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 is higher with a new drug. 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
What should be the null and alternative hypotheses?

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 is higher with a new drug. 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  
What should be the null and alternative hypotheses?

BlackTom AI · Accepted Answer

The scenario describes a hypothesis test about the population proportion p of cures with a new drug, where the survival (cure) rate under the existing treatment is known to be 0.40. We are testing whether the new drug increases the cure rate, with a one-sided alternative (p > 0.40) and a rejection region based on X, the number of cures out of 20.

Option 1: H0: p ≤ 0.4 versus H1: p > 0.4
- This matches a one-sided test where the null posits that the true proportion is at most 0.4, and the alternative asserts it is greater than 0.4. Since the rejection region is X ≥ 12 (i.e., observing 12 or more cures out of 20), this would provide support for p being larger than 0.4 if the observed X is in the tail to the right. This is consistent with assessing whether the new drug improves outcomes.

Option 2: H0: p ≤ 0.4 versus H1: p > 0.4
- This repeats the same structure as Option 1 but with different formatting in the prompt here. If presented as an option, it would be equivalent to Option 1 and would be a correct specification for a one-sided test with the stated rejection region.

Option 3: H0: p ≥ 0.4 versus H1: p < 0.4
- This describes a left-tailed test, where the alternative suggests a decrease in the cure rate. It does not align with the claim of higher survival with the new drug, which is a right-tailed (p > 0.4) alternative.

Option 4: H0: p = 0.4 versus H1: p ≠ 0.4
- This is a two-sided test. It does not reflect the directional claim that the new drug increases the survival rate. The rejection region X ≥ 12 corresponds to a one-sided test, not a two-sided one, so this option is inappropriate for the stated claim.

In summary, the question seeks a null and alternative that correspond to a one-sided increase in the cure rate (p > 0.4). The appropriate pairing is the one that states H0: p ≤ 0.4 and H1: p > 0.4, which matches the one-sided nature of the claim and the right-tailed rejection region. The other options either describe a left-tailed test, a two-sided test, or misstate the direction of the alternative, making them inconsistent with the scenario.

AMS 310 Section 7.2 Quiz

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