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Chi-square

\(\chi^2\) Proportion \(\chi^2\) Goodness Fit test \(\chi^2\) Tes of Independence

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Chi-Square Calculator for Comparing Two or More Proportions

Action	Category			Proportion

Significance Level (α):

Tips about Hypothesis testing for Mean when σ known:

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How to use

• Determine the number of catagories How many catagories you have, if you have above two catagories then you can click the add catagory button. If yow want to remove the unwanted catagory then you can click the remove action button respect to the catagory
• Change the name (optional): You can customize the name of your catagories and the success and failure factors as you want.
• Inputs Fill all the number of success and failure in each catagories, and enter the correct signifcance lable
• Calculate Click the calculate button to see the result of your case

Definations

• The Chi-Square Test for Two or More Proportions is used to determine whether there is a significant association between categorical variables in a contingency table. This test compares observed frequencies in each category to expected frequencies to assess whether deviations from expectation are due to chance or indicate a relationship between the variables.

General Steps

1. State Hypotheses:
   • Null Hypothesis (\(H_0\)): claimed that all proportions are equal
   • Alternative Hypothesis (\(H_1\)): assume that at least one of the proportion is different
   So based on this the hypothesis statement would like to be: \[H_0: {p_1} = {p_2} = {p_3} = ... = {p_k}\]\[H_1: at least- one -p- is- different\]
2. Specify the signifcance level \(\alpha\):
   • The common signifcance levels are 10$, 5%, and 1%
   • \(\alpha\) determine the widith of the rejection region
   Note; The \(\chi^2\) rejection region always placed on the right side of the distribution.
3. Compute the \(\chi^2\) test statistics:
   • First compute the expected frequency (\(f_e\)):
   • Expected Frequency always computed by assuming the null hypothesis is true:
   The \(f_e\) for each success cell in the contingency table is calculated as follows: \[f_e = {p_o} \times {n}\] and, the \(f_o\) for each failure cell is computed as \[f_e = (1- {p_o}) \times {n}\] where, \(f_e\) is the expected frequency for all cell (success and failure), and n is the total number observaition for each catagory.
   
   • Then, compute the chi-square statistics: \[\chi^2 = \sum \frac{(f_o - f_e)^2}{f_e}\]
4. Determine the critical value of \(\chi_\alpha ^2\):
   • The \(\chi_\alpha ^2\) will determin the degree of freedom. df is computed as follows
   \[df = (k - 1)\] where,
   \(df\) = degree of freedom
   \(k\) = Number of categories.
5. Decision and rejection region \(\chi_\alpha ^2\):
   Compare the computed Chi-Square statistic to the critical value from the Chi-Square distribution table with the appropriate degrees of freedom and significance level (𝛼),
   
   
   • If \(\chi^2 > \chi_\alpha ^2\), then reject \(H_0\) or
   • If \(\chi^2 ≤ \chi_\alpha ^2\), then do not reject the \(H_0\)
6. Draw Conclusion:
   • Consider the context when interpreting the results of the chi-square hypothesis testing

Common Errors:

1. Small Expected Frequencies:
   • If expected frequencies in any cell are less than 5, the Chi-Square approximation may not be valid. Use Fisher's Exact Test for small samples or combine categories to meet the assumption.
2. Incorrect Hypothesis Testing:
   • Misunderstanding whether the test is for homogeneity or independence can lead to incorrect conclusions. Ensure you use the appropriate test based on the research question.
3. Ignoring Assumptions:
   • IThe Chi-Square test assumes a large sample size and independent observations. Violating these assumptions can invalidate results.
4. Overlooking Table Size:
   • Ensure that the table size is appropriate for the Chi-Square test. For large tables with many categories, ensure the sample size is sufficiently large to meet the test assumptions.
5. Misinterpretation of Results:
   • The Chi-Square test indicates whether an association exists but does not specify the nature or direction of the association. Further analysis may be needed to understand the relationship.

Additional Tips

1. Verify Assumptions:
   • Check that the sample size is large enough and that expected frequencies are sufficiently high to use the Chi-Square test. For small samples, consider alternatives like Fisher's Exact Test.
2. Use Exact Tests for Small Samples:
   • For small sample sizes or tables with low expected frequencies, use Fisher’s Exact Test or Monte Carlo simulations instead of the Chi-Square test.
3. Analyze Residuals:
   • Examine standardized residuals to understand which cells contribute most to the Chi-Square statistic. This can provide insights into specific associations or patterns.
4. Combine Categories Wisely:
   • If some categories have very low frequencies, consider combining similar categories to meet the expected frequency requirement while maintaining meaningful analysis.
5. Check Independence:
   • Ensure that observations are independent. The Chi-Square test assumes that each observation is independent of others.