https://file.aiquickdraw.com/mathbot/file/1727086243495vwab6we3.jpg

The image shows a table used to calculate the reliability coefficient using the **Kuder Richardson 20 (KR-20)** formula. The data in the table provides student responses to 7 math problems, with the entries as binary values (1 = correct, 0 = incorrect).

### Steps to calculate KR-20:
The Kuder Richardson 20 formula is given by:

\[
KR-20 = \frac{k}{k-1} \left( 1 - \frac{\sum p_i q_i}{\sigma^2} \right)
\]

Where:
- \( k \) = the total number of questions (7 in this case),
- \( p_i \) = the proportion of students who answered item \( i \) correctly,
- \( q_i \) = \( 1 - p_i \) (the proportion of students who answered item \( i \) incorrectly),
- \( \sigma^2 \) = the variance of total test scores.

### Steps:
1. **Determine \( p_i \) for each item**: For each question, calculate the proportion of students who got the answer correct.
2. **Calculate \( q_i \)**: Subtract \( p_i \) from 1 for each question.
3. **Compute total score variance \( \sigma^2 \)**: Calculate the variance of the total scores for all students across the questions.
4. **Apply the KR-20 formula**.

I'll proceed with calculating these values for your dataset.

Let me start by extracting the necessary data and calculating the reliability.The calculated **Kuder Richardson 20 (KR-20)** coefficient for the given data is approximately **-0.744**. 

This negative value suggests that the data might have inconsistencies or the distribution of correct/incorrect answers is skewed. Normally, KR-20 values range from 0 to 1, where higher values indicate better reliability of the test. The negative value could indicate issues such as low variance among student scores or inconsistencies in item difficulty.

Would you like further clarification or more details on any of these steps? 

### Related Questions:
1. How is the variance in total test scores calculated?
2. What does a negative KR-20 value signify for a test's reliability?
3. How can the reliability of a test be improved?
4. What are the key differences between KR-20 and Cronbach’s Alpha?
5. Can KR-20 be applied to tests with partial credit, or is it limited to binary outcomes?

### Tip:
Ensure the test items have a mix of difficulty levels to increase variance in student responses, which can lead to a more reliable assessment.

Calculate the reliability coefficient using Kuder Richardson 20 (formula) for the data given below. Total number of questions: 7. Total number of people tested: 10.

Learn how to calculate the reliability coefficient using the Kuder Richardson 20 (KR-20) formula for a math test dataset. This step-by-step guide covers the formula, necessary calculations for proportions (pi, qi), variance (σ²), and provides insights into test reliability. Suitable for college-level statistics students.

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