The chi-square goodness-of-fit test helps determine if categorical data matches a specific distribution. It compares observed frequencies to expected ones, calculating a test statistic to assess the fit between sample data and hypothesized population distribution.
Interpreting results involves comparing the test statistic to a critical value or using the p-value . This helps decide whether to reject or fail to reject the null hypothesis, indicating if the sample data significantly differs from the expected distribution.
Chi-Square Goodness-of-Fit Test
Purpose of chi-square goodness-of-fit test
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Determines if a sample of categorical data (colors, types) comes from a population with a specific distribution
Compares observed frequencies of categories in the sample to expected frequencies based on the hypothesized distribution
Applicable when data is categorical or nominal, sample is randomly selected, and expected frequency of each category is at least 5
Calculation of chi-square test statistic
Calculate expected frequencies for each category by multiplying hypothesized probability of each category by total sample size
Chi-square test statistic calculated using formula χ 2 = ∑ i = 1 k ( O i − E i ) 2 E i \chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i} χ 2 = ∑ i = 1 k E i ( O i − E i ) 2
O i O_i O i represents observed frequency for category i i i (actual count)
E i E_i E i represents expected frequency for category i i i (calculated based on hypothesized distribution)
k k k represents number of categories (options, choices)
Degrees of freedom for chi-square tests
Degrees of freedom for chi-square goodness-of-fit test calculated as [ d f ] ( h t t p s : / / w w w . f i v e a b l e K e y T e r m : d f ) = k − 1 [df](https://www.fiveableKeyTerm:df) = k - 1 [ df ] ( h ttp s : // www . f i v e ab l eKey T er m : df ) = k − 1
k k k represents number of categories
Critical value determined by significance level (α \alpha α ), degrees of freedom
Found using chi-square distribution table or statistical software (Excel, R)
Interpretation of chi-square test results
Compare calculated chi-square test statistic to critical value
If test statistic greater than critical value, reject null hypothesis
Suggests sample data does not follow hypothesized distribution (significantly different)
If test statistic less than or equal to critical value, fail to reject null hypothesis
Suggests sample data consistent with hypothesized distribution (not significantly different)
P-value also used to make decision
If p-value less than significance level (α \alpha α ), reject null hypothesis
If p-value greater than or equal to significance level (α \alpha α ), fail to reject null hypothesis