Find Rejection Region Calculator – Critical Values

Find Rejection Region Calculator

Enter the details to find the rejection region for your hypothesis test.

Significance Level (α):

E.g., 0.05 for 5% significance. Must be between 0.001 and 0.999.

Test Type:

Distribution:

Degrees of Freedom (df):

For t-distribution, usually sample size minus 1 (n-1). Must be ≥ 1.

What is a Find Rejection Region Calculator?

A Find Rejection Region Calculator is a tool used in hypothesis testing to determine the range of values for a test statistic that would lead to the rejection of the null hypothesis (H₀). The rejection region, also known as the critical region, is defined by the critical value(s), which are boundary points derived from the chosen significance level (α) and the distribution of the test statistic (e.g., Z-distribution or t-distribution).

Researchers, statisticians, data analysts, and students use a Find Rejection Region Calculator to make objective decisions about their hypotheses based on sample data. If the calculated test statistic from the data falls within the rejection region, the null hypothesis is rejected in favor of the alternative hypothesis (H₁).

Common misconceptions include thinking that the rejection region tells us the probability of the null hypothesis being true or false; it only tells us whether the observed data is statistically significant at the chosen α level, warranting rejection of H₀ if it falls in the region.

Find Rejection Region Formula and Mathematical Explanation

The rejection region is determined by comparing the test statistic to the critical value(s). The critical value(s) depend on:

The significance level (α): The probability of making a Type I error (rejecting a true null hypothesis).
The type of test (left-tailed, right-tailed, or two-tailed).
The probability distribution of the test statistic (e.g., normal/Z, t).
Degrees of freedom (df), if using the t-distribution.

For a Z-test (Normal Distribution):

Right-tailed test: Reject H₀ if Z_statistic > Z_α (critical value)
Left-tailed test: Reject H₀ if Z_statistic < -Z_α (critical value)
Two-tailed test: Reject H₀ if |Z_statistic| > Z_α/2 (or Z_statistic > Z_α/2 or Z_statistic < -Z_α/2)

For a t-test (t-Distribution):

Right-tailed test: Reject H₀ if t_statistic > t_{α, df}
Left-tailed test: Reject H₀ if t_statistic < -t_{α, df}
Two-tailed test: Reject H₀ if |t_statistic| > t_{α/2, df}

The Find Rejection Region Calculator helps you find these critical values (Z_α, Z_α/2, t_{α, df}, t_{α/2, df}) and define the region(s).

Variables Table

Variable	Meaning	Unit	Typical Range
α	Significance Level	Probability	0.001 to 0.10 (commonly 0.05, 0.01, 0.10)
Z_critical	Critical Z-value	Standard Deviations	±1 to ±3.5 (depends on α and test type)
t_critical	Critical t-value	–	Depends on α, df, and test type
df	Degrees of Freedom	Integer	1 to ∞ (for t-distribution)

Variables used in finding the rejection region.

Practical Examples (Real-World Use Cases)

Example 1: Two-tailed Z-test

A company wants to test if the average fill volume of their soda bottles is 500ml. They take a large sample and calculate a Z-statistic of 2.15. They choose a significance level (α) of 0.05. Using a Find Rejection Region Calculator for a two-tailed Z-test with α = 0.05, the critical values are approximately ±1.96. The rejection region is Z < -1.96 or Z > 1.96. Since 2.15 > 1.96, the Z-statistic falls in the rejection region, and they reject the null hypothesis that the average volume is 500ml.

Example 2: Left-tailed t-test

A researcher is testing if a new drug lowers blood pressure more effectively than an old one. The null hypothesis is that it does not. They use a sample of 15 patients (df = 14) and get a t-statistic of -2.50. They set α = 0.01 for a left-tailed test. Using a Find Rejection Region Calculator for a left-tailed t-test with α = 0.01 and df = 14, the critical t-value is around -2.624. The rejection region is t < -2.624. Since -2.50 is not less than -2.624, the t-statistic does not fall in the rejection region, and they fail to reject the null hypothesis at the 0.01 significance level.

How to Use This Find Rejection Region Calculator

Enter Significance Level (α): Input the desired significance level, typically between 0.01 and 0.10.
Select Test Type: Choose whether you are performing a two-tailed, left-tailed, or right-tailed test based on your alternative hypothesis.
Select Distribution: Choose 'Z-distribution' if you know the population standard deviation or have a large sample size (often n > 30), or 't-distribution' if you are working with a small sample and the population standard deviation is unknown.
Enter Degrees of Freedom (df): If you selected 't-distribution', enter the degrees of freedom, which is usually the sample size minus 1 (n-1).
Calculate: The calculator automatically updates, or click "Calculate" if needed.
Read Results: The calculator will show the critical value(s) and define the rejection region(s). The chart visually represents this.

Based on the output, if your calculated test statistic from your data falls within the displayed rejection region, you reject the null hypothesis.

Key Factors That Affect Find Rejection Region Calculator Results

Significance Level (α): A smaller α (e.g., 0.01) leads to more extreme critical values and a smaller rejection region, making it harder to reject the null hypothesis. It reduces the chance of a Type I error but increases the chance of a Type II error.
Test Type (One-tailed vs. Two-tailed): A one-tailed test allocates all of α to one tail, making the critical value less extreme (closer to zero) than the critical values for a two-tailed test with the same α (where α is split into α/2 for each tail). This makes it easier to reject H₀ in the specified direction.
Choice of Distribution (Z vs. t): The t-distribution has heavier tails than the Z-distribution, especially for small degrees of freedom. This means t-critical values are more extreme (further from zero) than Z-critical values for the same α and tail, making the rejection region start further out.
Degrees of Freedom (df) for t-distribution: As df increases, the t-distribution approaches the Z-distribution, and the t-critical values get closer to Z-critical values. Smaller df values lead to more spread-out t-distributions and more extreme critical values.
Sample Size (indirectly): Sample size affects the degrees of freedom (for t-tests) and the standard error (used to calculate the test statistic, though not directly the critical value for Z/t given alpha), and the decision to use Z or t. Larger samples often allow the use of Z or lead to higher df for t, affecting critical values.
Underlying Data Distribution Assumption: The Z-test assumes data from a normal distribution or a large enough sample for the Central Limit Theorem to apply. The t-test assumes data from a normal distribution, especially for small samples. Violations can affect the validity of the calculated rejection region.

Frequently Asked Questions (FAQ)

What is a rejection region?: The rejection region (or critical region) is the set of values for the test statistic for which the null hypothesis is rejected in a hypothesis test.
What is a critical value?: A critical value is the point on the scale of the test statistic beyond which we reject the null hypothesis. It marks the boundary of the rejection region. The Find Rejection Region Calculator helps you find these.
How is the significance level (α) related to the rejection region?: The significance level α determines the size of the rejection region. For example, in a two-tailed test, α/2 is the area in each tail that constitutes the rejection region.
When do I use a Z-distribution vs. a t-distribution?: Use the Z-distribution when the population standard deviation is known or when the sample size is large (e.g., n > 30), assuming the Central Limit Theorem applies. Use the t-distribution when the population standard deviation is unknown and the sample size is small (and the sample data is approximately normally distributed). Our Find Rejection Region Calculator supports both.
What are degrees of freedom (df)?: Degrees of freedom refer to the number of independent values or quantities that can be assigned to a statistical distribution. In the context of a t-test, it's typically the sample size minus the number of parameters estimated (e.g., n-1 for a one-sample t-test).
What is a p-value and how does it relate to the rejection region?: The p-value is the probability of observing a test statistic as extreme as or more extreme than the one calculated from the sample data, assuming the null hypothesis is true. If the p-value ≤ α, the test statistic falls in the rejection region, and you reject H₀.
Can the Find Rejection Region Calculator be used for any hypothesis test?: This calculator is specifically for tests involving the Z or t distributions (like one-sample or two-sample mean tests under certain conditions). Other tests (like Chi-square or F-tests) have different distributions and critical values.
What if my test statistic falls exactly on the critical value?: If the test statistic is exactly equal to the critical value, the decision can go either way, although typically the null hypothesis is not rejected unless the statistic falls *strictly* within the rejection region (e.g., > critical value for right-tailed).

Related Tools and Internal Resources

P-Value Calculator: Calculate the p-value from a test statistic to compare against your alpha level.
Confidence Interval Calculator: Find the confidence interval for a mean or proportion.
Sample Size Calculator: Determine the sample size needed for your study.
Guide to Hypothesis Testing: An in-depth article explaining the concepts of hypothesis testing.
Z-Score Calculator: Calculate the Z-score for a given value, mean, and standard deviation.
t-Test Calculator: Perform one-sample or two-sample t-tests.

Results