Chi-Square Goodness of Fit Test

Chi-Square Goodness of Fit Test

• The Chi-Square Goodness of Fit test is a statistical method used to determine if a set of observed frequencies fits a specific theoretical distribution.
• The test is used to determine if there is a significant difference between the observed frequency distribution and the theoretical distribution.
• If there is no significant difference, it is said that the observed frequencies fit the expected frequencies.

Hypotheses

• The null hypothesis (HO) states that the observed frequencies fit the expected frequencies.
• The alternative hypothesis (H1) states that the observed frequencies do not fit the expected frequencies.
• Hypotheses must be stated in a relevant manner for each specific problem.

Assumptions

• The data must be obtained from a random sample.
• The expected frequency for each category must be 5 or more.

Chi-Squared Test Formula

• The chi-squared test statistic is calculated as: χ² = ∑[(Oi - Ei)² / Ei], where Oi is the observed frequency and Ei is the expected frequency.

Degrees of Freedom

• The degrees of freedom (df) is calculated as: df = n - 1, where n is the number of observations.

Critical Value

• The critical value is obtained using the Google Sheets formula: =CHISQ.INV.RT(α, df), where α is the significance level and df is the degrees of freedom.

P-Value

• The P-value is obtained using the Google Sheets formula: =CHISQ.DIST.RT(x², df), where x² is the calculated chi-squared statistic and df is the degrees of freedom.

Calculating Expected Frequencies

Equal Expected Frequencies

• When all expected frequencies are equal, the expected frequency (E) is calculated as: E = n/k, where n is the total number of observations and k is the number of categories.

Unequal Expected Frequencies

• When the expected frequencies are not equal, the expected frequency (E) is calculated as: E = n * p, where n is the total number of observations and p is the probability for that category.