A researcher wants to use split-half reliability to estimate the reliability of a 50-item test. They begin by splitting the test into two 25-item halves and correlating scores on the two halves. They then want to use the Spearman–Brown prophecy formula to estimate the reliability of the full 50-item test. What expansion factor should they use?

Question

BlackTom AI · Accepted Answer

Spearman–Brown prophecy formula uses an expansion factor k equal to the ratio of the target test length to the current test length. For a split-half situation where each half has 25 items and the researcher wants the reliability of the full 50-item test, k = (full length) / (half length).

1) Relevant knowledge points and concepts
- Split-half reliability: correlate scores on two equal halves of a test to estimate internal consistency for one half.
- Spearman–Brown prophecy formula: adjusts a reliability estimate for changing test length. In algebraic form: r_new = (k * r_old) / [1 + (k - 1) * r_old], where r_old is the reliability of the current length and k is the expansion factor (new length / old length).

2) Step-by-step reasoning
- Current half length = 25 items.
- Desired full test length = 50 items.
- Compute expansion factor k = 50 / 25.

3) Intermediate calculation
- k = 50 ÷ 25 = 2.

4) Why this is correct
- k = 2 means the full test is twice as long as one half, so the Spearman–Brown formula should double the number of items (and adjust the reliability accordingly). Substituting k = 2 into the formula gives r_full = (2 * r_half) / [1 + (2 - 1) * r_half] = (2 r_half) / (1 + r_half), which is the standard correction when going from half-length to full-length.

5) Why other options are incorrect
- k = 1 would imply no change in length (full test same length as half), which is false.
- k = 0.5 would imply the full test is half the length of the half-test, which is nonsensical here.
- Any other numeric factor not equal to 50/25 = 2 does not represent the actual length ratio and would give an incorrect Spearman–Brown adjustment.

6) Conclusion
The correct expansion factor is 2, matching the provided answer C. 2.

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