After we use the method of least-squares to calculate regression coefficients (b_{0} and b_{1}) and we validate the LINE assumptions, we next turn to evaluating the regression, specifically the slope, b_{1} and ask two questions:

1. Is it statistically significant?

2. What is the confidence interval for b_{1}?

The first question (we actually covered this after the second question in class), whether b_{1} 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, β_{1} is 0 and using our skills at hypothesis testing to determine whether we should reject that hypothesis.

H_{0}: β_{1} = 0

H_{1}: β_{1} ≠ 0

The t statistic that we use to test the hypothesis is:

t = (b_{1}-β_{1})/S_{b1}

where S_{b1} is the standard error of the slope.

In our case, β_{1} is 0 according to our hypothesis, so t reduces to:

t = b_{1}/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 b_{1}.

We finish our hypothesis testing by comparing the t-score for b_{1} 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, H_{0}, 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 H_{0}.

**Confidence interval for b_{1} will be covered in the next blog post.**

## Tuesday, March 4, 2008

### Lecture 8 - Inferences About the Regression Slope

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