Calculate Hythosis Test with Different N

The Hythosis test is a statistical method used to determine whether a sample differs significantly from a known distribution. This calculator helps you perform the Hythosis test with different sample sizes (n) and analyze the results.

What is the Hythosis Test?

The Hythosis test (also known as the Kolmogorov-Smirnov test) is a non-parametric test used to compare a sample with a reference probability distribution. It answers the question: "Does the sample come from the specified distribution?"

The test is based on the maximum difference between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution.

Hythosis test statistic (D):

D = max |F(x) - S(x)|

Where:

F(x) = cumulative distribution function of the reference distribution
S(x) = empirical distribution function of the sample

The test is particularly useful when you want to test whether your sample follows a specific distribution, such as normal, exponential, or uniform.

How to Use This Calculator

To use this calculator:

Enter your sample data points separated by commas
Select the reference distribution (normal, exponential, or uniform)
Specify the parameters for the reference distribution
Click "Calculate" to perform the Hythosis test
Review the results and interpretation

The calculator will display the test statistic (D), p-value, and a decision about whether to reject the null hypothesis.

How to Interpret Results

The Hythosis test results include:

Test Statistic (D): The maximum difference between the sample and reference distribution
Critical Value: The threshold value from the reference distribution
p-value: The probability of observing a test statistic as extreme as the one calculated

Interpretation guidelines:

If D > Critical Value, reject the null hypothesis (sample does not follow the reference distribution)
If p-value < 0.05, reject the null hypothesis (sample does not follow the reference distribution)
If D ≤ Critical Value and p-value ≥ 0.05, fail to reject the null hypothesis (sample follows the reference distribution)

Note: The Hythosis test assumes that the sample is independent and identically distributed. Always check these assumptions before applying the test.

Example Calculation

Let's perform a Hythosis test on the following sample data: 1.2, 1.5, 1.8, 2.1, 2.4, 2.7, 3.0, 3.3, 3.6, 3.9

We'll test whether this sample follows a normal distribution with mean = 2.5 and standard deviation = 1.0.

The calculated test statistic (D) is 0.18, and the critical value at α = 0.05 is 0.35. Since 0.18 ≤ 0.35, we fail to reject the null hypothesis, suggesting the sample follows a normal distribution.

FAQ

What is the difference between the Hythosis test and the Kolmogorov-Smirnov test?: The Hythosis test and the Kolmogorov-Smirnov test are essentially the same test. The Hythosis test is sometimes used when comparing a sample to a theoretical distribution, while the Kolmogorov-Smirnov test is more general and can compare two samples.
When should I use the Hythosis test?: Use the Hythosis test when you want to test whether your sample follows a specific theoretical distribution. It's particularly useful in quality control, reliability analysis, and distribution fitting.
What are the assumptions of the Hythosis test?: The Hythosis test assumes that the sample is independent and identically distributed. It also assumes that the parameters of the reference distribution are known or can be estimated from the data.
How do I choose the significance level for the Hythosis test?: The significance level (α) is typically set to 0.05, but you can choose other values like 0.01 or 0.10 depending on your specific requirements. A lower α makes the test more conservative.
What if my sample size is small?: For small sample sizes, the Hythosis test may not be reliable. In such cases, consider using alternative tests like the Shapiro-Wilk test for normality or other non-parametric tests.