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how are the expected counts calculated when a chi-square goodness-of-fit test is conducted?
how are the expected counts calculated when a chi-square goodness-of-fit test is conducted?
Answer
When conducting a chi-square goodness-of-fit test, the expected counts are calculated to determine how well the observed data fits a specific theoretical distribution. Here’s how they are typically calculated:
- Identify the Null Hypothesis: The null hypothesis for a chi-square goodness-of-fit test generally states that the observed frequencies (counts) follow a specific theoretical distribution (e.g., uniform, normal, etc.).
- Determine the Total Number of Observations: Calculate the total number of observations in your dataset. Let this total be denoted as NN.
- Identify the Expected Proportions: Determine the theoretical proportions for each category or outcome under the null hypothesis. These proportions represent the expected distribution if the null hypothesis is true.
\text{For each category, multiply the total number of observations } N \text{ by the theoretical proportion for that category. Mathematically, if } p_i \text{ is the theoretical proportion for the } i\text{-th category, the expected count } E_i \text{ for that category is calculated as:}
E_i = N \times p_i
\text{Formulate the Chi-Square Statistic:}
\text{After calculating the expected counts, the chi-square statistic is computed to compare the observed counts to these expected counts. The formula for the chi-square statistic } \chi^2 \text{ is:}
\chi^2 = \sum_{i} \frac{(O_i - E_i)^2}{E_i}
\text{where } O_i \text{ represents the observed count for category } i \text{, and } E_i \text{ represents the expected count for that category.}
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