AP Statistics (AP Stats): Interpreting Computer Output (SEb, t, p)

AP Statistics – Interpreting Computer Output (SEb, t, p)

AP Statistics: Interpreting Computer Output (SEb, t, p) – Exam-Ready Study Guide

What This Is

Computer output is how real statisticians report regression and t-test results. On the AP exam, you’ll be given tables or calculator screens showing standard error of the slope (SEb), t-ratio, and p-value for a linear regression or t-test. Your job is to interpret these values in context, check conditions, and make conclusions (e.g., "Does a new study method improve test scores?" or "Is there a linear relationship between ice cream sales and temperature?"). Mastering this skill lets you tackle FRQs efficiently and avoid losing points on misinterpretations.

Key Terms & Formulas

• Linear Regression t-test for β (slope):
• H₀: β = 0 (no linear relationship)
• Hₐ: β ≠ 0 (linear relationship exists)

• Test statistic: t = (b – β₀) / SEb, where:

  
  
  • b = sample slope
  • β₀ = hypothesized slope (usually 0)
  • SEb = standard error of the slope (from output)

• SEb (Standard Error of the Slope):
  Measures the variability of the sample slope b from the true slope β. Smaller SEb → more precise estimate.

• t-ratio (t-statistic):
  The number of standard errors b is from β₀. Use tcdf(lower, upper, df) on TI-84 to find the p-value.

• p-value (for slope):
  Probability of observing a slope as extreme as b if H₀ (β = 0) is true. Small p-value (≤ α) → reject H₀.

• Degrees of Freedom (df) for regression:
  df = n – 2 (for simple linear regression with 1 predictor).

• Confidence Interval for β:
  b ± t × SEb, where t is the critical value from invT(area, df) (e.g., invT(0.975, df) for 95% CI).

• LINER Conditions (for regression inference):

• Linear: Residual plot shows no pattern.
• Independent: 10% condition (if sampling without replacement).
• Normal: Histogram of residuals is roughly symmetric/unimodal.
• Equal variance: Residual plot shows consistent spread.

• Random: Data comes from a random sample/experiment.

• TI-84 Commands:

• LinRegTTest: For regression t-test (Menu → STAT → TESTS → LinRegTTest).
• tcdf(lower, upper, df): Finds p-value for a t-test.
• invT(area, df): Finds critical t-value for confidence intervals.

Step-by-Step / Process Flow

For a regression FRQ (e.g., "Is there a linear relationship between X and Y?"):
1. State Hypotheses:
- H₀: β = 0 (no linear relationship)
- Hₐ: β ≠ 0 (linear relationship exists)
Write in context! (e.g., "H₀: There is no linear relationship between ice cream sales and temperature.")

1. Check LINER Conditions:
2. Sketch residual plot (if not given) and check for linearity, equal variance, and outliers.

3. Verify independence (10% condition) and randomness.

4. Extract Values from Output:

5. Identify b (slope), SEb, t-ratio, and p-value.

6. Example output:
   Coef SE Coef T P
2.5 0.8 3.125 0.004
   Here, b = 2.5, SEb = 0.8, t = 3.125, p = 0.004.

7. Compute Test Statistic (if not given):

8. t = (b – 0) / SEb = 2.5 / 0.8 = 3.125.

9. Find p-value:

10. Use tcdf(3.125, 1E99, df) on TI-84 (df = n – 2).

11. Compare to α (usually 0.05).

12. Make Conclusion in Context:

13. If p ≤ α: "Reject H₀. There is convincing evidence of a linear relationship between [X] and [Y]."
14. If p > α: "Fail to reject H₀. There is not convincing evidence of a linear relationship."

Common Mistakes

• Mistake: Misinterpreting SEb as the slope.
  Correction: SEb is the standard error of the slope, not the slope itself. The slope is b (from the "Coef" column).

• Mistake: Forgetting to check LINER conditions.
  Correction: Always verify conditions before making conclusions. The AP exam will deduct points if you skip this.

• Mistake: Using z instead of t for regression.
  Correction: Regression inference uses t-tests (not z-tests) because we estimate σ with s (standard deviation of residuals).

• Mistake: Confusing p-value for slope with p-value for intercept.
  Correction: Focus on the p-value for the slope (usually the second row in output). The intercept’s p-value is rarely tested.

• Mistake: Ignoring units in interpretations.
  Correction: Always include units! (e.g., "For each additional degree of temperature, ice cream sales increase by 2.5 thousand cones.")

AP Exam Insights

• Tricky Distinction: The p-value tests whether the slope is 0, not whether the correlation is strong. A small p-value means the relationship is statistically significant, but r (correlation) measures strength/direction.

• Common FRQ Setup:

• Given a regression output, ask for:
  1. Interpretation of slope/SEb.
  2. Hypothesis test for β = 0.
  3. Confidence interval for β.

• Example: "Construct a 95% CI for the slope and interpret it in context."

• Calculator Pitfall: LinRegTTest on TI-84 gives two-tailed p-values by default. If the FRQ asks for a one-tailed test, divide the p-value by 2.

• AP Loves Residual Plots: Be ready to sketch or interpret residual plots to check LINER conditions.

Quick Check Questions

1. Multiple Choice:
   A regression output shows:
   Coef SE Coef T P
1.2 0.3 4.0 0.001
   What is the correct interpretation of the p-value?
   A) The probability that the slope is 0.
   B) The probability of observing a slope of 1.2 or more extreme if the true slope is 0.
   C) The probability that the sample slope is correct.
   Answer: B. The p-value is the probability of observing a slope as extreme as 1.2 if H₀ (β = 0) is true.

2. FRQ Part:
   A study reports a slope of 0.5 with SEb = 0.1 and n = 30. Construct a 95% confidence interval for the slope.
   Answer:

3. df = 30 – 2 = 28.
4. t* = invT(0.975, 28) ≈ 2.048.
5. CI = 0.5 ± 2.048 × 0.1 = (0.2952, 0.7048).
6. Interpretation: "We are 95% confident that for each additional [unit of X], [Y] increases by between 0.2952 and 0.7048 [units]."

Last-Minute Cram Sheet

1. Hypotheses for slope: H₀: β = 0 vs. Hₐ: β ≠ 0 (or >/< 0 for one-tailed).
2. t = (b – β₀) / SEb (β₀ is usually 0).
3. df = n – 2 for regression.
4. LINER conditions: Check before inference!
5. SEb ≠ slope (SEb is the "error" of the slope estimate).
6. p-value ≤ α → reject H₀ (evidence of a relationship).
7. CI for β: b ± t* × SEb.
8. TI-84: LinRegTTest for regression, tcdf for p-values.
9. ⚠️ Always interpret in context (units, variables, direction).
10. ⚠️ Residual plots matter! AP loves testing LINER conditions.