The survival rate of patients with a tumor using an existing medication is known to be 40%. A pharmaceutical company claims that the survival rate of a new drug is higher. The new drug is given to 20 patients to test for this claim. Let be the number of cures out of the 20 patients. Suppose the rejection region is
Determine the type of error that can occur when the true survival rate is 50%.

Question

The survival rate of patients with a tumor using an existing medication is known to be 40%. A pharmaceutical company claims that the survival rate of a new drug is higher. The new drug is given to 20 patients to test for this claim. Let  be the number of cures out of the 20 patients. Suppose the rejection region is  
Determine the type of error that can occur when the true survival rate is 50%.

BlackTom AI · Accepted Answer

The problem asks us to classify the type of error that could occur when the true survival rate is 50%.
First, identify the hypotheses and the decision rule: the company tests H0: p = 0.40 (survival rate 40%) against the alternative that p > 0.40, and rejects H0 if X ≥ 12, where X is the number of cures out of 20.
Next, consider the scenario given: the true rate is p = 0.50. Under this scenario, rejecting H0 when X ≥ 12 is the intended direction (since p > 0.40).
The type of error we are concerned with is what happens when we fail to reject H0 despite the alternative being true. In this setup, that corresponds to not exceeding the rejection region (i.e., X ≤ 11) when p = 0.50.
Therefore, the probability of this event is P(X ≤ 11) with X ~ Binomial(n = 20, p = 0.50). This probability is the Type II error probability (β).
Now assess the answer options:
- Option: Type II error. This correctly describes failing to reject H0 when the true p is 0.50, which is what the question asks for.
- Option: Type I error. This would be rejecting H0 when it is actually true (p = 0.40). But here the true rate is 0.50, not 0.40, so this is not the event in question.
In summary, since the situation involves not rejecting a false null when the true rate is higher (0.50) and the rejection region is X ≥ 12, the described error corresponds to a Type II error.

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