# Let X i </msub> be Gaussian random variables with &#x03BC;<!-- μ --> =

Let ${X}_{i}$ be Gaussian random variables with $\mu$ = $10$ and ${\sigma }^{2}$ = $1$. We decide to use the test statistic $\stackrel{^}{\mu }$ = $\frac{1}{20}\sum _{i=1}^{20}{X}_{i}$ and following tests:
$|\stackrel{^}{\mu }-10|>\tau$: Reject ${H}_{0}$
$|\stackrel{^}{\mu }-10|\le \tau$: Cannot reject ${H}_{0}$
1) Find $\tau$ if you want to have $95\mathrm{%}$ confidence in the test.
2) You find that $\stackrel{^}{\mu }=10.588$. Do you reject ${H}_{0}$? If so, what is the p-value?
Using textbook equations, I found out that for
1) $\tau$ = ${\sigma }_{\stackrel{^}{\mu }}$ = $1.96\sqrt{\frac{1}{20}}$ $=0.392$
2) $|10.588-10|=0.588>0.392\to$ Reject ${H}_{0}$
p-value $=0.003$
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Kharroubip9ej0
The $95\mathrm{%}$ confidence interval for $\mu$ is:

It implies that we are $95\mathrm{%}$ confident that the population mean will lie within $0.392$ distance from $\stackrel{^}{\mu }$. The null hypothesis states to reject if it is greater distance. With $\stackrel{^}{\mu }=10.588$, it is greater distance, so we reject ${H}_{0}$.
$p$-value is: