Introduction

In linear regression analysis, the t-statistic, p-value, and confidence interval are essential tools that help us assess the significance of regression coefficients and draw meaningful inferences from the model. These statistical measures provide insights into the reliability and validity of the estimated coefficients. In this chapter, we will explore the interconnected relationship between the t-statistic, its associated p-value, and the confidence interval.

In linear regression, understanding the relationship between the t-statistic, p-value, and confidence interval is crucial for interpreting the significance of regression coefficients. These concepts allow us to make informed decisions about the impact of explanatory variables on the response variable.

t-Statistic

The t-statistic is a measure of how many standard errors the estimated coefficient is away from the hypothesized population value (usually zero). Mathematically, the t-statistic is calculated as: $$t=\frac{\text { Coefficient Estimate }}{\text { Standard Error of the Estimate }}$$ A higher absolute value of the $t$-statistic indicates that the estimated coefficient is farther from zero, suggesting stronger evidence against the null hypothesis.

p-Value

The p-value associated with the t-statistic is a measure of the probability of observing a t-statistic as extreme as the one calculated, assuming the null hypothesis is true. A small p-value indicates that the observed t-statistic is unlikely to occur by random chance alone, leading us to reject the null hypothesis.

Confidence Interval

The confidence interval is a range of values within which the true population parameter (such as the coefficient) is likely to fall. A common confidence level is 95%, which means that if we were to replicate our analysis many times, about 95% of the resulting confidence intervals would contain the true population parameter.

Relationship between t-Statistic and p-Value

The relationship between these measures is closely connected. If the t-statistic is large (far from zero), the associated p-value will be small, indicating that the coefficient is statistically significant. A small p-value leads to rejecting the null hypothesis in favor of the alternative hypothesis, suggesting that the explanatory variable has a significant effect on the response variable.

Moreover, the confidence interval also helps in interpreting the t-statistic. If the confidence interval for a coefficient does not include zero, it implies that the coefficient is likely to be statistically significant. Conversely, if the interval includes zero, the coefficient’s significance is less certain.

Example: Let’s consider a linear regression model where we’re examining the relationship between years of experience (explanatory variable) and salary (response variable). If the t-statistic for the coefficient of years of experience is 4.23 and the associated p-value is 0.001, we have strong evidence against the null hypothesis. This means that years of experience significantly affects salary. Furthermore, if the 95% confidence interval for the coefficient (2.1, 3.7) does not include zero, it confirms the significance of the coefficient.