Chi Square Goodness Of Fit Test Example Pdf
In statistics, the Chi-Square Goodness of Fit Test is a tool that allows researchers to determine if a sample data fits or deviates from a theoretical distribution. It is used as a hypothesis testing method to establish the likelihood of an observed distribution, compared to a theoretical one. The test can be applied across a wide variety of fields and subjects, including biology, psychology, sociology, and medicine, among others.
How does the Chi Square Goodness of Fit Test work?
The Chi-Square Goodness of Fit Test is based on the comparison of two probability distributions. This includes an observed frequency distribution, which includes the actual numbers of events that have occurred, and an expected frequency distribution, which includes the expected numbers of events that should occur based on a theoretical or expected distribution.
The Chi Square statistic is then calculated by comparing the difference between the observed and expected frequencies, squared, divided by the expected frequency. This calculation is repeated for all categories or variables, and the sum of these values is used to determine the overall goodness of fit.
If the calculated Chi Square value is greater than the critical value of the distribution, it can be concluded that the hypothesis is not supported, and the observed data distribution is significantly different from the expected theoretical distribution. Conversely, if the calculated Chi Square value is less than the critical value, it can be concluded that the hypothesis is supported, and the observed data distribution is not significantly different from the expected theoretical distribution.
An example application of Chi Square Goodness of Fit Test
To gain a better understanding of the Chi Square Goodness of Fit Test, let's consider an example. Suppose a researcher wants to test whether a die is fair or not. A fair die should have an equal probability of rolling each of the six sides. The researcher rolls the die 100 times, and records the frequencies of each number rolled as follows:
1: 17 times, 2: 15 times, 3: 20 times, 4: 12 times, 5: 18 times, 6: 18 times.
The expected values for each number of rolls are 16.6 (100/6). To determine if the roll of the dice is fair or not, the researcher uses the Chi Square Goodness of Fit Test with a significance level of 0.05.
First, the researcher lists the observed and expected frequencies in a table.
| Number Rolled | Observed Frequency | Expected Frequency | (O - E)^2/E |
|---|---|---|---|
| 1 | 17 | 16.6 | 0.02 |
| 2 | 15 | 16.6 | 0.3 |
| 3 | 20 | 16.6 | 0.79 |
| 4 | 12 | 16.6 | 1.14 |
| 5 | 18 | 16.6 | 0.16 |
| 6 | 18 | 16.6 | 0.16 |
| 2.57 |
Next, the researcher calculates the Chi Square statistic by summing the values in the last column of the table. In this example, Chi Square = 2.57.
Finally, the researcher compares the calculated Chi Square value to the critical value at the desired significance level and degrees of freedom. For a significance level of 0.05 and five degrees of freedom (number of categories - 1), the critical value is 11.07. As the calculated Chi Square value is less than the critical value, the researcher accepts the null hypothesis that the die is fair.
Advantages and limitations of Chi Square Goodness of Fit Test
The Chi Square Goodness of Fit Test is a useful statistical tool with several advantages. Firstly, it is easy to understand and apply, making it accessible to researchers with limited statistical expertise. Additionally, it allows for the testing of multiple categories simultaneously, providing a comprehensive analysis of the data.
However, there are some limitations to the Chi Square Goodness of Fit Test. One major issue is that the test assumes the independence of the data, which may not always be the case in real-world scenarios. Additionally, the test is sensitive to sample size, with larger sample sizes producing more accurate results. Finally, the test is only applicable for nominal or categorical data, and cannot be used for continuous data.
Conclusion
The Chi Square Goodness of Fit Test is a versatile and widely used statistical method, with applications across a variety of fields and subjects. By comparing observed and expected frequencies, researchers can determine the likelihood of an observed distribution, and make inferences about the underlying theoretical distribution. While there are limitations to the test, it remains a valuable tool for hypothesis testing and data analysis, providing insights into the relationships between variables and outcomes.