Slope of a Regression Model (Setup) - AP Statistics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Linearity condition. Scatterplot should show a roughly linear pattern.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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