Question

Suppose that X₁,…,Xₙ is a random sample from N(μ,σ² ), where σ² is known.

(a) Construct a two-sided 1−α confidence interval for μ.
(b) Find an unbiased estimator for μ.

Answer 1

(a) The two-sided 1−α confidence interval for μ, given a known σ², can be constructed as:
`\(\bar{X} \pm Z_{(\alpha)/(2)} * (\sigma)/(√(n))\)` , where
`\(\bar{X}\)` is the sample mean,
`\(Z_{(\alpha)/(2)}\)` is

the critical value from the standard normal distribution at
`\((\alpha)/(2)\)` , σ is the

known standard deviation, and n is the sample size.

(b) An unbiased estimator for μ is simply the sample mean,
`\(\bar{X}\)` .

In statistics, the sample mean
`\(\bar{X}\)` is an unbiased estimator for the population mean μ.

Step-by-step explanation:

(a) To construct a two-sided confidence interval for μ, we use the

formula
`\(\bar{X} \pm Z_{(\alpha)/(2)} * (\sigma)/(√(n))\)` ,

where
`\(\bar{X}\)` denotes the sample mean. This interval provides a range within which we're confident the population mean μ lies, with a

confidence level of 1−α. The critical value
`\(Z_{(\alpha)/(2)}\)` is determined based on

the chosen confidence level and is extracted from the standard normal

distribution. Utilizing the known σ², the formula helps ascertain the interval's width around the sample mean to encompass the true population mean.

(b) An unbiased estimator for μ is crucial in statistics. For a normal distribution with a known σ², the sample mean
`\(\bar{X}\)` is an unbiased estimator for the population mean μ. Unbiased estimators are those where the expected value of the estimator equals the parameter being estimated. In this case, the expected value of the sample mean
`(\(\bar{X}\))` is μ, making it an unbiased estimator for the population mean.