How do the critical values for a two-tailed test change as alpha decreases?

Verified step by step guidance

1. Understand the concept of alpha (α): Alpha represents the significance level of a hypothesis test, which is the probability of rejecting the null hypothesis when it is actually true. Common values for alpha are 0.05, 0.01, etc.
2. Recognize the relationship between alpha and critical values: In a two-tailed test, the critical values define the boundaries of the rejection region. As alpha decreases, the rejection region becomes smaller, meaning the critical values move further away from the center of the distribution.
3. Recall the distribution used: For most hypothesis tests, the critical values are determined using a standard normal (Z) distribution or a t-distribution, depending on the sample size and whether the population standard deviation is known.
4. Determine the new critical values: To find the critical values for a smaller alpha, calculate the z-scores or t-scores that correspond to the new alpha level. For a two-tailed test, divide alpha by 2 to account for both tails, and then find the values that leave the remaining probability in the center of the distribution.
5. Interpret the change: As alpha decreases, the critical values become more extreme (larger in magnitude for positive and negative values), indicating that it becomes harder to reject the null hypothesis because the rejection region is smaller.

Key Concepts

Critical Values

Critical values are the threshold points that define the boundaries for rejecting the null hypothesis in hypothesis testing. In a two-tailed test, critical values are determined based on the significance level (alpha) and the distribution of the test statistic. They mark the points beyond which the test statistic is considered extreme enough to reject the null hypothesis.

Alpha Level (α)

The alpha level, often denoted as α, represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common alpha levels are 0.05, 0.01, and 0.10. As alpha decreases, the criteria for rejecting the null hypothesis become stricter, leading to a higher threshold for critical values.

Two-Tailed Test

A two-tailed test is a statistical test that evaluates whether a sample mean is significantly different from a population mean in either direction (greater or less). This type of test is used when the alternative hypothesis does not specify a direction of the effect. As alpha decreases, the critical values move further from the mean, reflecting the need for stronger evidence to reject the null hypothesis.