MHB Confidence Interval for Mean $\mu$: Let $X_1, X_2, \dots, X_n$

[evinda]

Hello! (Wave)

Let $X_1, X_2, \dots, X_n$ a random sample with $E(X_i)=\mu$, $Var(X_i)=\sigma^2 \forall i$. For $0<a<0.5$:

show that for any $k \in [0,1]$, the interval
$$\left( \overline{X}-z_{k=a} \frac{\sigma}{\sqrt{n}}, \overline{X}+z_{(1-k)=a} \frac{\sigma}{\sqrt{n}}\right)$$
is a 100(1-a)% confidence interval for the mean $\mu$.

How could we do this? (Thinking)

 

[I like Serena]

Hello! (Wave)

Let $X_1, X_2, \dots, X_n$ a random sample with $E(X_i)=\mu$, $Var(X_i)=\sigma^2 \forall i$. For $0<a<0.5$:

show that for any $k \in [0,1]$, the interval
$$\left( \overline{X}-z_{k=a} \frac{\sigma}{\sqrt{n}}, \overline{X}+z_{(1-k)=a} \frac{\sigma}{\sqrt{n}}\right)$$
is a 100(1-a)% confidence interval for the mean $\mu$.

How could we do this? (Thinking)

Hey! (Smile)

What is $k$? And what is $z_{k=a}$? (Wondering)