Step 1 :Calculate the total observed frequency, which is the sum of the observed frequencies for all categories: \(22 (A) + 6 (B) + 12 (C) + 23 (D) + 22 (E) = 85\).
Step 2 :Since we are testing the claim that all 5 categories are equally likely to be selected, the expected frequency for each category is the total observed frequency divided by the number of categories: \(Expected frequency = \frac{Total observed frequency}{Number of categories} = \frac{85}{5} = 17\).
Step 3 :Complete the table with the observed and expected frequencies, and the calculation of \(\frac{(O-E)^{2}}{E}\) for each category.
Step 4 :Calculate the chi-square test-statistic, which is the sum of the values in the last column of the table: \(\chi^{2} = \frac{(22-17)^{2}}{17} + \frac{(6-17)^{2}}{17} + \frac{(12-17)^{2}}{17} + \frac{(23-17)^{2}}{17} + \frac{(22-17)^{2}}{17}\).
Step 5 :Simplify the chi-square test-statistic: \(\chi^{2} = \frac{25}{17} + \frac{121}{17} + \frac{25}{17} + \frac{36}{17} + \frac{25}{17} = 1.471 + 7.118 + 1.471 + 2.118 + 1.471 = 13.649\).
Step 6 :So, the chi-square test-statistic for this data is \(\boxed{13.649}\) (rounded to three decimal places).