Question

A 30 question test has a reliability of 0.60. Using the Spearman-Brown Prophecy formula, by how much does this test need to be increased in order to raise the reliability to 0.80 ?

Answer 1

By Spearman-Brown Prophecy Formula,

r_kk = k(r₁₁) / [1 + (k – 1)* r₁₁]

• r_kk = reliability of a test “k” times as long as the original test,
• r₁₁ = reliability of the original test(e.g. Cronbach’s Alpha),
• k = factor by which the length of the test is changed.

Given,

r₁₁ = 0.60

r_kk = 0.80

0.80 = 0.60 k / [1 + (k – 1)* 0.60]

0.8  [1 + (k – 1)* 0.60] = 0.6 k

0.8 (1 + 0.6k - 0.6) = 0.6k

0.8(0.4 + 0.6k) = 0.6k

0.32 + 0.48k = 0.6k

(0.6 - 0.48)k = 0.32

0.12k = 0.32

k = 0.32/ 0.12 = 8/3

The test need to be increased by 8/3 times in order to raise the reliability to 0.80

The number of questions to be increased by (8/3) * 30 = 80