AP Statistics (AP Stats): Confidence Interval for Slope (b ± t* × SEb)

AP Statistics – Confidence Interval for Slope (b ± t* × SEb)

AP Statistics: Confidence Interval for Slope (b ± t* × SEb) – Exam-Ready Study Guide

What This Is

A confidence interval for the slope (?) estimates the true average change in the response variable (y) for each one-unit increase in the explanatory variable (x). This is essential for determining whether a linear relationship exists between two quantitative variables and quantifying its strength. For example, a real estate analyst might use this interval to estimate how much home prices increase (on average) for each additional square foot of living space, with 95% confidence.

Key Terms & Formulas

• Linear Regression Model: ? = a + bx, where b is the sample slope (estimates ?, the true population slope).

• Standard Error of the Slope (SEb): Measures the variability of the sample slope b from sample to sample. Given in regression output or calculated as: [ SE_b = \frac{s}{\sqrt{\sum (x_i - \bar{x})^2}} ] where s = residual standard deviation, x? = individual x-values, x? = mean of x.

• Confidence Interval for Slope (?): [ b \pm t^* \times SE_b ]

• b = sample slope (from regression output).
• t = critical t-value for C% confidence, with df = n – 2 (where n* = sample size).

• SEb = standard error of the slope (from regression output).

• Degrees of Freedom (df): df = n – 2 (for linear regression with one predictor).

• t* for Confidence Interval: Use invT(area to left, df) on TI-84. For a 95% CI, use invT(0.975, df).
• Hypotheses for Slope Test:
• H?:-= 0 (no linear relationship).

• H?:-? 0 (two-tailed), ? > 0 (one-tailed), or ? < 0 (one-tailed).

• Conditions for Inference (LINER):

• Linear: Scatterplot shows a roughly linear pattern.
• Independent: Observations are independent (check 10% condition if sampling without replacement).
• Normal: Residuals are approximately normal (check histogram or Normal probability plot).
• Equal Variance: Residuals have roughly equal spread for all x-values (check residual plot).

• Random: Data comes from a random sample or randomized experiment.

• Residual (e): e = y – ? (observed y – predicted y).

• Residual Plot: Scatterplot of residuals vs. x (or ?). Used to check Equal Variance and Linear conditions.
• TI-84 Regression Output:
• STAT-CALC-8:LinReg(a+bx)-Store RegEQ (Y1) for residual plots.
• STAT-EDIT-L3 = RESID (after running regression) for residual analysis.

Step-by-Step / Process Flow

How to Construct a Confidence Interval for the Slope (AP FRQ Style):

1. State the Parameter and Confidence Level:

2. "We want to estimate the true slope (?) of the population regression line relating [explanatory variable] to [response variable] with [C]% confidence."

3. Check Conditions (LINER):

4. Linear: Scatterplot shows a roughly linear pattern.
5. Independent: Data comes from a random sample or experiment (check 10% condition if sampling without replacement).
6. Normal: Residuals are approximately normal (check histogram or Normal probability plot).
7. Equal Variance: Residual plot shows no clear pattern (random scatter around 0).

8. Random: Data is collected randomly.

9. Compute the Interval:

10. Run regression on TI-84 (STAT-CALC-8:LinReg(a+bx)).
11. Note b (slope) and SEb (standard error of slope) from output.
12. Find t using invT( (1 + C)/2, df ) where df = n – 2*.

13. Calculate interval: b ± t × SEb*.

14. Interpret the Interval in Context:

15. "We are [C]% confident that the true slope of the population regression line relating [x] to [y] is between [lower bound] and [upper bound]. This means that for each additional [unit of x], the average [y] increases/decreases by between [lower bound] and [upper bound] [units of y]."

16. Link to Hypothesis Test (if asked):

17. If the interval contains 0, fail to reject H?:-= 0 (no significant linear relationship).
18. If the interval does not contain 0, reject H? (significant linear relationship).

Common Mistakes

• Mistake: Forgetting to check the Equal Variance condition (residual plot).

• Correction: Always examine the residual plot for random scatter. If residuals fan out or form a pattern, the condition is violated.

• Mistake: Using df = n – 1 instead of df = n – 2.

• Correction: For regression, df = n – 2 (two parameters: slope and intercept). Using n – 1 inflates t* and makes the interval too narrow.

• Mistake: Misinterpreting the interval as "the slope will be in this interval 95% of the time."

• Correction: The correct interpretation is about the method: "If we repeated this process many times, 95% of the intervals would capture the true slope."

• Mistake: Ignoring the 10% condition for independence.

• Correction: If sampling without replacement, ensure n-0.10N (where N = population size).

• Mistake: Using z instead of t for small samples.

• Correction: Always use t for regression slopes (since SEb is estimated from data). Use z only for proportions.

AP Exam Insights

• FRQ Setup: Expect a scatterplot, regression output, and residual plot. You’ll need to:
• Check LINER conditions (especially Equal Variance and Normal).
• Construct and interpret a confidence interval for the slope.

• Link the interval to a hypothesis test (e.g., "Since 0 is not in the interval, we reject H?:-= 0").

• Tricky Distinction: Confidence Level vs. Confidence Interval

• Confidence level (e.g., 95%) = success rate of the method.

• Confidence interval = specific range (e.g., 0.5 to 1.2).

• Calculator Pitfall: Students often forget to store residuals (L3 = RESID) after running regression, making residual plots impossible.

• Common Trap: The AP exam may give a non-significant slope (interval contains 0). Don’t assume the slope is meaningful just because the problem asks for an interval!

Quick Check Questions

1. Multiple Choice: A regression of y = house price (in $1000s) on x = square footage yields b = 0.15 and SEb = 0.03. For a 95% confidence interval with df = 28, which of the following is the correct margin of error?
2. (A) 0.03 × 1.701
3. (B) 0.03 × 2.048
4. (C) 0.15 × 2.048
5. (D) 0.15 ± 0.03 × 2.048

Answer: (B) 0.03 × 2.048. The margin of error is t × SEb, where t = invT(0.975, 28)-2.048.

1. FRQ Part: A study examines the relationship between x = hours studied and y = exam score (out of 100) for 30 students. The regression output is below: Predictor Coef SE Coef T P Constant 45.2 3.1 14.58 0.000 Hours 2.8 0.5 5.60 0.000 S = 8.2, R-sq = 54.3%
2. a. Construct a 95% confidence interval for the true slope.
3. b. Interpret the interval in context.
4. c. Does this interval provide convincing evidence of a linear relationship? Explain.

Answer: - a. df = 30 – 2 = 28, t = invT(0.975, 28)-2.048. Interval: 2.8 ± 2.048 × 0.5? (1.776, 3.824). - b. "We are 95% confident that for each additional hour studied, the average exam score increases by between 1.776 and 3.824 points." - c. Yes, because 0 is not in the interval, so we reject H?:-= 0* at the 5% significance level.

Last-Minute Cram Sheet

1. Formula: b ± t × SEb* (for slope confidence interval).
2. df = n – 2 (always for regression slope).
3. LINER conditions: Check all 5 (especially Equal Variance and Normal).
4. TI-84: invT( (1+C)/2, df ) for t* (e.g., invT(0.975, 28) for 95% CI).
5. Residuals: L3 = RESID after regression to check conditions.
6. Interpretation: "We are [C]% confident that the true slope is between [LB] and [UB]."
7. Hypothesis Test Link: If 0 is in the interval-fail to reject H?:-= 0.
8. 10% condition: Check if sampling without replacement (n-0.10N).
9. Don’t use z*: Always use t* for regression slopes.
10. Residual plot: Must show random scatter (no patterns) for Equal Variance.