Final answer:
If the calculated chi-square value is less than the critical value, you would fail to reject the null hypothesis. This decision is based on whether the p-value is greater than the significance level, indicating insufficient evidence to support the alternative hypothesis.
Step-by-step explanation:
If the calculated value of chi-square is less than the critical value of chi-square, you would fail to reject the null hypothesis. In hypothesis testing, failing to reject the null hypothesis indicates that there is not enough evidence to support the alternative hypothesis over the null hypothesis. This decision is typically based on a comparison between the calculated p-value and a predetermined level of significance, often denoted as α (alpha). If the p-value is greater than α, the results are not considered statistically significant, and the null hypothesis is not rejected.
An example to illustrate this would be if your level of significance, α, is set at 0.05 and the p-value is calculated to be 0.07. Since the p-value is higher than the level of significance, you would not reject the null hypothesis. On the other hand, if the p-value were, for instance, 0.03, which is lower than the significance level, you would reject the null hypothesis, as the data suggests that the observed results are statistically significant and that the alternative hypothesis may be correct.
Remember that a chi-square test is one of several statistical tests used to determine if there is a significant difference between observed and expected frequencies in one or more categories. In each case, the final decision to reject or not reject the null hypothesis is determined by whether the calculated test statistic is greater than the critical value associated with the chosen level of significance in the chi-square distribution table.