Fail to Reject

Definition

Fail to reject is a statistical decision made during hypothesis testing where the evidence from the sample data is not strong enough to reject the null hypothesis. This means the null hypothesis is retained, or not rejected, even though it may not be entirely true in the population.

5 Must Know Facts For Your Next Test

1. Failing to reject the null hypothesis does not mean the null hypothesis is true, only that the evidence is not strong enough to conclude it is false.
2. The probability of failing to reject the null hypothesis when it is false is known as the Type II error rate, denoted as β.
3. The power of a statistical test is the probability of rejecting the null hypothesis when it is false, which is equal to 1 - β.
4. In the context of hypothesis testing for a single mean or single proportion, failing to reject the null hypothesis suggests the sample data is consistent with the hypothesized population parameter value.
5. The decision to fail to reject the null hypothesis is made by comparing the test statistic (e.g., z-score or t-statistic) to the critical value from the appropriate probability distribution.

Review Questions

• Explain the meaning of failing to reject the null hypothesis in the context of hypothesis testing for a single mean.
  • When conducting a hypothesis test for a single mean, failing to reject the null hypothesis means the sample data does not provide sufficient evidence to conclude the true population mean is different from the hypothesized value. This indicates the observed sample mean is consistent with the null hypothesis, and the researcher cannot reject the claim that the population mean is equal to the hypothesized value. Failing to reject the null hypothesis does not imply the null hypothesis is true, only that the data does not provide strong enough evidence to conclude it is false.
• Describe how the probability of a Type II error relates to the decision to fail to reject the null hypothesis in hypothesis testing for a single proportion.
  • In the context of hypothesis testing for a single proportion, failing to reject the null hypothesis means the sample proportion is not significantly different from the hypothesized population proportion. The probability of making a Type II error, denoted as β, is the likelihood of failing to reject the null hypothesis when it is actually false. A higher Type II error rate corresponds to a lower probability of detecting a difference that truly exists, which is known as the test's power (1 - β). When the null hypothesis is failed to be rejected, it suggests the sample data is consistent with the hypothesized population proportion, but it does not conclusively prove the null hypothesis is true.
• Analyze the implications of failing to reject the null hypothesis in the context of hypothesis testing and discuss how it relates to the concepts of statistical significance and practical significance.
  • Failing to reject the null hypothesis in hypothesis testing indicates the sample data does not provide statistically significant evidence to conclude the null hypothesis is false. However, this does not necessarily mean the null hypothesis is true or that the effect size is negligible from a practical standpoint. Failing to reject the null hypothesis could be due to insufficient sample size, high variability in the data, or a truly small effect size that is difficult to detect. It is important to consider both statistical significance (the likelihood the observed result occurred by chance) and practical significance (the magnitude of the effect and its real-world importance) when interpreting the implications of failing to reject the null hypothesis. A non-significant result may still have important practical implications, and further investigation or a larger sample size may be needed to determine the true nature of the relationship or effect.

Related terms

Null Hypothesis: The null hypothesis (H0) is a statistical hypothesis that states there is no significant difference or relationship between the variables being studied.

Alternative Hypothesis:

The alternative hypothesis (H1) is a statistical hypothesis that contradicts the null hypothesis, stating there is a significant difference or relationship between the variables.

Type II Error:

A Type II error occurs when the null hypothesis is false, but the test fails to reject it, meaning the test did not detect a difference that actually exists.