Hey I have some questions about this problem:
You order a large load of balloons for a school festival. You order round and heart-shaped balloons in a ratio of $4:1$. When the delivery arrives, you will see that all the balloons are mixed up in one big card. They suspect the manufacturer shipped fewer of the more expensive heart-shaped balloons to save money.
Make a test for the hypothesis $H_0:p=p_0=0.2$ against the alternative $H_1: p=p_1=0.1$ with maximum allowed error of 1st kind of $\alpha=0.1$ and the Sample size $n=20$.
So what I have done is:
We have $X=\{X_1,...,X_n\}$ with $n=20$ we have $F=2^{\Omega}$ and every $X_i$ has Bernoulli distribution therefore X has Binomial distribution.
We have that the
Zero Hypothesis is $H_0:p=p_0=0.2$
And the alternative $H_1: p=p_1=0.1$
I think that a test we could choose is $\phi (x)=1_{(|x-n/5|\geq k_{\alpha})}$ with $P_{1/5}(\phi=1) \leq \alpha$ which in our case is $\alpha=0.1$
we have that
$P_{1/5}(|X-n/5|\geq k_{\alpha})=P_{1/5}(x-n/5 \leq -k_{\alpha} \cup x-n/5 \geq k_{\alpha})=P_{1/5}(x-n/5 \leq -k_{\alpha})+P_{1/5}(x-n/5 \geq k_{\alpha})=P_{1/5}(x-n/5 \geq k_{\alpha})=P_{1/5}(x \geq k_{\alpha}+n/5)=1-(\sum_{i=0}^{k_{\alpha}+n/5}\binom{n}{i} p^i(1-p)^{n-i})\approx0,1$
so we find a $k_{\alpha}$ with $n=20$ and after that we are ready for doing our test.
Have I done any mistake? Tomorrow I will have my stochastic exam and this is the last problem where I need to know if I understood the concept correctly. Please help me :)