercept in the regression line. Understanding this concept is foundational because it affects subsequent calculations, such as hypothesis tests and confidence intervals.

A t-Test in regression is used to determine if there is a significant relationship between variables. It is especially useful in testing hypotheses about the regression coefficients.

In our case, the null hypothesis \( H_0 : \beta = 0 \) suggests that there is no relationship between the independent variable and the dependent variable. We use the t-distribution to examine if we should reject this hypothesis.

To achieve this, we calculate the t-statistic using the critical values for our desired significance level. For example, a significance level \( \alpha = 0.10 \) splits into two tails with 0.05 each in a two-tailed test. With \( df = 30 \), the critical t-values found (through a t-table or software) are approximately \( t = \pm 1.697 \).

This means that if the calculated t-statistic is less than \(-1.697\) or more than \(1.697\), we reject the null hypothesis. These boundary values play an essential role in decision-making regarding the relationships in data.

Creating a confidence interval in regression gives us a range of plausible values for the regression slope \( \beta \). For a 90% confidence level, we express the degree of certainty that this range includes the true value of \( \beta \).

The general form of a confidence interval for the slope is:

• \( \text{Slope estimate} \pm (t \times \text{Standard Error}) \)

where \( t \) is the critical value correlated with the desired confidence level and degrees of freedom.

In our exercise with 30 degrees of freedom, the critical t-value for the middle 90% of the t-distribution is about \( 1.697 \). This t-score is pivotal for determining the margin of error around our slope estimate.

By multiplying this t-score with the standard error of the estimate, we get the margin of error, providing insightful information about the precision and stability of our regression model's slope.