AP Statistics Flashcards: Slope Of A Regression Model Setup

What this deck covers

This deck focuses on Slope Of A Regression Model Setup, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What null hypothesis is most common for a test of a regression slope?

Answer: $H_0: \beta = 0$. Tests whether there's no linear relationship between variables.

Flashcard 2: What is the usual null hypothesis for a test of slope in linear regression?

Answer: $H_0: \beta = 0$. Tests whether there's no linear relationship between variables.

Flashcard 3: Which condition checks that residuals are approximately normally distributed for inference?

Answer: Normality of residuals. Required for valid $t$ distribution and p-values.

Flashcard 4: What are the degrees of freedom for a slope test when $n=18$ data pairs are used?

Answer: $df = 16$. Calculate as $n-2 = 18-2 = 16$.

Flashcard 5: Compute the test statistic when $b = 2.4$, $SE_b = 0.6$, and $H_0: \beta = 0$.

Answer: $t = 4$. Substitute: $t = \frac{2.4 - 0}{0.6} = 4$.

Flashcard 6: Which condition checks that residuals have constant spread across $x$ values?

Answer: Equal variance (homoscedasticity). Residual plot should show consistent vertical spread.

Flashcard 7: What alternative hypothesis matches testing for a positive linear association?

Answer: $H_a: \beta > 0$. One-sided test for when $y$ increases as $x$ increases.

Flashcard 8: What parameter is tested when you test the slope in a linear regression model?

Answer: The population slope $\beta$. The sample slope $b$ estimates the population slope $\beta$.

Flashcard 9: What alternative hypothesis matches the claim "as $x$ increases, $y$ tends to increase"?

Answer: $H_a: \beta > 0$. One-sided test for a positive linear relationship.

Flashcard 10: Identify the correct conclusion format when $p < \alpha$ in a slope test.

Answer: Reject $H_0$; evidence supports $H_a$ about $\beta$. Small $p$-value provides evidence against null hypothesis.

Flashcard 11: What is the sample statistic used to estimate the slope parameter $\beta$?

Answer: The least-squares slope $b$. Sample slope $b$ estimates the population slope $\beta$.

Flashcard 12: What value of $\beta_0$ is used in the test statistic for the common null $H_0: \beta=0$?

Answer: $\beta_0 = 0$. Under the null hypothesis, the population slope equals zero.

Flashcard 13: Which sampling distribution is used for the slope test statistic when assumptions hold?

Answer: A $t$ distribution with $df = n - 2$. The $t$ distribution accounts for estimating variance from the sample.

Flashcard 14: What distribution is used for the slope test statistic when conditions are met?

Answer: A $t$ distribution. The standardized slope follows $t$ when regression conditions hold.

Flashcard 15: Identify the correct conclusion format when $p \ge \alpha$ in a slope test.

Answer: Fail to reject $H_0$; insufficient evidence for $H_a$. Large $p$-value means data consistent with null hypothesis.

Flashcard 16: What alternative hypothesis matches testing for a negative linear association?

Answer: $H_a: \beta < 0$. One-sided test for when $y$ decreases as $x$ increases.

Flashcard 17: Compute the test statistic if $b=2.4$, $SE_b=0.6$, and $H_0: \beta=0$.

Answer: $t = 4$. Calculate $t = \frac{2.4-0}{0.6} = 4$.

Flashcard 18: What is the parameter tested when you test the slope in a linear regression model?

Answer: The population slope $\beta$. We test the true population slope, not the sample slope.

Flashcard 19: State the test statistic formula for testing a regression slope using sample slope $b$.

Answer: $t = \frac{b - \beta_0}{SE_b}$. Standardizes the difference between sample and hypothesized slope.

Flashcard 20: Identify the correct response variable and explanatory variable symbols in regression.

Answer: Response $y$, explanatory $x$. $y$ is predicted by $x$ in regression notation.

Flashcard 21: Find the degrees of freedom for a slope test when $n = 18$ data pairs are used.

Answer: $df = 16$. Apply $df = n - 2 = 18 - 2 = 16$.

Flashcard 22: What are the degrees of freedom for the $t$ test of a regression slope with sample size $n$?

Answer: $df = n - 2$. Loses 2 degrees: one for slope, one for intercept.

Flashcard 23: Which assumption is checked by a histogram or normal probability plot of residuals?

Answer: Nearly Normal residuals. Normality of residuals validates $t$ distribution inference.

Flashcard 24: Which condition checks that the relationship between $x$ and $y$ is approximately linear?

Answer: Linearity condition. Scatterplot should show a roughly linear pattern.

Flashcard 25: In the common slope test, what value is used for $\beta_0$ in $t = \frac{b-\beta_0}{SE_b}$?

Answer: $\beta_0 = 0$. Testing for no linear relationship means $\beta_0 = 0$.

Flashcard 26: Which condition addresses how the data were collected to justify a slope inference?

Answer: Independence from random sampling or random assignment. Random selection ensures observations are independent.

Flashcard 27: What alternative hypothesis matches the claim "there is a linear relationship"?

Answer: $H_a: \beta \ne 0$. Two-sided test for any linear relationship, positive or negative.

Flashcard 28: Find and correct the parameter error: Using $H_0: b=0$ to test for linear association.

Answer: Correct: $H_0: \beta = 0$. Must test population parameter $\beta$, not sample statistic $b$.

Flashcard 29: What alternative hypothesis matches a two-sided slope test for linear association?

Answer: $H_a: \beta \ne 0$. Two-sided test checks for any linear relationship, positive or negative.

Flashcard 30: Which assumption is checked by plotting residuals versus $x$ for a slope test?

Answer: Linearity (no curved pattern in residuals). Curved patterns violate the linear model assumption.

Flashcard 31: What is the test statistic form for testing a regression slope?

Answer: $t = \frac{b-\beta_0}{SE_b}$. Standardizes the difference between sample and hypothesized slopes.

Flashcard 32: Identify the correct hypotheses for testing a positive association between $x$ and $y$ via slope.

Answer: $H_0: \beta = 0;\ H_a: \beta > 0$. Tests if slope is positive (right-tailed test).

Flashcard 33: What is the correct wording of the parameter in context for a slope test?

Answer: $\beta$ is the true change in mean $y$ per $1$ unit increase in $x$. Interprets slope as the rate of change in the mean response.

Flashcard 34: What alternative hypothesis matches the claim "as $x$ increases, $y$ tends to decrease"?

Answer: $H_a: \beta < 0$. One-sided test for a negative linear relationship.

Flashcard 35: Which assumption is checked by looking for a roughly equal vertical spread in residual plots?

Answer: Constant variance (homoscedasticity). Equal spread ensures consistent error variance across $x$ values.

Flashcard 36: What are the degrees of freedom for the $t$ test of a regression slope with sample size $n$?

Answer: $df = n-2$. Lose 2 degrees of freedom for estimating intercept and slope.

Flashcard 37: Identify the correct hypotheses for testing any linear association between $x$ and $y$ via slope.

Answer: $H_0: \beta = 0;\ H_a: \beta \ne 0$. Tests if slope differs from zero in either direction.

Flashcard 38: Which condition addresses whether observations are independent for regression inference?

Answer: Independence of observations. Data collection method should ensure no systematic dependencies.