Step 1 :First, calculate the expected frequency for each category. Since the lottery is fair, each category should occur with the same frequency. There are 5 categories and 86 observations, so the expected frequency for each category is \(\frac{86}{5} = 17.2\).
Step 2 :Complete the table with the observed and expected frequencies for each category.
Step 3 :Next, calculate the chi-square test statistic. The formula for the chi-square test statistic is \(\chi^{2} = \sum \frac{(O_i - E_i)^2}{E_i}\), where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency.
Step 4 :Calculate the chi-square test statistic: \(\chi^{2} = \frac{(23 - 17.2)^2}{17.2} + \frac{(21 - 17.2)^2}{17.2} + \frac{(6 - 17.2)^2}{17.2} + \frac{(21 - 17.2)^2}{17.2} + \frac{(15 - 17.2)^2}{17.2}\).
Step 5 :Simplify the chi-square test statistic: \(\chi^{2} = \frac{33.64}{17.2} + \frac{14.44}{17.2} + \frac{125.44}{17.2} + \frac{14.44}{17.2} + \frac{4.84}{17.2}\).
Step 6 :Further simplify the chi-square test statistic: \(\chi^{2} = 1.956 + 0.839 + 7.297 + 0.839 + 0.281\).
Step 7 :Finally, add up all the values to get the chi-square test statistic: \(\boxed{\chi^{2} = 11.212}\).