Final answer:
The obtained chi-square value of 2.62 does not exceed the critical value for one degree of freedom at the 0.05 significance level, thus the null hypothesis should not be rejected. The degrees of freedom for a test fitting a 3:1 ratio would be one.
Step-by-step explanation:
When evaluating whether to accept a null hypothesis in a chi-square test of goodness-of-fit, the obtained chi-square value must be compared to a critical value from a chi-square distribution table at a specified significance level. Assuming the common significance level of 0.05, for one degree of freedom, the critical value from the chi-square distribution table is approximately 3.84. Since the obtained chi-square value of 2.62 is less than 3.84, the null hypothesis should not be rejected. This means that there is not enough evidence to suggest that the observed ratios significantly deviate from the expected 3:1 ratio.
For a chi-square goodness-of-fit test with a 3:1 ratio, the degrees of freedom are calculated as the number of categories minus one. Since a 3:1 ratio implies two categories (3 parts one category, 1 part another), the degrees of freedom would be 2-1=1. Therefore, the test would have one degree of freedom.