According to a recent study, 15% of adults who take a certain medication experience side effects. To further investigate this finding, a researcher selects a separate random sample of 150 adults, of which 32 experience side effects. The researcher would like to determine if there is convincing statistical evidence that the true proportion of adults who would experience side effects from this medication is greater than 0.15 using a significance level of α = 0.05. Complete the "state" and "plan" steps.

Which statements are true? Check all that apply.

A. [tex]H_0: p = 0.15[/tex]; [tex]H_a: p < 0.15[/tex]
B. The random condition is met.
C. The 10% condition is met.
D. The large counts condition is met.
E. The test is a z-test for one proportion

D. The large counts condition is met. The sample size (150) is greater than 10 times the expected number of successes (based on the null hypothesis of 15% experiencing side effects) and the expected number of failures (calculated as 85). 150 * 0.15 = 22.5 and 150 * 0.85 = 127.5, both exceeding 10.

Statements that are not true:

A. H0: p = 0.15; Ha: p < 0.15 The alternative hypothesis should state that the proportion is greater than 0.15, as the research aims to investigate if the proportion is higher than what the study previously found. Therefore, the correct null and alternative hypotheses are:

H0: p ≤ 0.15

Ha: p > 0.15

C. The 10% condition is not met. This refers to the condition that at least 10% of observations fall in each category (success and failure). While the large counts condition ensures sufficient numbers for both categories overall, we haven't been given information about the specific number of individuals who did not experience side effects.

E. The test is not necessarily a z-test for one proportion. While a z-test is commonly used for this scenario, the researcher might choose a different test depending on the specific underlying assumptions and distribution of the data.

Therefore, the correct options are B) The random condition is met. and D) The large counts condition is met.

inshase60 inshase60 · Answer 2 · 2025-01-06T09:34:50-05:00

Final answer:

The null hypothesis is that the proportion of adults experiencing side effects from the medication is 0.15, while the alternative hypothesis is that the proportion is greater than 0.15. The conditions for the hypothesis test are met and the appropriate test is a z-test for one proportion.

Explanation:

The null hypothesis (H0) is that the true proportion of adults who would experience side effects from the medication is 0.15. The alternative hypothesis (Ha) is that the true proportion is greater than 0.15. So the correct statement is A: H0: p = 0.15; Ha: p > 0.15.

For the “plan” step, we need to check the conditions and choose the appropriate test. The random condition is met because the sample is selected randomly. The 10% condition is met because the sample size is less than 10% of the population. The large counts condition is also met because both np and n(1-p) are greater than 10 since 0.15 x 150 = 22.5 and 0.85 x 150 = 127.5. Therefore, the test is a z-test for one proportion, confirming statement E is true.