#AP Statistics: Hypothesis Testing for Regression Slope 🚀

Hey there, future AP Stats superstar! Let's nail down hypothesis testing for regression slopes. You've got this! 💪

#T-Distribution for Slope Estimation

• This is because the slope estimate is a linear combination of the observations, and it's a result of the Central Limit Theorem. Think of it as the sampling distribution of the slope.

#Calculating the T-Score

• The t-score helps us understand how far our sample slope is from the null slope (usually zero). It's our way of measuring the evidence against the null hypothesis. ⛰️

• Formula:

$$t = \frac{b - \beta}{SE_{b}}$$

Where:

- *b* is the sample slope.
- *β* is the hypothesized population slope (usually 0).
- *SE<sub>b</sub>* is the standard error of the sample slope.

• Degrees of Freedom: n - 2 (sample size minus the number of parameters estimated, which is 2 for a slope and intercept)

#Understanding the P-Value

• The p-value is the probability of observing a sample slope as extreme as (or more extreme than) the one we got, assuming the null hypothesis is true. 🚗

• A small p-value means our sample is unlikely if the null is true, giving us evidence to reject it.

• Example:

• If your t-score is 2.0791 with 21 degrees of freedom, and you're doing a two-tailed test, the p-value might be around 0.05. This means there's about a 5% chance of seeing a sample like yours if the true slope was 0. - If our significance level is also 0.05, we would reject the null hypothesis.

*Image: Visual represe...