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 35% (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 35% (round off to second decimal place).

BlackTom AI · Accepted Answer

We are given a binomial setup. Let X be the number of cures out of 20 patients when the true survival rate is 0.35, so X ~ Binomial(n=20, p=0.35).

The rejection region is defined as X ≥ 12. To understand the error in the specified scenario (survival rate 35%), we examine the probability of observing X in the rejection region under p = 0.35:

- Compute P(X ≥ 12) for X ~ Binomial(20, 0.35).
  - A direct calculation uses the binomial tail: P(X ≥ 12) = sum_{k=12}^{20} C(20, k) (0.35)^k (0.65)^{20-k}.
  - This tail probability is small because 12 or more successes is far above the mean np = 7.

A practical approximation uses a normal approximation with continuity correction: for X ~ Binomial(20, 0.35), mean μ = np = 7 and variance σ^2 = np(1-p) = 20 × 0.35 × 0.65 ≈ 4.55, so σ ≈ 2.13. The standardized z-value for x = 11.5 is z ≈ (11.5 − 7) / 2.13 ≈ 2.11. The upper-tail probability beyond z ≈ 2.11 is about 0.0175, which rounds to approximately 0.02.

Therefore, P(X ≥ 12) under p = 0.35 is roughly 0.02. In plain terms, the chance of observing 12 or more cures when the true rate is 0.35 is about 2%. This is the likelihood of the test statistic landing in the rejection region given the true rate is 0.35.

Note: Depending on the framing, this probability is sometimes referred to as the test's power at p = 0.35 (the probability of rejecting H0 when the true p is 0.35). If one instead considers the Type II error, it would be 1 minus this value, i.e., approximately 0.98, representing the probability of failing to reject H0 when p = 0.35.

AMS 310 Section 7.2 Quiz

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