Question

(Inference on the mean of a distribution with the variance unknown) Consider the random variable x, with unknown mean μ and known variance σ2 . Suppose that a random sample of n observations is taken, e.g. x1, x2, ... , xn and we compute x¯. Then a 100(1-α)% two sided Confidence Interval of μ is based on what distribution? x¯ = Z α/2 σ/sqrt(n) ≤ μ ≤ x¯ + Z α/2 σ/sqrt(n)

Answer 1

When estimating the mean of a distribution with an unknown variance, a 100(1-α)% two-sided confidence interval for the mean (μ) is based on the t-distribution.

Step-by-step explanation:

This is applicable when we have a random sample of n observations (x₁, x₂, ..., xn), and we calculate the sample mean (x¯) and the sample standard deviation (s).

The formula for the confidence interval is:

x¯ - t(α/2, n-1) * (s/√n) ≤ μ ≤ x¯ + t(α/2, n-1) * (s/√n),

where t(α/2, n-1) represents the critical value from the t-distribution with n-1 degrees of freedom and a significance level of α/2.

The t-distribution is used when the population standard deviation is unknown and needs to be estimated using the sample standard deviation. It takes into account the variability in the sample and provides a wider confidence interval compared to the z-distribution (which is used when the population standard deviation is known).

By using the t-distribution, we can construct a confidence interval that captures the range within which we expect the true population mean to lie.