AP - Statistics:: Inference for Quantitative Data - Slopes

In inference for regression, the TRUE (population) regression line is written as \(\mu_y = \alpha + \beta x\), while the SAMPLE (estimated) regression line is written as \(\hat{y} = a+bx\). What is the relationship between \(b\) (sample slope) and \(\beta\) (true population slope)?

- **A)** \(\beta\) is calculated directly from the sample data
- **B)** \(b\) represents the true population slope, and \(\beta\) is the sample estimate
- **C)** \(b\) and \(\beta\) are always exactly equal for any given sample
- **D)** \(b\) is a SAMPLE STATISTIC used to estimate the unknown, fixed POPULATION PARAMETER \(\beta\), and like any statistic, \(b\) will vary from sample to sample

The SAMPLING DISTRIBUTION of the sample slope \(b\) describes:

- **A)** A distribution that never varies from sample to sample
- **B)** The distribution of residuals in one specific sample
- **C)** How the value of \(b\) would vary across many different random samples taken from the same population, if the regression were repeated over and over
- **D)** The distribution of the raw x and y values in a single sample

For inference about the true slope \(\beta\), what critical conditions (sometimes remembered by the mnemonic 'LINER') must be checked?

- **A)** The correlation must equal exactly 1
- **B)** No conditions are needed for slope inference
- **C)** Linear relationship (between x and y), Independent observations, Normal distribution of residuals (at each x-value), Equal variance of residuals across all x-values (constant spread), and Random sample/assignment
- **D)** Only that the sample size exceeds 1000

Which diagnostic graph is primarily used to check the LINEAR and EQUAL VARIANCE conditions for slope inference?

- **A)** A two-way table
- **B)** A pie chart
- **C)** A stem-and-leaf plot
- **D)** A residual plot (residuals vs. the explanatory variable), checking for the absence of a curved pattern (linearity) and consistent vertical spread across all x-values (equal variance)

Which diagnostic is used to check the NORMALITY condition (that residuals are approximately Normally distributed) for slope inference?

- **A)** A residual plot showing residuals vs. x
- **B)** The correlation coefficient \(r\) alone
- **C)** A histogram or Normal quantile plot of the RESIDUALS (not the raw x or y data), checking for approximate symmetry/Normal shape
- **D)** The sample size alone, with no graph needed

The STANDARD ERROR of the sample slope \(b\), denoted \(SE_b\), measures:

- **A)** The exact value of the true population slope \(\beta\)
- **B)** The y-intercept of the regression line
- **C)** The correlation coefficient
- **D)** The typical amount that the sample slope \(b\) would vary from sample to sample, if repeated samples were taken

The CONFIDENCE INTERVAL for the true slope \(\beta\) has the general form:

- **A)** \(r \pm t^* \times SE_b\)
- **B)** \(a \pm t^* \times SE_b\)
- **C)** \(b \pm z^* \times \sigma\)
- **D)** \(b \pm t^* \times SE_b\)

The DEGREES OF FREEDOM for slope inference procedures (confidence intervals and significance tests) is calculated as:

- **A)** \(df = n-1\)
- **B)** \(df = n+2\)
- **C)** \(df = n-2\)
- **D)** \(df = n\)