It is suspected that a coin is not balanced (not fair). Let p be the probability

Question

It is suspected that a coin is not balanced (not fair). Let p be the probability of tossing a head. To test H_0: p = 0.5 against the alternative hypothesis H_a: p >0.5, a coin is tossed 15 times. Let Y equal the number of times a head is observed in the 15 tosses of this coin. Assume the rejection region to be (Y greaterthanorequalto 10]. (a) Find alpha. (b) Find for beta for p = 0.7. (c) Find for beta for p = 0.6. (d) Find the rejection region for (Y greaterthanorequalto K) for alpha = 0.01, and alpha = 0.03. (e) For the alternative H_a: p = 0.7, find beta for the values of alpha given in (d).

Explanation / Answer

a) alpha =the probability of rejecting the null hypothesis when it is true

P(Y>= 10 when p =0.5) = 0.15087890

n =15 ,p=0.5

b)beta = the probability of failing to reject the hypothesis tested when that hypothesis is false and a specific alternative hypothesis is true.

p =0.7

P(Y< 10 when p = 0.7)

=0.278378559795636

c) when p= 0.6

=0.596784449585152

d)for alpha= 0.01

P(Y>=K| p =0.5) = 0.01

P(Y>=11) = 0.0592

P(Y>=12) = 0.0179

hence for alpha = 0.01 , K = 12

for alpha= 0.03

K = 11

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