Significance Level to Critical Value Calculator with N

This calculator helps you find the critical value for a given significance level (α) and sample size (n). Critical values are essential in hypothesis testing to determine whether to reject the null hypothesis.

What is Significance Level?

The significance level (α) is the probability of rejecting the null hypothesis when it is true. Common significance levels are 0.05 (5%) and 0.01 (1%). A lower significance level means stricter criteria for rejecting the null hypothesis.

For example, if α = 0.05, there's a 5% chance of incorrectly rejecting a true null hypothesis. This is known as a Type I error.

How to Find Critical Value

The critical value depends on:

The significance level (α)
The sample size (n)
The type of distribution (t-distribution or z-distribution)
The tails of the test (one-tailed or two-tailed)

For small sample sizes (n < 30), use the t-distribution. For larger samples, the z-distribution is appropriate.

Note: The critical value is the threshold value that the test statistic must exceed to reject the null hypothesis.

Critical Value Formula

The formula for the critical value depends on the distribution:

For t-distribution: Critical Value = t_{α/2, n-1} (for two-tailed test) Critical Value = t_{α, n-1} (for one-tailed test) For z-distribution: Critical Value = z_α/2 (for two-tailed test) Critical Value = z_α (for one-tailed test)

Where:

α is the significance level
n is the sample size
t and z are the critical values from the t-distribution and standard normal distribution tables

Worked Example

Let's find the critical value for α = 0.05, n = 20, two-tailed test using t-distribution.

Calculate degrees of freedom: df = n - 1 = 19
Find the critical value from t-distribution table for α/2 = 0.025 and df = 19
The critical value is approximately 2.093

This means if your test statistic is greater than 2.093 or less than -2.093, you would reject the null hypothesis at the 0.05 significance level.

FAQ

What is the difference between significance level and critical value?: The significance level (α) is the threshold probability for rejecting the null hypothesis. The critical value is the threshold value that the test statistic must exceed to reject the null hypothesis at that significance level.
When should I use t-distribution vs. z-distribution?: Use t-distribution for small sample sizes (n < 30) when the population standard deviation is unknown. Use z-distribution for large samples or when the population standard deviation is known.
What does a one-tailed vs. two-tailed test mean?: A one-tailed test looks for significant differences in one direction only. A two-tailed test looks for significant differences in either direction. The critical value is different for each type of test.
How do I interpret the critical value?: The critical value is the threshold that your test statistic must exceed to reject the null hypothesis. If your test statistic is more extreme than the critical value, you can reject the null hypothesis at your chosen significance level.