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
Compute the type of error that can occur when the survival rate is 50% (round off to second decimal place).

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  
Compute the type of error that can occur when the survival rate is 50% (round off to second decimal place).

BlackTom AI · Accepted Answer

First, restate the scenario to ensure clarity: we have a test with n = 20 patients, under the null hypothesis p0 = 0.40 (survival rate of existing medication). The rejection region is X ≥ 12, where X ~ Binomial(n = 20, p). The question asks for the type of error that can occur when the true survival rate is p = 0.50, i.e., when the alternative is true.

Step 1: Identify which error this corresponds to.
- Type I error (alpha) occurs when we reject the null hypothesis even though p = p0 (false positive).
- Type II error (beta) occurs when we fail to reject the null hypothesis when the true p is the alternative (p = 0.50 in this case).
Since the true survival rate is 50% (the alternative is true) and the rejection region is X ≥ 12, the error that can occur in this situation is a Type II error, i.e., failing to reject H0 when p = 0.50.

Step 2: Compute beta = P(fail to reject H0 | p = 0.50).
- The test rejects H0 if X ≥ 12. Therefore, failing to reject H0 corresponds to X ≤ 11.
- Under p = 0.50, X ~ Binomial(n = 20, p = 0.50).
- So beta = P(X ≤ 11) = sum_{x=0}^{11} C(20, x) (0.50)^20.

Step 3: Approximate the probability.
- A quick approximation uses the normal approximation to the binomial: with n = 20, p = 0.5, mu = np = 10 and sigma = sqrt(np(1-p)) = sqrt(5) ≈ 2.236.
- Apply continuity correction for P(X ≤ 11): P(X ≤ 11) ≈ P(Z ≤ (11.5 - 10)/2.236) = P(Z ≤ 0.67).
- From standard normal tables, P(Z ≤ 0.67) ≈ 0.75.

Step 4: Round to two decimals.
- 0.75 already has two decimals, so beta ≈ 0.75.

Conclusion (conceptual): When the true survival rate is 50% and the rejection region is X ≥ 12, the type II error probability is approximately 0.75 (75%). This represents the chance of not observing enough cures to reject the null despite the drug having a true 50% survival rate.

AMS 310 Section 7.2 Quiz

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