# 17.3: Testing Distribution Demonstration

Skills to Develop

- Develop a basic understanding of the properties of a sampling distribution based on the properties of the population

### Instructions

In this simulation, \(100\) numbers are either sampled from a normal distribution or a uniform distribution. The frequencies in each of \(10\) "bins" is then displayed in the "observed" column. The expected frequencies based on both a normal distribution (on the left) or a uniform distribution (on the right) are shown just to the left of the observed frequencies. For each bin the value \(\frac{(E-O)^2}{E}\) is computed where \(E\) is the expected frequency and \(O\) is the observed frequency. The sum of these quantities is the value of Chi Square shown at the bottom.

- The default is to sample from a normal distribution. Click the sample button and \(100\) values will be sampled from a normal distribution. Compare the observed values in the "From a Normal Distribution" section to the expected values. Is the Chi Square test significant at the \(0.05\) level? How often would you expect it to be significant.
- Compare the observed frequencies from the "From a Uniform Distribution" section to the expected frequencies. In what way are they different? Is the difference significant? If so, then the null hypothesis that the numbers were sampled from a uniform distribution could be rejected. Of course, in this simulation, you know where the numbers were sampled so you know the null hypothesis is false.
- Simulate several experiments and see if the significance for the test of a uniform distribution is always significant.
- Make the actual distribution a uniform distribution and do more simulated experiments. Compare the results to when the actual distribution was normal.

### Illustrated Instructions

This simulation samples \(100\) values from a normal or uniform distribution and calculates the the Chi Square value. As can be seen from the image below, the simulation begins by displaying a table with expected frequencies.

**Figure \(\PageIndex{1}\): **Testing Distribution Simulation

Clicking on the "Sample" button, samples \(100\) values from a normal distribution (by default) and displays the observed frequencies as well as the results of the Chi Square tests.

**Figure \(\PageIndex{2}\): **Testing Distribution Simulation

### Contributor

Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.