Inference for Regression Slopes¶

Sampling Distributions for Sample Slopes¶

Population Least-Squares Regression Line¶

Theoretical, best-fitting straight line that described the true linear relationship between two variables for an entire population

$\hat{y} = \alpha + \beta x$

• $\hat{y}$: predicted response
• $\alpha$: population $\hat{y}$-intercept
• $\beta$: population slope

Least-squares indicate that the sum of squared residuals are minimized

Sample Least-Squares Regression Line¶

The line of best fit for a set of data points that minimize the sum of squared residuals

$\hat{y} = a+bx$

• Different samples produce different sample least-square regression lines. These all have different sample sloped $b$. This means $b$ has a sampling distribution.

Mean and Standard Deviation of Sampling Distribution for Sample Slopes¶

For the sample slopes $b$:

• $\mu = \beta$
• $\sigma = \dfrac{\sigma}{\sigma_x \sqrt{n}}$
  • $n$: sample size
  • $\sigma$: $\sigma$ of all population residuals
  • $\sigma_x = \sqrt{\dfrac{\sum (x_i - \overline{x})^2}{n}}$: $\sigma$ of the $x$-values only
• $\dfrac{\sigma}{\sigma_x \sqrt{n}}$ is unknown in practice and must be estimated from the sample (standard error)
  • $s_b = \dfrac{s}{s_x \sqrt{n-1}}$: the standard error of sample slopes
  • $s = \sqrt{\dfrac{\sum (y_i - \hat{y_i})^2}{n-2}}$
    • Divided by $n-2$ as two parameters, $\alpha$ and $\beta$, are estimated
  • $s_x = \sqrt{\dfrac{\sum (x_i - \overline{x})^2}{n-1}}$

Sampling Distributions for Standardized Sample Slopes¶

• $t = \dfrac{b - \beta}{s_b}$: standardized sample slope
• $t$-distribution with $dof=n-2$

Hypothesis Tests for Slopes of Regression Lines¶

Conditions for A t-Test for A Slope¶

• Relationship between $x$ and $y$ is linear
• $\sigma_y$ cannot vary with $x$
• Residuals are independent
  • Data is collected by random sampling/assignment
  • If sampling without replacement, $n < 0.1N$
• For a given value of $x$, $y$-values follow an approximate normal distribution
• If $n<30$, $y$-values distribution have no strong skew and no outliers

Computer Output Table¶