In the problem I am given :

$\displaystyle

H_0 : u=5 ; H_1:u\neq5$

n = 8 (sample size)

The random variable follows a normal law with a standard deviation of 0.25

The problem also states that we need to reject [tex]H_0[/Math] if $\displaystyle \overline{x}<4.85$ or $\displaystyle \overline{x}>5.15.$

I need to find the probability of a type I error.

What I've tried so far: (alpha represents my Type I error)

$\displaystyle \alpha=P(reject H_0|H_0 true)

=P(\overline{x}<4.85 or \overline{x}>5.15 | u=5)$

So I tried to start and find the probability of $\displaystyle \overline{x}$ not in that interval.

$\displaystyle \overline{x} \not \in [u-0.15;u+0.15] $

$\displaystyle [u-\frac{s*Z_{\alpha/2}}{\sqrt{n}},u+\frac{s*Z_{\alpha/2}}{\sqrt{n}}]$

which gives me $\displaystyle \frac{s*Z_{\alpha/2}}{\sqrt{n}} = 0.15$

when I plug my numbers and isolate $\displaystyle Z_{\alpha/2}$, i get $\displaystyle Z_{\alpha/2}=0.5*\sqrt{8}/0.25^2=22.62

$

It seems to me that this value of Z can't be right since my table only goes to a maximum of 3.09.

As for the rest of the problem, I could use a tip or two about how to find P(u=5) and how to calculate the conditionnal probability.

Thanks