Question

To test the hypothesis that a coin is fair, you toss it 100 times. Your decision
rules allow you to accept the hypothesis only if you get between 40 and 60 tails in
100 tosses. What is the probability of committing Type II error when the actual
probability of tails is 0.7?

Answer 1

let X is number of tails out 100 tosses

we have to test that

where P is probability of getting a tail

Here we accept H0 if number of tails falls between 40 and 60 that is 40<X<60

we have to find P(type 2 error)

so under Ha: p=0.7 (given)

Mean =n*p=100*0.70=70

Var(X)=n*p*(1-p)=100*0.70*0.3=21

since number of tosses "n" is large so we will approximate it by normal

so continuity correction for acceptance region we accept H0 if 39.5<X<60.5

now

So answer is 0.02