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What is the general form of the least-squares regression line equation?
$\hat{y} = a + bx$, where $\hat{y}$ is the predicted value, a is the y-intercept, and b is the slope.
How do you calculate the correlation coefficient (r) from R-squared?
$r = \pm \sqrt{R^2}$. The sign of r matches the sign of the slope.
What is the formula for the test statistic (t) in a t-test for the slope?
$t = \frac{b - 0}{SE_b}$, where b is the sample slope and $SE_b$ is the standard error of the slope.
What is the formula for the confidence interval for the slope (b)?
$b \pm t^*SE_b$, where b is the sample slope, $t^*$ is the critical t-value, and $SE_b$ is the standard error of the slope.
How to calculate degrees of freedom (df) for t-tests and t-intervals for slopes?
$df = n - 2$, where n is the number of data points.
What is the definition of explanatory variable?
The explanatory variable (independent variable) is plotted on the x-axis and explains the patterns seen in a scatterplot; consider it the 'cause'.
What is the definition of response variable?
The response variable (dependent variable) is plotted on the y-axis and responds to the explanatory variable; consider it the 'effect'.
What is the definition of inference?
Inference uses sample data to make predictions or test claims about a population parameter, moving from describing data to making informed decisions.
What is a t-interval for slopes?
A confidence interval used to estimate the true slope of the population regression line, providing a range of plausible values.
What is a t-test for a slope?
A hypothesis test used to determine if there is a significant linear relationship between two variables by testing if the slope is significantly different from zero.
What are the differences between a t-interval for slopes and a t-test for a slope?
T-interval: Estimates the range of plausible values for the true slope. | T-test: Tests a hypothesis about the value of the slope (typically if it is zero).
What are the differences between the null and alternative hypotheses in a t-test for a slope?
Null Hypothesis ($H_0$): Assumes there is no linear relationship (slope = 0). | Alternative Hypothesis ($H_a$): Claims there is a linear relationship (slope ≠ 0, slope > 0, or slope < 0).
What are the differences between correlation (r) and the slope (b) of a regression line?
Correlation (r): Measures the strength and direction of a linear relationship, but is unitless and does not predict values. | Slope (b): Predicts the change in the response variable for a one-unit change in the explanatory variable and has units.
What are the differences between deterministic and predictive language when interpreting regression results?
Deterministic language: Implies a certain outcome (e.g., 'studying one more hour *will* increase the score by 5 points'). | Predictive language: Acknowledges variability and makes predictions (e.g., 'studying one more hour is *predicted* to increase the score by 5 points').
What are the differences between the standard deviation of the residuals (s) and the standard error of the slope ($SE_b$)?
Standard deviation of residuals (s): Measures the typical distance of the observed values from the regression line. | Standard error of the slope ($SE_b$): Measures the variability of the sample slopes around the true population slope.