If a chi-square test produces a chi-square value of 7.83 with 4 degrees of freedom,

Is the result sufficient to reject the chance hypothesis?

Verified step by step guidance

1. Identify the null hypothesis: The observed data fits the expected distribution (chance hypothesis).
2. Determine the degrees of freedom: In this case, it is given as 4.
3. Find the critical value for chi-square at the desired significance level (commonly 0.05) with 4 degrees of freedom using a chi-square distribution table.
4. Compare the calculated chi-square value (7.83) to the critical value from the table.
5. If the calculated chi-square value is greater than the critical value, reject the null hypothesis; otherwise, do not reject it.

Key Concepts

Chi-Square Test

The chi-square test is a statistical method used to determine if there is a significant association between categorical variables. It compares the observed frequencies in each category to the expected frequencies, which are calculated under the assumption that the null hypothesis is true. A higher chi-square value indicates a greater discrepancy between observed and expected values.

Degrees of Freedom

Degrees of freedom (df) in a chi-square test refer to the number of independent values that can vary in the analysis. It is calculated as the number of categories minus one for goodness-of-fit tests or as the product of (rows - 1) and (columns - 1) for contingency tables. In this case, with 4 degrees of freedom, it indicates the complexity of the data being analyzed.

Critical Value

The critical value in a chi-square test is the threshold that the chi-square statistic must exceed to reject the null hypothesis. This value is determined based on the chosen significance level (commonly 0.05) and the degrees of freedom. For 4 degrees of freedom, the critical value is approximately 9.488, meaning a chi-square value of 7.83 would not be sufficient to reject the chance hypothesis.