Slope of a Regression Model (Test) - AP Statistics

1. Find the degrees of freedom for a slope test when the sample size is $n = 18$.

   $df = 16$. $df = 18 - 2 = 16$ for simple linear regression.

2. What does a small $p$-value in a slope test indicate about the slope parameter?

   Evidence that $\beta \ne 0$ (or matches $H_a$). Small $p$ suggests slope differs from hypothesized value.

3. Identify the correct interpretation of $\beta$ in context for a regression slope test.

   Mean change in $y$ for a $1$-unit increase in $x$. Slope measures average $y$ change per unit $x$ change.

4. What value is used for $\beta_0$ when testing for linear association between $x$ and $y$?

   $\beta_0 = 0$. Testing for any association means hypothesized slope is zero.

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

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

6. What sampling distribution is used for the slope test statistic under the null hypothesis?

   $t$ distribution with $df = n - 2$. Test statistic follows $t$ when null is true and conditions are met.

7. What is the typical two-sided alternative hypothesis for a test of slope in linear regression?

   $H_a: \beta \ne 0$. Tests for any linear relationship, positive or negative.

8. What scatterplot feature supports the linearity condition for a regression slope test?

   Approximately linear relationship between $x$ and $y$. Linear model appropriate when pattern is straight.

9. What alternative hypothesis matches the claim that $y$ increases as $x$ increases?

   $H_a: \beta > 0$. Positive slope means $y$ increases with $x$.

10. What value of $\beta_0$ is used in the test statistic when testing for no linear relationship?

    $\beta_0 = 0$. Testing for no relationship means the hypothesized slope is zero.

11. What degrees of freedom are used for the $t$ test of slope in simple linear regression?

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

12. Identify the correct conclusion when $p=0.12$ and $\alpha=0.05$ for $H_a: \beta > 0$.

    Fail to reject $H_0$; insufficient evidence that $\beta>0$. Since $0.12 > 0.05$, fail to reject null hypothesis.

13. What is the correct one-sided alternative hypothesis for a negative slope test?

    $H_a: \beta < 0$. Tests if slope is negative (downward trend).

14. What conclusion wording is correct regarding causation after a significant slope test?

    Association only; do not claim causation. Regression shows association, not cause-and-effect.

15. Identify the correct conclusion when $p=0.03$ and $\alpha=0.05$ for $H_a: \beta \ne 0$.

    Reject $H_0$; conclude evidence of nonzero slope. Since $0.03 < 0.05$, reject null hypothesis.

16. What are the parameter and null hypothesis for a test of regression slope?

    Parameter: $\beta$; $H_0: \beta = 0$. Tests whether the true slope $\beta$ equals zero (no linear relationship).

17. Identify the correct decision at $\alpha = 0.05$ if the slope test $p$-value is $0.03$.

    Reject $H_0$. $p = 0.03 < 0.05$, so reject at 5% significance level.

18. What does it mean, in context, to reject $H_0: \beta = 0$ in a slope test?

    Evidence of a linear relationship between $x$ and $y$. Rejecting means the data supports a non-zero slope.

19. What are the degrees of freedom for the $t$ test of the slope in simple linear regression?

    $df = n - 2$. Lose 2 df: one for estimating slope, one for intercept.

20. State the test statistic formula for testing the population slope in simple linear regression.

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

21. Which quantity is the standard error of the estimated slope used in the slope test?

    $SE_b$. Measures variability in the slope estimate.

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

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

23. What is the typical alternative hypothesis for testing whether a linear relationship exists?

    $H_a: \beta \ne 0$. Two-sided test checks if slope differs from zero in either direction.

24. What is the correct one-sided alternative hypothesis for a positive slope test?

    $H_a: \beta > 0$. Tests if slope is positive (upward trend).

25. State the test statistic formula for testing the slope in simple linear regression.

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

26. What alternative hypothesis matches the claim that $y$ decreases as $x$ increases?

    $H_a: \beta < 0$. Negative slope means $y$ decreases with $x$.

27. Identify the correct decision at $\alpha = 0.01$ if the slope test $p$-value is $0.03$.

    Fail to reject $H_0$. $p = 0.03 > 0.01$, so fail to reject at 1% significance level.

28. What is the condition on residuals that supports the Normality requirement for a slope test?

    Residuals are approximately Normal. Normal residuals support $t$ distribution assumption.

29. What plot is used to assess the linear and equal-variance conditions for regression inference?

    Residual plot. Shows residuals vs. fitted values to check assumptions.

30. What does it mean, in context, to fail to reject $H_0: \beta = 0$ in a slope test?

    Insufficient evidence of a linear relationship. Cannot conclude a linear relationship exists.

31. Identify the parameter being tested when you perform a regression slope test.

    Population slope $\beta$. Testing the true slope of the regression line.

32. Find the degrees of freedom for a slope test when the sample size is $n=18$.

    $df = 16$. Apply formula: $df = 18-2 = 16$.

33. Which sampling distribution is used for the slope test statistic when conditions are met?

    $t$ distribution with $df = n-2$. Slope test statistic follows $t$ when conditions are met.

34. Identify the four inference conditions for a regression slope test in one-variable regression.

    Linear, Independent, Normal, Equal variance. LINE conditions ensure valid inference for regression.

35. What is the decision rule using significance level $\alpha$ for a slope test?

    Reject $H_0$ if $p\text{-value} < \alpha$. Standard hypothesis test decision rule.

36. Choose the correct two-sided $p$-value expression for a slope test given observed $t$ and $df$.

    $p = 2P(T_{df} \ge |t|)$. Two-sided test doubles the one-tail probability beyond $|t|$.

37. What data-collection condition supports independence for a regression slope test?

    Observations are independent (random sample/assignment). Independence ensures valid probability calculations.

38. What residual plot feature supports the equal variance condition for a regression slope test?

    Roughly constant spread of residuals across $x$. Equal variance ensures consistent standard error.

39. What is the correct null hypothesis for a test of linear association in a regression model?

    $H_0: \beta = 0$. Tests no linear relationship between variables.

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