Confidence Intervals: Slope of Regression Models - AP Statistics

1. What is the point estimate for the population slope $\beta$ in a regression model?

   $b$ (the least-squares slope). Sample statistic estimates population parameter.

2. Compute $df$ for a slope confidence interval when the sample size is $n=18$.

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

3. Identify the degrees of freedom for slope inference when $n=18$ paired observations are used.

   $df=16$. Use $df = n - 2 = 18 - 2$ for regression slope inference.

4. What is the general form of a $C\%$ confidence interval for the population slope $\beta$?

   $b \pm t^*\,SE_b$. Uses sample slope plus/minus critical value times standard error.

5. What does $s$ represent in $SE_b=\frac{s}{\sqrt{\sum (x_i-\bar{x})^2}}$?

   $s$ is the residual standard deviation. Measures typical size of residuals.

6. Compute the margin of error if $t^*=2.10$ and $SE_b=0.40$.

   $0.84$. Margin of error equals $t^* \times SE_b = 2.10 \times 0.40$.

7. Which condition for a slope CI requires residuals to be approximately Normal for inference?

   Normality of residuals. Required for valid t-distribution inference about the slope.

8. State the formula for the standard error of the slope in simple linear regression.

   $SE_b=\frac{s}{\sqrt{\sum (x_i-\bar{x})^2}}$. Measures variability of slope estimate.

9. Find the $95\%$ CI for $\beta$ if $b=3.2$, $t^*=2.00$, and $SE_b=0.50$.

   $(2.2,\,4.2)$. Calculate $3.2 \pm 2.00(0.50) = 3.2 \pm 1.0$.

10. Identify the condition about residuals needed for a slope confidence interval when using $t$ procedures.

    Residuals are approximately Normal. Required for valid $t$ distribution inference.

11. What is the parameter estimated by the slope confidence interval in simple linear regression?

    $\beta$ (the true population slope). Greek letter beta represents the true slope in the population regression line.

12. What is the meaning of $b$ in the interval $b \pm t^*SE_b$?

    $b$ is the least-squares sample slope. Calculated from sample data using least squares method.

13. Choose the correct interpretation of a CI for slope: what does $95\%$ confidence refer to?

    $95\%$ of intervals from repeats capture $\beta$. Long-run proportion containing true slope.

14. If a $C\%$ CI for $\beta$ contains $0$, what conclusion should you make about a linear association?

    Slope not statistically different from $0$ at level $\alpha$. Cannot reject null hypothesis of no linear relationship.

15. State the formula for the residual standard deviation $s$ in simple linear regression.

    $s=\sqrt{\frac{\sum e_i^2}{n-2}}$. Sum of squared residuals divided by df.

16. Compute the $95\%$ CI for $\beta$ if $b=2.4$, $SE_b=0.5$, and $t^*=2.10$.

    $\left(1.35,\ 3.45\right)$. $2.4 \pm 2.10(0.5) = 2.4 \pm 1.05$.

17. Identify whether there is evidence of a positive linear relationship if the CI for $\beta$ is $(0.2,\ 1.1)$.

    Yes, because the entire interval is $>0$. All plausible values are positive.

18. Which condition for a slope CI is assessed using a residual plot for constant spread?

    Equal variance (homoscedasticity). Residuals should have roughly constant spread across all x values.

19. If a $C\%$ CI for $\beta$ is entirely above $0$, what does that imply about the association?

    Positive linear association; reject $H_0:\beta=0$. All plausible slopes are positive, indicating x and y increase together.

20. Which condition for a slope CI requires that observations are independent (often from random sampling)?

    Independence of observations. Ensures one observation doesn't influence another's error term.

21. What does $s$ represent in the slope standard error formula $SE_b=\frac{s}{\sqrt{\sum (x_i-\bar{x})^2}}$?

    $s=\sqrt{\frac{\sum e_i^2}{n-2}}$ (residual SD). Square root of sum of squared residuals divided by degrees of freedom.

22. What parameter does a confidence interval for the slope in linear regression estimate?

    $\beta$ (the true population slope). CIs estimate population parameters, not sample statistics.

23. Identify whether $\beta=0$ is plausible if the $90\%$ CI for $\beta$ is $(-0.8,\ 1.6)$.

    Yes, because $0$ is in the interval. CI contains 0, so no linear relationship proven.

24. If a $C\%$ CI for $\beta$ is entirely below $0$, what does that imply about the association?

    Negative linear association; reject $H_0:\beta=0$. All plausible slopes are negative, indicating x increases as y decreases.

25. What degrees of freedom are used for a $t$ interval for the slope in simple linear regression?

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

26. Choose the correct units for $\beta$ in a regression of $y$ (dollars) on $x$ (hours).

    dollars per hour. Slope units are always y-units divided by x-units.

27. Which distribution is used to find the critical value $t^*$ for a slope confidence interval?

    $t$ distribution with $df=n-2$. Used when population SD is unknown.

28. What is the margin of error for a $C\%$ confidence interval for slope $\beta$?

    $t^*\,SE_b$. How far CI extends from point estimate.

29. Identify what happens to the width of a slope CI when $n$ increases (all else equal).

    It decreases (interval becomes narrower). Larger $n$ reduces $SE_b$ and $t^*$.

30. Identify the condition about the residuals' spread needed for a slope confidence interval.

    Constant variance (homoscedasticity). Equal spread at all $x$ values.

31. Find the margin of error if $t^*=2.00$ and $SE_b=0.30$ for a slope interval.

    $0.60$. $ME = 2.00 \times 0.30 = 0.60$.

32. Identify the condition about the relationship between $x$ and $y$ needed for a slope confidence interval.

    Linearity of the mean response. Ensures the model form is appropriate.

33. Which condition checks that the relationship between $x$ and $y$ is approximately straight for a slope CI?

    Linearity (scatterplot shows linear trend). Ensures the model $y = \beta_0 + \beta x + \epsilon$ is appropriate.

34. State the general form of a $C\%$ confidence interval for the regression slope $\beta$.

    $b \pm t^*\,SE_b$. Sample slope ± critical value × standard error.

35. Identify the critical value symbol used in a $C\%$ CI for slope $\beta$.

    $t^*=t_{\frac{\alpha}{2},\,n-2}$. Critical value from t-distribution with $n-2$ df and area $\alpha/2$ in each tail.

36. What data-collection condition is required to generalize a slope confidence interval to a population?

    Random sample or random assignment. Ensures results apply beyond the sample.

37. State the degrees of freedom used for a $t$ interval for the regression slope $\beta$.

    $df=n-2$. Subtract 2 from sample size for slope inference.

38. What is the correct interpretation template for a $C\%$ CI for $\beta$ in context?

    $C\%$ confident $\beta$ is between L and U (units of $y$ per $x$). States confidence that true slope falls within calculated bounds.

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