We have two coins, C0 and C1, that looks identical but have different probabilities
(0.5 and 0.7 respectively) of showing Heads. A coin is chosen at random and then is tossed X
times until a Head comes up. Denote by H0 and H1 the hypotheses that the chosen coin is C0 and
C1 respectively.
(a) Construct the rejection region of the likelihood ratio test for H0 vs H1 of significance level α
(b) What is the significance level of a test that rejects H0 if X ≥ 8?
(c) Find the p-value of the likelihood ration test for observation X = k.
(a) To construct the rejection region of the likelihood ratio test for H0 vs H1 of significance level α, we need to compare the likelihood ratios for H0 and H1.
The likelihood ratio is given by:
LR(X) = (Probability of observing X given H1) / (Probability of observing X given H0)
In this case, the probability of observing X given H1 is (0.7)^X, and the probability of observing X given H0 is (0.5)^X.
To find the rejection region, we need to determine the threshold value of the likelihood ratio that corresponds to the significance level α. This threshold value is given by:
Threshold = (Probability density function of the likelihood ratio) = α.
The likelihood ratio test statistic can be approximated by a chi-square distribution with one degree of freedom. Using this approximation, we can find the value of the likelihood ratio at which the chi-square distribution with one degree of freedom has a cumulative probability of α.
(b) To determine the significance level of a test that rejects H0 if X ≥ 8, we need to find the probability of observing a value of X greater than or equal to 8, assuming H0 is true.
P-value = (Probability of observing a test statistic as extreme or more extreme than the observed test statistic)
In this case, the observed test statistic is X = 8. To calculate the p-value, we need to find the probability of observing a value greater than or equal to 8, assuming H0 is true.
P-value = P(X ≥ 8 given H0) = 1 - P(X < 8 given H0) = 1 - ΣP(X = k given H0) for k = 0 to 7.
(c) To find the p-value of the likelihood ratio test for observation X = k, we need to find the probability of observing a test statistic as extreme or more extreme than the observed test statistic, assuming H0 is true.
P-value = (Probability of observing a test statistic as extreme or more extreme than the observed test statistic)
In this case, the observed test statistic is X = k. To calculate the p-value, we need to find the probability of observing a value greater than or equal to k, assuming H0 is true.
P-value = P(X ≥ k given H0) = 1 - P(X < k given H0) = 1 - ΣP(X = i given H0) for i = 0 to k-1.