Significance Test for the Slope

Full AP Regression Inference: Test and Interval

By the end of this lesson, you will be able to execute a complete four-step regression inference procedure from computer output, including both the hypothesis test and confidence interval for the slope.

The Full Regression Inference Framework

When given computer regression output on the AP exam, you need to be able to:

1. State hypotheses and check conditions (LINER)
2. Compute or read the test statistic and p-value
3. Construct a confidence interval for β
4. Write complete, contextual conclusions

Complete Example: Sea Level Rise

Predictor   Coef    SE Coef     T       P
Constant    1.2     0.84      1.43    0.170
Year        3.4     0.41      8.29   <0.001

S = 2.13   R-Sq = 79.3%   n = 20

A climate scientist records annual sea level measurements over 20 years (x = year since 2000, y = sea level rise in mm).

Computer output:

Step 1: State

"We will test $H_0: \beta = 0$ vs. $H_a: \beta > 0$ at the α = 0.05 level, where β is the true slope of the linear relationship between year and sea level rise."

Step 2: Check Conditions (LINER)

Assume we've been told:

• Linear: Scatterplot is approximately linear; residual plot shows random scatter ✓
• Independent: Measurements in different years are independent ✓
• Normal: Histogram of residuals is approximately normal ✓
• Equal variance: Residual plot shows constant spread ✓
• Random: Data collected by consistent measurement method ✓

Step 3: Do the Test

From output: t = 8.29, p < 0.001 (one-sided)

Step 4: Conclude the Test

"Since p < 0.001 < α = 0.05, we reject $H_0$. There is convincing evidence of a positive linear relationship between year and sea level rise — sea levels are rising over time."

Confidence Interval for β

df = n − 2 = 20 − 2 = 18; $t^$ for 95% CI with df = 18: $t^ = 2.101$

$$3.4 \pm 2.101(0.41) = 3.4 \pm 0.861 = (2.54, , 4.26)$$

Interpretation: "We are 95% confident that, on average, sea levels rise by between 2.54 and 4.26 mm per year."