I have 10 independent coin tosses with probability p for heads. I'm testing the null hypothesis $H_0: p=1/2$ Versus the alternative $H_1 : p > 1/2 $
$W$ is the number of heads in the 10 coin tosses. We accept $H_0$ if $W \leq c$ else reject $H_0$ Decide c so that $\alpha = 0.05$
I started off saying $W$ is $binomial(10,½)$ so the probability of type I errors is:
$P(Type I Error)=P(Reject H_0 \mid H_0) = P(RejectH_0 \mid p=½) = P(W >c\mid p=½)=\sum\limits_{k=c+1}^{\infty} P(W=k)$
$And because of W being binomial(10,½)
$=\sum\limits_{k=c+1}^{10} 0.5^k 0.5^{10-k}={(10-c)}/1024 $
And so because of us wanting $\alpha =0.05$ we choose a c so that $(10-c)/1024 \leq \alpha $
So I get $c=10-\alpha*1024=-41.2$
Which seems wrong to me, where did i go wrong in my thought process? Because if c=-41.2 then i would never be able to say the coin is fair and accept the null hypothesis