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

Correct option is A

The Chi-Square (χ²) test is a statistical method used to determine if there is a significant difference between observed and expected frequencies. It is commonly used in hypothesis testing, particularly in goodness-of-fit tests and independence tests.

Understanding the Null Hypothesis (H₀):

• The null hypothesis (H₀) states that there is no significant difference between the observed and expected values.

• The alternative hypothesis (H₁) suggests that there is a significant difference.

Acceptance of the Null Hypothesis:

• When performing a Chi-Square test, we compare the calculated Chi-Square value (χ² cal) with the critical Chi-Square value from the table (χ² table) at a given significance level (α, usually 0.05).

• If χ² cal ≤ χ² table, the null hypothesis (H₀) is accepted because the observed differences are not statistically significant.

• This means that the sample data does not provide enough evidence to reject H₀.

Key Decision Rule:

• If χ² cal ≤ χ² table → Fail to reject H₀ (Accept the null hypothesis).

• If χ² cal > χ² table → Reject H₀ (Accept the alternative hypothesis).

Information Booster:

• The Chi-Square test measures the deviation between observed and expected frequencies.

• A low χ² cal value means that observed data is close to expected data, making it more likely that the null hypothesis is true.

• The critical value (χ² table) is determined based on degrees of freedom (df) and significance level (α).

• When χ² cal is lower than or equal to χ² table, the deviation is statistically insignificant, leading to acceptance of H₀.

Additional Knowledge:

1. (b) χ² cal > χ² table:

   • If χ² cal is greater than χ² table, the null hypothesisis rejected, not accepted.

   • A high calculated χ² value suggests a significant difference between observed and expected data.

2. (c) χ² cal < χ² table:

   • This option is partially correct but does not include cases where χ² cal is exactly equal to χ² table.

   • The correct condition is χ² cal ≤ χ² table, which covers both cases.

3. (d) χ² cal = χ² table:

   • This condition alone does not fully define acceptance of H₀.

   • If χ² cal is exactly equal to χ² table, it is at the borderline of rejection, but we still fail to reject H₀.

   • However, acceptance of H₀ occurs when χ² cal is less than or equal to χ² table.