Reject

Definition

In the context of hypothesis testing, the term 'reject' refers to the decision made when the test statistic falls within the critical region, indicating that the null hypothesis should be rejected in favor of the alternative hypothesis. This decision is based on the observed data and the predetermined significance level.

5 Must Know Facts For Your Next Test

1. The decision to reject the null hypothesis is based on the p-value, which represents the probability of obtaining a test statistic as extreme or more extreme than the observed value, given that the null hypothesis is true.
2. If the p-value is less than the predetermined significance level (usually denoted as $\alpha$), the null hypothesis is rejected, indicating that the observed data provides sufficient evidence to support the alternative hypothesis.
3. Rejecting the null hypothesis does not necessarily mean that the alternative hypothesis is true; it only suggests that the observed data is unlikely to have occurred if the null hypothesis were true.
4. The consequences of rejecting the null hypothesis when it is true (a Type I error) and not rejecting the null hypothesis when it is false (a Type II error) should be considered when making the decision to reject or not reject the null hypothesis.
5. The power of a hypothesis test is the probability of rejecting the null hypothesis when it is false, and it is an important consideration in determining the appropriate sample size and significance level for a given hypothesis test.

Review Questions

• Explain the decision-making process involved in rejecting the null hypothesis in the context of hypothesis testing of a single mean.
  • In the context of hypothesis testing of a single mean, the decision to reject the null hypothesis is made when the test statistic (e.g., the sample mean) falls within the critical region. The critical region is determined by the significance level (α) and the sampling distribution of the test statistic. If the p-value, which represents the probability of obtaining a test statistic as extreme or more extreme than the observed value under the null hypothesis, is less than the significance level, the null hypothesis is rejected in favor of the alternative hypothesis. This decision indicates that the observed data provides sufficient evidence to suggest that the population mean is different from the hypothesized value.
• Describe the consequences of rejecting the null hypothesis when it is true (a Type I error) and not rejecting the null hypothesis when it is false (a Type II error) in the context of hypothesis testing of a single proportion.
  • In the context of hypothesis testing of a single proportion, a Type I error occurs when the null hypothesis is rejected when it is actually true. This means that the researcher concludes that the population proportion is different from the hypothesized value, when in reality, it is not. A Type II error occurs when the null hypothesis is not rejected when it is actually false. This means that the researcher fails to detect a difference in the population proportion when there is one. The consequences of these errors depend on the context of the hypothesis test, but generally, a Type I error can lead to unnecessary actions or interventions, while a Type II error can result in missed opportunities or failure to detect an important effect. The significance level (α) and the power of the test (1 - β) are used to balance the risks of these two types of errors.
• Analyze the relationship between the decision to reject the null hypothesis and the concept of statistical significance in the context of hypothesis testing.
  • The decision to reject the null hypothesis is directly related to the concept of statistical significance. Statistical significance is determined by the p-value, which represents the probability of obtaining the observed data (or more extreme data) under the assumption that the null hypothesis is true. If the p-value is less than the predetermined significance level (α), the null hypothesis is rejected, indicating that the observed data is unlikely to have occurred by chance if the null hypothesis were true. This suggests that the difference or effect observed in the sample is statistically significant, meaning that it is unlikely to have occurred due to random chance alone. The smaller the p-value, the stronger the evidence against the null hypothesis, and the more confident the researcher can be in rejecting it. The significance level (α) represents the maximum acceptable probability of a Type I error, which is the decision to reject the null hypothesis when it is actually true.

Related terms

Null Hypothesis: The null hypothesis, denoted as $H_0$, is the statement being tested and is assumed to be true unless there is strong evidence to the contrary.

Alternative Hypothesis: The alternative hypothesis, denoted as $H_a$ or $H_1$, is the statement that is accepted if the null hypothesis is rejected.

Critical Region: The critical region, also known as the rejection region, is the set of values of the test statistic that lead to the rejection of the null hypothesis.