A survey of 100 randomly selected motorist reveals that 57 are unhappy with the state of the roads. test the null hypothesis that half of motorists in the population are unhappy with the state of the roads.
$H_0:P=0.5$
$H_1:P\neq 0.5$
at which levels of 1, 5 and 10 percent significance can $H_0$ be rejected?
can someone explain this. i know its a binomial(bernoulli) but it lacks the standard deviation.... thanks for solution example how to define.
$n=100$
$X_1,\dots X_n$ i.i.d. $B(p)$.
Let's compute the p-value, which is $P_{H_0}(\sum_{i=1}^n X_i\geq 57)=P_{H_0}(Y\geq 57)$
We know that under $H_0$, $Y=\sum_{i=1}^nX_i$ is approximately normal with mean $np_0=50$ and variance $np_0(1-p_0)=25$.
So $P_{H_0}(Y\geq 57)=P(\mathcal{N}(50,5)\geq 57)=P(\mathcal{N}(0,1)\geq (57-50)/5)=1-\Phi(1.4)\simeq 0.08$ where $\Phi$ is the cdf of a standard gaussian distribution.
So if the level of the test is $1\%$ or $5\%$ you don't reject. If it's $10\%$, you reject.