Which of the following will make a true score confidence interval wider?

Question

BlackTom AI · Accepted Answer

Confidence-interval width is proportional to the standard error of estimation, so increasing that standard error makes the interval wider.

1. Relevant knowledge points and concepts
- Classical test theory: observed score X = true score T + error E. Reliability r is the proportion of variance in X due to T.
- Standard deviation of test scores (σX) and reliability determine the standard error of measurement (SEM): SEM = σX * sqrt(1 − r).
- A confidence interval (CI) for a true score is centered on the predicted true score and has half-width = z * SEM (for z corresponding to the desired confidence level). Total width = 2 * z * SEM.

2. Step-by-step reasoning
- The CI width depends directly on SEM: width ∝ SEM.
- If SEM increases, the multiplier z stays the same for a given confidence level, so the half-width and total width both increase.
- SEM itself increases if either σX increases or r decreases (since SEM = σX * sqrt(1 − r)).

3. Intermediate calculations (example)
- Suppose σX = 10 and r = 0.80. SEM = 10 * sqrt(1 − 0.80) = 10 * sqrt(0.20) = 10 * 0.4472 = 4.472.
- 95% CI half-width = 1.96 * SEM = 1.96 * 4.472 ≈ 8.764; total width ≈ 17.528.
- If SEM rises to 6, half-width = 1.96 * 6 = 11.76; total width = 23.52 — clearly wider.

4. Why the correct answer is correct
- Option C states “The standard error of estimation being higher.” Because CI width = 2 * z * SEM, a higher SEM directly increases the CI half-width and total width. Thus C makes the true score CI wider.

5. Why the other options are incorrect
- A (higher reliability): Increases r → decreases SEM (SEM = σX * sqrt(1 − r)), so the CI becomes narrower, not wider.
- B (lower standard deviation): Decreases σX → decreases SEM, so CI becomes narrower.
- D (predicted true score being lower): Shifts the center of the interval but does not change SEM or the interval’s width; width remains the same.

6. Conclusion
Because CI width is directly proportional to the standard error of estimation, a higher standard error of estimation will make the true score confidence interval wider. This confirms the correct answer: C.

Which of the following will make a true score confidence interval wider?Single choice

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