Calculating New Test Reliability with Spearman-Brown Formula

The question asks us to predict the new reliability coefficient of a test when its length is increased by adding similar items. We start with a test having 20 items and a reliability coefficient of 0.60. We are adding 80 more similar items, which means the new test will have a total of $20 + 80 = 100$ items.

Understanding the Spearman-Brown Prophecy Formula

To predict the reliability of a test after changing its length, we use the Spearman-Brown prophecy formula. This formula assumes that the items added are similar in quality (difficulty and discrimination) to the original items and that the test is homogeneous.

The formula is given by:

$\text{R}_{new} = \frac{n \cdot \text{R}_{old}}{1 + (n-1) \cdot \text{R}_{old}}$

Where:

• $\text{R}_{new}$ is the predicted reliability of the new, lengthened test.
• $\text{R}_{old}$ is the reliability of the original test (given as 0.60).
• $n$ is the factor by which the test length is increased. It is calculated as the ratio of the new test length to the original test length.

Step-by-Step Calculation

Step 1: Determine the new test length

Original number of items = 20

Number of added items = 80

New total number of items = Original items + Added items = $20 + 80 = 100$ items.

Step 2: Calculate the length factor 'n'

The length factor 'n' is the ratio of the new test length to the original test length.

$n = \frac{\text{New Length}}{\text{Original Length}}$

$n = \frac{100 \text{ items}}{20 \text{ items}}$

$n = 5$

This means the new test is 5 times longer than the original test.

Step 3: Apply the Spearman-Brown formula

We have $\text{R}_{old} = 0.60$ and $n = 5$. Substitute these values into the formula:

$\text{R}_{new} = \frac{5 \cdot 0.60}{1 + (5-1) \cdot 0.60}$

Step 4: Perform the calculation

First, calculate the numerator:

$5 \cdot 0.60 = 3.00$

Next, calculate the term $(n-1) \cdot \text{R}_{old}$ in the denominator:

$(5-1) \cdot 0.60 = 4 \cdot 0.60 = 2.40$

Now, complete the denominator:

$1 + (n-1) \cdot \text{R}_{old} = 1 + 2.40 = 3.40$

Finally, calculate $\text{R}_{new}$:

$\text{R}_{new} = \frac{3.00}{3.40}$

$\text{R}_{new} \approx 0.88235$

Rounding the result to two decimal places, we get 0.88.

Conclusion

By adding 80 similar items to a test with 20 items and a reliability of 0.60, the new reliability coefficient is predicted to be approximately 0.88 according to the Spearman-Brown prophecy formula. Increasing the test length generally increases its reliability, assuming the added items are of similar quality.

Revision Table: Test Reliability Concepts

Additional Information: Factors Affecting Reliability

Reliability is a crucial psychometric property of a test. Besides test length, several other factors can influence test reliability:

• Quality of Items: Poorly written, ambiguous, or tricky items reduce reliability. Items should be clear and well-constructed.
• Range of Scores: Reliability tends to be higher in groups with a wider range of abilities or traits being measured. A restricted range can lower reliability estimates.
• Test Administration Conditions: Standardized testing environments (clear instructions, adequate time, minimal distractions) help ensure consistency and improve reliability.
• Scoring Objectivity: Objective scoring (e.g., multiple-choice tests with a clear answer key) leads to higher reliability compared to subjective scoring (e.g., essays where different raters might assign different scores).
• Test Format: The type of questions (e.g., multiple-choice, true/false, essay) and their structure can impact reliability.
• Sample Size and Characteristics: The group on which reliability is estimated affects the coefficient. Characteristics of the sample (e.g., homogeneity) are important.

The Spearman-Brown formula is a powerful tool for test developers to understand the trade-offs between test length and desired reliability, but it's important to remember its underlying assumptions, particularly the similarity of items.