Goodness-of-Fit Test

Definition

The goodness-of-fit test is a statistical hypothesis test used to determine whether a sample of data fits a particular probability distribution. It evaluates how well the observed data matches the expected data under a specified distribution model.

5 Must Know Facts For Your Next Test

1. The goodness-of-fit test uses the chi-square statistic to measure the discrepancy between the observed and expected frequencies.
2. The test evaluates whether the observed frequencies in each category are significantly different from the expected frequencies under the null hypothesis.
3. The degrees of freedom for the chi-square test are calculated as the number of categories minus the number of parameters estimated from the data.
4. The p-value from the goodness-of-fit test represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.
5. The goodness-of-fit test is commonly used to assess the fit of a probability distribution, such as the normal, Poisson, or binomial distribution, to a set of observed data.

Review Questions

• Explain how the goodness-of-fit test is used to evaluate the fit of a probability distribution to a set of observed data.
  • The goodness-of-fit test is used to determine whether a sample of data follows a particular probability distribution. It compares the observed frequencies in each category to the expected frequencies under the null hypothesis, which states that the data follows the specified distribution. The test calculates a chi-square statistic that measures the discrepancy between the observed and expected frequencies. If the p-value from the test is less than the chosen significance level, the null hypothesis is rejected, indicating that the data does not fit the proposed distribution.
• Describe the relationship between the goodness-of-fit test and the chi-square distribution.
  • The goodness-of-fit test utilizes the chi-square distribution to evaluate the statistical significance of the difference between the observed and expected frequencies. The chi-square statistic calculated in the test follows a chi-square distribution with degrees of freedom equal to the number of categories minus the number of parameters estimated from the data. The p-value from the test is determined by comparing the calculated chi-square statistic to the critical values of the chi-square distribution, which allows for the assessment of whether the observed data fits the proposed probability distribution.
• Analyze the role of the null and alternative hypotheses in the goodness-of-fit test and explain how the test results are interpreted.
  • In the goodness-of-fit test, the null hypothesis (H₀) states that the observed data follows the specified probability distribution, while the alternative hypothesis (H₁) states that the data does not follow the proposed distribution. The test calculates a chi-square statistic that measures the discrepancy between the observed and expected frequencies. If the p-value from the test is less than the chosen significance level, the null hypothesis is rejected, indicating that the data does not fit the proposed distribution. Conversely, if the p-value is greater than the significance level, the null hypothesis is not rejected, suggesting that the data is consistent with the specified probability distribution.

Related terms

Chi-Square Distribution: The chi-square distribution is a probability distribution used in hypothesis testing, particularly in goodness-of-fit tests, to determine if a sample data fits a specified distribution.

Hypothesis Testing: Hypothesis testing is a statistical method used to determine whether a claim or hypothesis about a population parameter is likely to be true or false based on sample data.

Null Hypothesis: The null hypothesis is a statement that there is no significant difference between a sample statistic and a population parameter or that there is no significant difference between two samples.