There is a 100-item test called the "Original Voight-Kampff Test" which has a reliability of 0.5. You have been asked to improve the reliability of this test to 0.6 by adding additional items, and the new test will be called the "Longer Voight-Kampff Test". According to the expansion factor formula, the "Longer Voight-Kampff Test" should have how many items IN TOTAL?

Question

BlackTom AI · Accepted Answer

Applying the Spearman–Brown prophecy formula for test lengthening: this formula predicts how reliability changes when you multiply the number of parallel items by a factor k. The relevant knowledge points are classical test theory (reliability increases with more parallel items) and the Spearman–Brown formula:
r_new = (k * r_old) / [1 + (k - 1) * r_old].

Start with r_old = 0.50 (original reliability) and desired r_new = 0.60. Solve for the expansion factor k.

Step-by-step algebra:
r_new = (k r_old) / [1 + (k - 1) r_old]
Multiply both sides by the denominator:
r_new [1 + (k - 1) r_old] = k r_old
Expand left side:
r_new + r_new (k - 1) r_old = k r_old
Collect terms with k on one side:
k r_new r_old - k r_old = r_new r_old - r_new
Factor k on left:
k (r_new r_old - r_old) = r_new (r_old - 1)
Recognize negatives and simplify to:
k r_old (1 - r_new) = r_new (1 - r_old)
Thus:
k = [r_new (1 - r_old)] / [r_old (1 - r_new)]

Plug in numbers:
Numerator = 0.60 * (1 - 0.50) = 0.60 * 0.50 = 0.30
Denominator = 0.50 * (1 - 0.60) = 0.50 * 0.40 = 0.20
k = 0.30 / 0.20 = 1.5

Total items required = original items * k = 100 * 1.5 = 150.

Why the correct answer is correct: a 1.5× increase in length raises reliability from 0.50 to 0.60 per the Spearman–Brown relation, so the Longer Voight-Kampff Test must have 150 items.

Why other options are incorrect:
- 100: no increase, reliability remains 0.50.
- 200: k = 2 would produce r_new = (2*0.5)/(1+0.5) = 1/1.5 ≈ 0.667, which is higher than 0.60.
- 250: k = 2.5 yields r_new ≈ 0.714, also not 0.60.

Final confirmation: the required total number of items is 150, matching the provided answer.

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