8.9
Expected Frequencies in Goodness-of-Fit Tests
A goodness-of-fit test is conducted to determine whether the observed frequency values are statistically similar to the frequencies expected for the d…
The goodness-of-fit test is performed to see if the observed results are statistically similar to the expected ones.
The goodness-of-fit test uses a null hypothesis - which assumes that the distribution is as claimed, and a contradicting alternative hypothesis.
If all the expected frequencies in a distribution are equal, such as when predicting the color of traffic lights, the expected frequency - E is expressed as the ratio of the total number of operations, six and k, the number of categories, three.
However, for unequal expected frequencies such as finding women with different hair color, E is calculated by multiplying the sum of the observed frequencies by the probability for each category.
In the earlier example of child birth, the expected and observed frequencies are used to calculate the chi-square value. Using the chi-square table, one can discover whether the difference between the expected and observed frequencies is statistically significant.
If this difference and the test statistic is large, with a small P-value, this means that the test statistic falls in the critical region. Hence, the null hypothesis is rejected. If not, fail to reject the null hypothesis.
View the full transcript and gain access to JoVE Core videos
Q1: What is the purpose of a goodness-of-fit test?
A goodness-of-fit test determines whether observed frequency values are statistically similar to the frequencies expected for a dataset. It uses a null hypothesis assuming the distribution is as claimed and an alternative hypothesis to test this assumption. By comparing observed and expected frequencies, researchers can determine if differences are statistically significant.
Q2: How do you calculate expected frequency when all categories are equally likely?
When expected frequencies are equal, such as predicting any number on a die, the expected frequency equals the total number of observations divided by the number of categories. For example, rolling a die has six possible outcomes, so each number's expected frequency is one-sixth. This ratio applies whenever each category has an identical probability of occurring.
Q3: How is expected frequency calculated for unequal probability distributions?
For unequal expected frequencies, multiply the total number of observations by the probability for each category. This method applies when categories have different probabilities, such as finding women with different hair colors. The determination of expected frequency depends on knowing the probability associated with each specific category in your dataset.
Q4: What does a small p-value indicate in a goodness-of-fit test?
A small p-value indicates that the difference between expected and observed frequencies is statistically significant, meaning the test statistic falls in the critical region. This leads to rejecting the null hypothesis, suggesting the observed data does not match the expected distribution. Conversely, a large p-value means you fail to reject the null hypothesis.
Q5: How do you use the chi-square table in a goodness-of-fit test?
The chi-square table helps determine whether the difference between expected and observed frequencies is statistically significant. After calculating the chi-square value from your data, you reference the table to find the critical value. If your calculated chi-square exceeds the critical value, the result is statistically significant and you reject the null hypothesis.
Q6: What are the null and alternative hypotheses in a goodness-of-fit test?
The null hypothesis assumes the distribution is as claimed, while the alternative hypothesis contradicts this assumption. These competing hypotheses form the foundation of the goodness-of-fit test. The test determines which hypothesis is supported by comparing observed frequencies to expected frequencies and calculating whether their difference is statistically significant.
Q7: When should you reject the null hypothesis in a goodness-of-fit test?
Reject the null hypothesis when the test statistic is large and the p-value is small, indicating the test statistic falls in the critical region. This means the observed data significantly differs from the expected distribution. If the test statistic is not large enough or the p-value is large, you fail to reject the null hypothesis instead.