Your workplace makes important decisions about individuals on the basis of the results of a certain test. You were the designer of the test, and all 100 items in it were originally devised by you. The test in question has a reliability of 0.89. Your boss has heard that reliability is a good thing, and has asked you to investigate the practicality of increasing the number of test items and thereby getting the reliability of this test to 0.99.

Which of the following is the best response?

Question

Your workplace makes important decisions about individuals on the basis of the results of a certain test. You were the designer of the test, and all 100 items in it were originally devised by you. The test in question has a reliability of 0.89. Your boss has heard that reliability is a good thing, and has asked you to investigate the practicality of increasing the number of test items and thereby getting the reliability of this test to 0.99.

Which of the following is the best response?

BlackTom AI · Accepted Answer

Spearman–Brown prophecy formula predicts how reliability changes with test length: r_new = (k * r_old) / [1 + (k − 1) * r_old], where k is the factor by which the number of items is multiplied. Relevant concepts: classical test theory reliability, Spearman–Brown formula for lengthening a test, diminishing returns as reliability approaches 1, and practical constraints (time, fatigue, content breadth, cost).

Step-by-step calculation to reach r_new = 0.99 from r_old = 0.89:
1. Set up the formula: 0.99 = (k * 0.89) / [1 + (k − 1) * 0.89].
2. Multiply both sides by the denominator: 0.99[1 + 0.89(k − 1)] = 0.89k.
3. Simplify inside the brackets: 1 + 0.89(k − 1) = 1 + 0.89k − 0.89 = 0.11 + 0.89k.
4. Multiply out the left side: 0.99(0.11 + 0.89k) = 0.1089 + 0.8811k.
5. Equate to the right side: 0.1089 + 0.8811k = 0.89k.
6. Solve for k: 0.1089 = 0.89k − 0.8811k = 0.0089k, so k = 0.1089 / 0.0089 ≈ 12.24.
7. Multiply by current items: new number of items ≈ 12.24 × 100 ≈ 1,224 items.

Why this makes the provided answer correct: reaching 0.99 reliability would require multiplying the test length by over twelvefold—raising a 100-item test to about 1,224 items. That is impractical because of examinee fatigue, longer administration time, higher costs, potential loss of content validity (overemphasis on item sampling from the same item writer), and logistical infeasibility. Moreover, reliability of 0.89 is already high; increasing reliability has diminishing returns, and alternative strategies (improving item quality, removing noisy items, combining multiple measures) are usually more practical than adding >1,000 items.

Why other responses (e.g., “simply add a few items” or “don’t attempt any change at all”) would be incorrect: small increases in length won’t approach 0.99 (Spearman–Brown calculation shows enormous increase needed), and saying no change is needed ignores feasible improvements to item quality. Thus the best practical response is the provided answer: D. “We should increase the reliability of the test, but in this situation it’s not practical to try to get reliability all the way up to 0.99 by increasing the number of items.” This matches the given correct answer.

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