Step 1 :Given values are: Test statistic, \(X^{2} = 27.35\), sample size, \(n = 31\), and sample standard deviation, \(s = 2.11\).
Step 2 :Since the sample standard deviation, \(s\), is smaller than the population standard deviation, \(\sigma = 2.21\), this will be a left-tail set up.
Step 3 :Calculate the degrees of freedom, \(df = n - 1 = 31 - 1 = 30\).
Step 4 :Use the chi-square cumulative distribution function (CDF) to calculate the p-value for the left tail, \(p_{left} = \text{chi2.cdf}(X^{2}, df)\).
Step 5 :Since it's a two-tailed test, multiply the p-value by 2, \(p_{value} = 2 \times p_{left}\).
Step 6 :Final Answer: The p-value for the two-tailed chi-square test is \(\boxed{0.7903}\).