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 new test length to the original test length.

1. Relevant knowledge points and concepts
- Split-half reliability compares two equal halves of a test and yields a correlation (r_half) between the half-scores.
- The Spearman–Brown prophecy formula adjusts that half-test correlation to estimate the reliability of a test of a different length.
- The expansion factor k is defined as (new test length) / (original test length used to compute the correlation).

2. Complete reasoning process step by step
- The researcher split the 50-item test into two halves of 25 items each and correlated the two 25-item scores. That correlation is based on a 25-item test length for each half.
- To estimate reliability of the full 50-item test from the half-test correlation, we must determine how many times longer the full test is compared with the half-test. That ratio is the expansion factor k.

3. Calculations and intermediate steps
- New test length = 50 items.
- Original (half) test length = 25 items.
- k = (new length) / (original length) = 50 / 25 = 2.

4. Why the correct answer is correct
- The Spearman–Brown formula uses k = 2 when you double test length (going from one half to the full test). Plugging k = 2 into the formula r_full = (k * r_half) / (1 + (k - 1) * r_half) correctly converts the split-half correlation into an estimate of the full 50-item reliability.

5. Why other possible choices are incorrect
- k = 1 would imply no change in length, which is false because the full test is twice as long as a half.
- k = 0.5 would imply the new test is half as long as the original, which is the opposite of this situation.
- Any k other than 2 contradicts the simple length ratio 50/25 = 2 and would produce an incorrect adjusted reliability.

6. Conclusion and confirmation
- The expansion factor to use is 2. This matches the provided correct answer: C. 2.

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