Critical Value Calculator

The choice of distribution depends on your test statistic and sample characteristics:

• Z-distribution: Use when you have a large sample size (n ≥ 30) and know the population standard deviation, or when testing proportions
• t-distribution: Use when you have a small sample size and don't know the population standard deviation
• Chi-square distribution: Use for tests of independence, goodness-of-fit tests, and tests of variance
• F-distribution: Use for ANOVA tests, comparing variances, and testing overall significance in regression

One-tailed tests are used when you're only interested in detecting an effect in one direction (either positive or negative). They have more statistical power but can only detect effects in the specified direction.

Two-tailed tests are used when you want to detect an effect in either direction. They're more conservative and require stronger evidence to reject the null hypothesis, but they can detect effects in both directions.

The most commonly used significance levels are:

• α = 0.05 (5%): Most common in social sciences and general research
• α = 0.01 (1%): More conservative, used when you want to be very confident in your results
• α = 0.10 (10%): Less conservative, used in exploratory research or when sample sizes are small

Choose based on the consequences of making a Type I error (rejecting a true null hypothesis) in your specific context.

Degrees of freedom (df) represent the number of independent pieces of information in your data:

• For t-tests: df = n - 1 (where n is sample size)
• For chi-square tests: df = (rows - 1) × (columns - 1) for contingency tables, or k - 1 for goodness-of-fit tests
• For F-tests: df1 = numerator degrees of freedom, df2 = denominator degrees of freedom

As degrees of freedom increase, t-distributions approach the normal distribution, and critical values become more stable.

The choice depends on your alternative hypothesis:

• Left-tailed: Use when H₁ states the parameter is less than some value (e.g., μ < μ₀)
• Right-tailed: Use when H₁ states the parameter is greater than some value (e.g., μ > μ₀)
• Two-tailed: Use when H₁ states the parameter is not equal to some value (e.g., μ ≠ μ₀)

Here are some commonly used critical values for Z-tests with α = 0.05:

• Two-tailed test: ±1.96
• Right-tailed test: 1.645
• Left-tailed test: -1.645

For α = 0.01, the two-tailed critical value is ±2.576.

Critical values and p-values are two different approaches to hypothesis testing:

• Critical value approach: Compare your test statistic to the critical value(s)
• P-value approach: Calculate the probability of observing a test statistic as extreme as yours under the null hypothesis

Both approaches lead to the same conclusion: reject H₀ if your test statistic falls in the rejection region OR if your p-value is less than α.