Finding the critical value associated with a percentage (often used as an alpha level in hypothesis testing) involves calculating the critical probability and then using that probability to find the corresponding value from a relevant distribution (e.g., Z-distribution, t-distribution).
Here's a breakdown of the steps:
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Determine the Alpha Level (α): This is the significance level of your test, representing the probability of rejecting the null hypothesis when it's actually true (Type I error). Common values are 0.05, 0.01, and 0.10. This is the percentage referenced in the question.
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Determine the Type of Test (One-tailed or Two-tailed):
- Two-tailed test: You're interested in deviations in either direction from the null hypothesis. Divide the alpha level by 2.
- One-tailed test: You're only interested in deviations in one direction. The alpha level remains as is.
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Calculate the Critical Probability (p*):
- Two-tailed test: p* = 1 - (α / 2)
- One-tailed test: p* = 1 - α
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Find the Critical Value: Use the calculated critical probability (p*) and the appropriate statistical distribution to find the critical value. This usually involves using a statistical table (e.g., Z-table, t-table), a statistical calculator, or software (e.g., R, Python, Excel).
- Z-distribution: Used for large sample sizes or when the population standard deviation is known. Look up the z-score corresponding to the critical probability in a Z-table.
- t-distribution: Used for small sample sizes or when the population standard deviation is unknown. You'll need the degrees of freedom (df = n-1, where n is the sample size) in addition to the critical probability to look up the critical t-value in a t-table.
Example:
Let's say you have an alpha level (α) of 0.05 and are conducting a two-tailed test.
- Alpha Level (α): 0.05
- Type of Test: Two-tailed
- Critical Probability (p*): p* = 1 - (0.05 / 2) = 1 - 0.025 = 0.975
- Find the Critical Value: Assuming you're using a Z-distribution, you would look up 0.975 in a Z-table. The corresponding Z-score (critical value) is approximately 1.96. This means values beyond +/- 1.96 standard deviations from the mean would lead you to reject the null hypothesis.
In summary, finding the critical value of a percentage involves calculating the critical probability based on the alpha level and the type of test, then using that probability and the appropriate statistical distribution to determine the critical value.
