Mastering chi-square goodness of fit
The Chi-square goodness of fit test is a non-parametric statistical test used to determine the extent to which observed values deviate from expected values. This test enables the comparison of an observed sample distribution with an expected probability distribution, assessing how well theoretical distributions such as normal, binomial, or Poisson fit the empirical data. By dividing sample data into intervals and comparing the number of data points falling into each interval with the expected values, one can ascertain the goodness of fit.
Procedure for Conducting the Chi-Square Goodness of Fit Test:
Formulating the Hypotheses
Null Hypothesis: The null hypothesis for the Chi-Square goodness of fit test posits that there is no significant difference between the observed and expected values.
Alternative Hypothesis: Conversely, the alternative hypothesis assumes a significant difference between the observed and expected values.
Calculating the Chi-Square Statistic
The Chi-Square goodness of fit test computes the value of the Chi-Square statistic using the following formula:
Chi-Square = Σ [(O – E)^2 / E]
Where: O represents the observed value,
E represents the expected value.
Determining Degrees of Freedom
The degrees of freedom for the Chi-Square goodness of fit test depend on the distribution of the sample.
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Interpretation of Chi-square goodness of Fit
The process of hypothesis testing in the Chi-Square goodness of fit test is similar to other tests such as t-tests and ANOVA. The calculated Chi-Square statistic is compared against the critical value from the Chi-Square distribution table.
If the calculated value exceeds the critical value, the null hypothesis is rejected, indicating a significant difference between the observed and expected frequencies.
Conversely, if the calculated value is lower than the critical value, we fail to reject the null hypothesis, implying no significant difference between the observed and expected values.
By employing the Chi-Square goodness of fit test, researchers and statisticians can evaluate the conformity of observed data to expected distributions. This test provides valuable insights into the statistical significance of deviations, facilitating informed conclusions about the fit between empirical and theoretical distributions.
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