Hypothesis Tests for Slopes of Regression Lines

• Population vs. Sample Slopes: The population regression line is represented by the equation $\hat{y} = \alpha + \beta x$, where $\beta$ is the true, unknown slope. We estimate this using a sample regression line $\hat{y} = a + bx$, where $b$ is the calculated sample slope.

• The Null Hypothesis ($H_0$): This hypothesis typically assumes that there is no linear relationship between the variables, stated as $H_0: \beta = 0$. However, it can also test if the slope equals a specific non-zero value $\beta_0$.

• The Alternative Hypothesis ($H_a$): This reflects the researcher's suspicion, such as $H_a: \beta \neq 0$ (two-tailed), $H_a: \beta > 0$ (positive relationship), or $H_a: \beta < 0$ (negative relationship).

• Standard Error of the Slope ($SE_b$): This value measures the estimated standard deviation of the sampling distribution of $b$. It quantifies how much the sample slope is expected to vary from one random sample to another.

• Sampling Distribution of $b$: If the conditions for inference are met, the sampling distribution of the sample slope $b$ is approximately normal. Its mean is the true population slope $\beta$, making $b$ an unbiased estimator.

• The t-Distribution: Because we must estimate the population standard deviation using the sample standard deviation of residuals ($s$), we use the $t$-distribution rather than the normal distribution. This accounts for the extra variability introduced by using the sample estimate $SE_b$.

• Degrees of Freedom: For a regression slope test, the degrees of freedom are calculated as $df = n - 2$. This is because we estimate two parameters from the data (the intercept $a$ and the slope $b$) before calculating the variability of the residuals.

Step-by-Step Procedure

• 1. State Hypotheses: Define the population parameter $\beta$ in context and state $H_0$ (usually $\beta = 0$) and $H_a$ (e.g., $\beta \neq 0$).
• 2. Check Conditions (LINER): Verify Linearity (scatter plot/residual plot), Independence (10% rule), Normality of residuals (histogram/Q-Q plot), Equal Variance (residual plot spread), and Randomness (random sampling/assignment).
• 3. Calculate the Test Statistic: Use the formula for the $t$-statistic:

$t = \frac{b - \beta_0}{SE_b}$

• 4. Determine the P-value: Use the $t$-distribution with $df = n - 2$ to find the probability of observing a test statistic as extreme as the one calculated, assuming $H_0$ is true.
• 5. Make a Conclusion: Compare the P-value to the significance level $\alpha$. If $P < \alpha$, reject $H_0$ and conclude there is evidence of a linear relationship.

• Slope vs. Correlation: While testing $H_0: \beta = 0$ is mathematically equivalent to testing if the population correlation $\rho = 0$, the slope test provides information about the rate of change, whereas correlation only measures the strength and direction of the linear bond.

• Reading Computer Output: Exams often provide a regression table. Always look for the row corresponding to the explanatory variable (not the 'Constant' or 'Intercept' row) to find the slope $b$, its standard error $SE_b$, the $t$-statistic, and the $p$-value.

• Defining Parameters: When writing hypotheses, always define $\beta$ as the 'true slope of the population regression line relating [Response Variable] to [Explanatory Variable]'. Failure to include the population context often results in lost marks.

• Two-Tailed vs. One-Tailed: If the question asks if there is a 'relationship' or 'change', use a two-tailed test ($H_a: \beta \neq 0$). If it asks if the relationship is 'positive' or 'negative', use a one-tailed test.

• Sanity Check: Ensure your $t$-statistic matches the sign of your sample slope $b$. If $b$ is negative, your $t$-value must also be negative (assuming $\beta_0 = 0$).