Finding the critical value is a crucial step in hypothesis testing, allowing you to determine whether to reject or fail to reject the null hypothesis. Here's a breakdown of how to find it, based on your provided references:
Steps to Find the Critical Value
The process involves a few key steps:
-
Determine the Alpha Level (α):
- The alpha level, denoted as α, represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Common alpha levels are 0.05 (5%) and 0.01 (1%).
-
Calculate the Area for Your Test:
- Two-Tailed Test: If you are conducting a two-tailed test (where you are interested in whether the sample mean is different in either direction from the population mean), subtract the alpha level directly from 1.
- Formula for Two-Tailed Test Area:
Area = 1 - α - One-Tailed Test: If you are conducting a one-tailed test (where you are interested in whether the sample mean is greater than or less than the population mean), subtract the alpha level from 0.5.
- Formula for One-Tailed Test Area:
Area = 0.5 - α
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Look Up the Critical Value:
- Once you've calculated the relevant area, use the z-distribution table (also known as the standard normal table) to find the corresponding z-score. This z-score represents the critical value.
- The z-distribution table provides the area to the left of a given z-score. You'll need to locate the area you calculated in step 2 and then find the corresponding z-score.
Example
Let’s illustrate with an example. Suppose you have a two-tailed test with an alpha level of 0.05:
- Step 1: α = 0.05
- Step 2: Area = 1 - 0.05 = 0.95. Now we need to find the areas on each tail which means dividing the alpha by two (0.05/2 =0.025). Therefore, the area in each tail = 0.025 and the area to the left of the critical value is 0.975 (1-0.025=0.975).
- Step 3: Locate 0.975 on a z-table which will correspond to the z-score or critical value of 1.96.
Summary
| Step | Description |
|---|---|
| 1. Find Alpha Level (α) | Determine the acceptable risk of a Type I error (e.g., 0.05, 0.01). |
| 2. Calculate Test Area | Subtract α from 1 for a two-tailed test, or subtract α from 0.5 for a one-tailed test. |
| 3. Look Up Critical Value | Use the z-distribution table to find the z-score corresponding to the calculated area. This is your critical value. |
By following these steps, you can accurately find the critical value, an important element in your hypothesis testing.
