Final answer:
x2 left and x2 right for a 95% confidence interval using the chi-square distribution with 14 degrees of freedom are the critical values that correspond to the 2.5th and 97.5th percentiles, leaving 2.5% of the probability in each tail of the two-tailed distribution.
Step-by-step explanation:
To find x2 left and x2 right for a 95% confidence interval using the chi-square distribution with 14 degrees of freedom, we're looking at the distribution's extreme values that leave 2.5% of the total area in each tail since it's a two-tailed interval. The chi-square distribution is skewed to the right, meaning that the critical values are not equidistant from the mean.
When constructing a two-sided 95 percent confidence interval, 2.5% of the probability will be in the left tail and another 2.5% will be in the right tail of the distribution. This is because the total area representing the extreme 5% of the probability distribution is split between the two tails.
To find these critical values, you would typically use a chi-square distribution table or a statistical software calculator. You would look up the value where the cumulative distribution function (CDF) equals 0.025 for the left critical value (x2 left), and the value where the CDF equals 0.975 for the right critical value (x2 right).
These correspond to the 2.5th percentile and the 97.5th percentile of the chi-square distribution with 14 degrees of freedom.