1. ## confidence interval

Let X1, X2,....,Xn be a random sample of size n from the normal distribution $\displaystyle N(\mu, \sigma^{2})$. Calculate the expected length of a 95% confidence interval for $\displaystyle \mu$ assuming that n=5 and the variance is:

a. known
b. unknown

Thank you for any help!

2. Originally Posted by redpack
Let X1, X2,....,Xn be a random sample of size n from the normal distribution $\displaystyle N(\mu, \sigma^{2})$. Calculate the expected length of a 95% confidence interval for $\displaystyle \mu$ assuming that n=5 and the variance is:

a. known
b. unknown

Thank you for any help!
let $\displaystyle \bar{X}=\frac{1}{n}(\sum_{k=1}^n X_k)$. If X_k independent with other, then we have $\displaystyle \bar{X}$~ $\displaystyle N(\mu, \sigma^2/\sqrt{n})$.
In the case \sigma is known
we use $\displaystyle \frac{\bar{X}-\mu}{\sigma/ \sqrt{n}}$~N(0,1).
In the case \sigma is unknown
we use $\displaystyle \frac{\bar{X}-\mu}{s/ \sqrt{n}}$~t-distribution.
From here, we get the confident interval.