Final answer:
A 95% two-sided confidence interval for the standard deviation will have 2.5% probability in each tail of the chi-square distribution.
Step-by-step explanation:
When constructing a confidence interval for a population standard deviation, it's important to decide on a confidence level. Typically, 95% and 99% are common choices. Since you are constructing a two-sided confidence interval, the level of confidence you choose will dictate the probability in each tail of the distribution. For a 95% confidence interval, 5% of the probability is excluded, which means that there is 2.5% in each tail (0.025 in the lower tail and 0.025 in the upper tail).
The next step involves using a chi-square distribution because you are dealing with the standard deviation (a measure of spread which is a squared value), not the mean. The chi-square distribution is used to build a confidence interval around the standard deviation given a sample variance and sample size. This will require calculating chi-square values which correspond to the probabilities at the tails, based on the degrees of freedom, which is one less than the sample size (n - 1).
Once these chi-square values are obtained, they are used to calculate the lower and upper bounds of the confidence interval for the standard deviation by the following formulas:
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- Lower bound = \(\sqrt{(n - 1)\times s^2 / \chi^2_{\text{upper}})}\)
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- Upper bound = \(\sqrt{(n - 1)\times s^2 / \chi^2_{\text{lower}})}\)
Where 'n' is the sample size, 's' is the sample standard deviation, and \(\chi^2_{\text{lower}}\) and \(\chi^2_{\text{upper}}\) are the chi-square values for the lower and upper tails, respectively.
For a 99% confidence interval, 1% of the probability is excluded, with 0.005 in each tail. However, these wider intervals would lead to a larger range for the standard deviation. Comparatively, 95% confidence intervals are narrower and hence give a more precise estimate of the population standard deviation but with a slightly lower confidence level.