Lecture 8 - Inferences About the Regression Slope

After we use the method of least-squares to calculate regression coefficients (b₀ and b₁) and we validate the LINE assumptions, we next turn to evaluating the regression, specifically the slope, b₁ and ask two questions:
1. Is it statistically significant?
2. What is the confidence interval for b₁?

The first question (we actually covered this after the second question in class), whether b₁ is statistically significant, is determined by asking: Is it any better than a flat horizontal line through the data?

We answer this question by making a hypothesis that the true relationship slope, β₁ is 0 and using our skills at hypothesis testing to determine whether we should reject that hypothesis.

H₀: β₁ = 0
H₁: β₁ ≠ 0

The t statistic that we use to test the hypothesis is:
t = (b₁-β₁)/S_b1
where S_b1 is the standard error of the slope.

In our case, β₁ is 0 according to our hypothesis, so t reduces to:
t = b₁/S_b1

The standard error of the slope, S_b1, is defined as:
S_b1 = S_XY/SQRT(SSX)
where S_XY is the standard error of the estimate.

The standard error of the estimate, S_XY, is defined as:
S_XY = SQRT(SSE/n-2)

So, if we have our calculations of SSX and SSE, we can do the math and find S_b1 and the t-score for b₁.

We finish our hypothesis testing by comparing the t-score for b₁ to t_{α/2, n-2}, where α is our level of significance.
If t is beyond t_{α/2, n-2} (either on the positive or negative end), we conclude that the hypothesis, H₀, must be rejected.
We could also make the conclusion based on the p-value of the t-score. If the p-value is less than α/2, then we reject H₀.

**Confidence interval for b₁ will be covered in the next blog post.**