Confidence interval when both μ and σ2 are unknownI have the following problem in my problem book:Let X∼N(μ,σ2) where both μ and σ2 are unknown. I have to find a confidence interval for the mean.What I have so far:I know that when σ2 is unknown I can use t-distribution for finding a confidence interval, i.e.:x―±tn−1⋅sn,where tn−1⋅ is a t-distribution with n−1 degrees of freedom and x― is the sample mean from a sample of size n.The whole problem seems kind of vague to me. Is this enough for describing the confidence interval? I don't know the sample mean x― so is it okay to describe the solution in such a way?

Question

Confidence interval when both μ and σ2 are unknownI have the following problem in my problem book:Let X∼N(μ,σ2) where both μ and σ2 are unknown. I have to find a confidence interval for the mean.What I have so far:I know that when σ2 is unknown I can use t-distribution for finding a confidence interval, i.e.:x―±tn−1⋅sn,where tn−1⋅ is a t-distribution with n−1 degrees of freedom and x― is the sample mean from a sample of size n.The whole problem seems kind of vague to me. Is this enough for describing the confidence interval? I don&#039;t know the sample mean x― so is it okay to describe the solution in such a way?

Eve Larson · Accepted Answer

When the standard deviation is unknown but the distribution is Gaussian, the trick is to use the fact that the reduced variablet―=x―−μds2nwhere s2 is the corrected variance follows a Student&#039;s distribution (of n−1 dof).Knowing the probability of this variable to be in a certain range, you have your confidence interval for μ.

Confidence interval when both \(\displaystyle\mu\) and

Answered question

Answer & Explanation

New Questions in College Statistics