Which one of the following is correct when a null hypothesis is accepted?

Correct option is C

Introduction:

In hypothesis testing, specifically using the Chi-Square ($$\chi^2$$​) test, we compare the calculated value of the test statistic ($$\chi^2_{\text{cal}}$$​) with the critical value obtained from the statistical table ($$\chi^2_{\text{table}}$$​) at a specific level of significance and degrees of freedom. A null hypothesis ($$H_0$$​) is accepted (or fails to be rejected) when the calculated sample statistic is less than the critical table value, meaning the observed differences are purely due to chance.

Therefore, the correct answer is option (C).

Information Booster:

• The Decision Rule: The fundamental rule in Chi-Square testing states that if $$\chi^2_{\text{cal}} < \chi^2_{\text{table}}$$​, we fail to reject (accept) the null hypothesis. Conversely, if $$\chi^2_{\text{cal}} \ge \chi^2_{\text{table}}$$​, we reject the null hypothesis and accept the alternative hypothesis.

• Acceptance Region: When the calculated value falls below the critical table value, it lands in the "acceptance region" (non-rejection region) of the sampling distribution, indicating that any variation between observed and expected frequencies is statistically insignificant.

• Chi-Square Formula: The calculated value is derived using the formula:

  ​$$\chi^2 = \sum \frac{(O - E)^2}{E}$$​​

  Where $$O$$​ represents Observed Frequencies and $$E$$​ represents Expected Frequencies.

• UGC NET Exam Tip: Always remember that table values represent the maximum allowable threshold of chance variation. If your sample's calculated variance exceeds this threshold, it cannot be ignored as mere chance, forcing the rejection of $$H_0$$​.

Additional Knowledge:

• Why Option (A) is incorrect: While $$\chi^2_{\text{cal}} \le \chi^2_{\text{table}}$$​ technically includes the less-than condition, the strict statistical threshold for absolute acceptance in standard textbook multiple-choice questions typically prioritizes the clean inequality $$\chi^2_{\text{cal}} < \chi^2_{\text{table}}$$​.

• Why Option (B) is incorrect: When $$\chi^2_{\text{cal}} > \chi^2_{\text{table}}$$​, the calculated value falls deep into the critical region (rejection region). This means the result is statistically significant, and the null hypothesis must be rejected.

• Why Option (D) is incorrect: An exact equality ($$\chi^2_{\text{cal}} = \chi^2_{\text{table}}$$​) represents the exact boundary line (critical point) of rejection. In standard practice, hitting the exact critical value usually leads to rejecting the null hypothesis or requires further fractional precision, making it an incorrect choice for general acceptance.

• Common Trap: Students often confuse the direction of the inequality sign. Remember: Calculated < Table = Accept $$H_0$$​; Calculated > Table = Reject $$H_0$$​.