Final answer:
To calculate the critical values for a 90% confidence interval using a chi-square distribution with 15 degrees of freedom, 5% of the probability is allocated to each tail. Critical values are found using a table or software, corresponding to the cumulative probabilities of 0.950 and 0.050 for the upper and lower tails, respectively.
Step-by-step explanation:
To find the chi-square distribution critical values (x² left and x² right) for a 90% confidence interval with 15 degrees of freedom, we need to split the remaining 10% probability into the two tails of the distribution. Hence, there will be 5% in each tail.
Consulting a chi-square distribution table or using statistical software, we look up the critical values that correspond to the upper and lower 5% of the chi-square distribution with 15 degrees of freedom.
The critical value for the upper tail (x² right) corresponds to a cumulative probability of 0.950 (central 90% + lower 5%), and the critical value for the lower tail (x² left) corresponds to the cumulative probability of 0.050.
Referencing the standard chi-square distribution tables will provide the exact values needed for the confidence interval calculation.