Questions related to sufficient statistics and confidence interval

[StudentW]

Homework Statement

Suppose X1, X2, ..., Xn constitute a random sample from a population following N(μ,θ) where μ is known.

(a)Find a sufficient statistic for θ.
(b)Use the maximum likelihood estimator of θ to construct a confidence interval for θ with confidence level 1-α.

Homework Equations

The Attempt at a Solution

For part (a), I have tried to use the factorization criterion to find the sufficient statistic for θ but I have difficulty in separating θ from the exponential function as θ is the denominator. Can someone teach me how to get a function that depends on Xi only!

For part (b), the maximum likelihood estimator of θ is (1/n)*Σ(Xi-μ)^2. I am not sure about whether the information that Σ(Xi-μ)^2/θ follows chi square distribution with n degrees of freedom can help us to find the confidence interval! Can someone teach me how to get the confidence interval as well?

 

[Ray Vickson]

Homework Statement

Suppose X1, X2, ..., Xn constitute a random sample from a population following N(μ,θ) where μ is known.

(a)Find a sufficient statistic for θ.
(b)Use the maximum likelihood estimator of θ to construct a confidence interval for θ with confidence level 1-α.

Homework Equations

The Attempt at a Solution

For part (a), I have tried to use the factorization criterion to find the sufficient statistic for θ but I have difficulty in separating θ from the exponential function as θ is the denominator. Can someone teach me how to get a function that depends on Xi only!

For part (b), the maximum likelihood estimator of θ is (1/n)*Σ(Xi-μ)^2. I am not sure about whether the information that Σ(Xi-μ)^2/θ follows chi square distribution with n degrees of freedom can help us to find the confidence interval! Can someone teach me how to get the confidence interval as well?

Use a chi-squared table to find points $a$ and $b$ such that
$$P(\chi^2(n) \leq a) = \frac{\alpha}{2}, \; P(\chi^2(n) \geq b) = \frac{\alpha}{2}$$
Thus
$$P\left( a \leq \frac{\sum(X_i - \mu)^2}{\theta} \leq b \right) = 1 - \alpha.$$
Now turn that into a $1-\alpha$ probability statement about $\theta$.