Unit 9: Inference for Quantitative Data: Slopes

Least-squares regression line

The line that minimizes the sum of squared residuals and summarizes the linear relationship between an explanatory variable x and a response variable y in a sample.

Explanatory variable (x)

The variable used to explain or predict changes in the response; typically placed on the horizontal axis in regression.

Response variable (y)

The outcome variable being predicted or explained by x; typically placed on the vertical axis in regression.

Sample slope ($b_1$)

The slope of the least-squares regression line from a sample; estimates how the predicted y changes for a 1-unit increase in x.

Population slope ($\beta_1$ or $\beta$)

The true slope parameter in the population regression model; represents how the population mean response changes with x.

Sample intercept ($b_0$)

The intercept of the sample regression line; the predicted value of y when x = 0 (may not be meaningful if x=0 is outside the data’s context).

Population intercept ($\beta_0$ or $\alpha$)

The true intercept parameter in the population regression line; the population mean response when x = 0.

Population regression line

The population model for the mean response: $\mu_y = \beta_0 + \beta_1 x$ (equivalently $\mu_y = \alpha + \beta x$).

Sample regression line

The fitted line from sample data: $\hat{y} = b_0 + b_1 x$.

Slope inference

Using sample regression results to draw conclusions about the population slope parameter $\beta_1$ (e.g., testing $\beta_1=0$ or estimating $\beta_1$ with a confidence interval).

Null hypothesis for slope

A statement about the population slope, most commonly $H_0: \beta_1 = 0$ (no linear relationship in the population).

Alternative hypothesis for slope

The competing claim about the population slope, such as $H_a: \beta_1 \ne 0$, $H_a: \beta_1 > 0$, or $H_a: \beta_1 < 0$ (chosen based on context).

Slope of 0 (flat population line)

A population slope of 0 means the population mean response does not change as x changes; no linear relationship is supported.

Linear relationship (regression context)

A relationship where the mean of y changes approximately linearly with x; required for valid linear regression slope inference.

Correlation–slope test equivalence (simple linear regression)

With one explanatory variable, testing for a linear relationship via regression is equivalent to testing whether the population correlation is 0, but regression questions should be phrased in slope terms.

Association

A relationship between variables where changes in one are related to changes in the other; regression slope inference primarily supports association in a population.

Causation

A cause-and-effect relationship; can be concluded from a significant slope only when the study design is a randomized experiment (within its scope).

Randomized experiment

A study where treatments are randomly assigned; supports cause-and-effect conclusions when conditions are met and results are significant.

Observational study

A study where variables are observed without random assignment; a significant slope supports association only, not causation.

Lurking variable

An unmeasured variable that may influence both x and y, potentially explaining an observed association in an observational study.

Sampling distribution of the slope

The distribution of sample slopes $b_1$ that would be obtained from repeated samples (or repetitions of an experiment) from the same population.

Mean of the sampling distribution ($\mu_b$)

The average of all possible sample slopes; under conditions, this mean equals the true population slope $\beta_1$.

Standard deviation of the sampling distribution ($\sigma_b$)

The true spread of sample slopes $b_1$ across repeated samples; typically unknown in practice.

Standard error of the slope ($SE_{b_1}$)

An estimate of the standard deviation of the sampling distribution of $b_1$, computed from sample data; used for t inference about the slope.

t distribution (for slope inference)

The distribution used for slope tests/intervals because the true variability is unknown and must be estimated from sample residuals.

Residual

The vertical difference between an observed y value and its predicted value: $y - \hat{y}$.

Residual standard deviation (s)

A measure of typical prediction error around the fitted line: $s = \sqrt{\Sigma(y-\hat{y})^2 / (n-2)}$.

Degrees of freedom ($df = n - 2$)

The df used in regression slope t procedures; $n-2$ because both slope and intercept are estimated from the data.

t statistic for slope

Standardizes the difference between the sample slope and hypothesized slope: $t = (b_1 - \beta_1) / SE_{b_1}$ (often with $\beta_1=0$).

p-value (slope test)

The probability, assuming $H_0$ is true, of observing a sample slope (or t statistic) at least as extreme as the one obtained, in the direction(s) of $H_a$.

Significance level ($\alpha$)

The cutoff probability for deciding whether evidence is strong enough to reject $H_0$ (e.g., $\alpha = 0.05$).

Statistical significance

A result is statistically significant if the p-value is less than $\alpha$, indicating evidence against $H_0$ beyond random sampling variation.

Practical importance

Whether an effect is large enough to matter in context; statistical significance does not guarantee practical importance.

Confidence interval for the population slope

A range of plausible values for $\beta_1$, typically computed as $b_1 \pm t^* SE_{b_1}$, and interpreted as change in the population mean response per 1 unit of x.

Critical value ($t^*$)

The t multiplier from the t distribution (with $df = n-2$) that matches the desired confidence level for an interval.

Margin of error (ME)

The amount added/subtracted in a confidence interval: $ME = t^* \times SE_{b_1}$.

Mean response ($\mu_y$)

The population average value of y at a given x; regression inference targets how $\mu_y$ changes with x, not individual outcomes.

Predicted value ($\hat{y}$)

The value of y predicted by the sample regression line for a given x.

Linearity condition

Condition that the relationship between x and the mean of y is approximately linear; checked with a scatterplot and/or residual plot (no curved pattern).

Independence condition

Condition that observations are independent; supported by random sampling or random assignment and by avoiding situations with dependence (e.g., related subjects).

10% condition

When sampling without replacement, independence is plausible if the sample size n is less than 10% of the population.

Time correlation

Dependence across observations collected over time (e.g., daily prices/temperatures) that can violate the independence assumption.

Normality of residuals condition

Condition that residuals are approximately normally distributed around the line; checked with a histogram or normal probability plot of residuals (not y itself).

Equal variance (constant spread) condition

Condition that the variability of residuals is roughly constant across x; checked by looking for no fanning/funneling in a residual plot.

Funnel (fan) pattern

A residual plot pattern where residual spread increases or decreases with x, indicating nonconstant variance (violating equal variance).

Influential point

A data point that strongly affects the fitted line (slope, SE, p-value) and can change conclusions; often associated with extreme x or large residuals.

High leverage point

A point with an extreme x-value compared to the rest of the data that can “pull” the regression line.

Outlier (large residual)

A point with an unusually large vertical deviation from the regression line; can distort regression results, especially if also high leverage.

Extrapolation

Using a regression model to predict for x-values far outside the observed range; predictions and inference are less trustworthy there.

r-squared (coefficient of determination)

The proportion of variation in y explained by the linear model with x; unitless and distinct from interpreting slope or establishing causation.