Question

For testing goodness of fit, suppose the observed frequencies for two classes are 28 and 12. It is expected that these occur in the ratio 3:1. The value of Chi-square statistic for testing this is:

• A. 0.26
• B. 0.40
• C. 0.53
• D. 0.75

Hint:

Exam Tip:
For goodness-of-fit tests:

• Compute expected frequencies based on the hypothesized ratios.
• Use the formula \(\chi^2 = \sum \frac{(O - E)^2}{E}\).
• The degrees of freedom are \((k - 1)\), where \(k\) is the number of classes.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to calculate the chi-square statistic for a goodness-of-fit test.
The expected frequencies are based on the ratio 3:1.

Step 2: Key Formula or Approach:
Total observed frequency = \(28 + 12 = 40\).
Expected frequencies:
• For class 1: \(\frac{3}{4} \times 40 = 30\).
• For class 2: \(\frac{1}{4} \times 40 = 10\). The chi-square statistic is: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \]

Step 3: Detailed Explanation:
\[ \chi^2 = \frac{(28 - 30)^2}{30} + \frac{(12 - 10)^2}{10} = \frac{4}{30} + \frac{4}{10} = \frac{2}{15} + \frac{2}{5} = \frac{2}{15} + \frac{6}{15} = \frac{8}{15} \approx 0.5333 \] So, \(\chi^2 \approx 0.53\).

Step 4: Final Answer:
Therefore, option (C) is correct.