College

According to a recent study, [tex]$15\%$[/tex] 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 [tex]$\alpha=0.05$[/tex].

Complete the "State" and "Plan" steps. Which statements are true? Check all that apply.

- [tex]$H_0: p = 0.15$[/tex]
- [tex]$H_a: p \ \textgreater \ 0.15$[/tex]
- The random condition is met.
- The 10% condition is met.
- The large counts condition is met.
- The test is a [tex]$z$[/tex]-test for one proportion.

Answer :

To address this hypothesis test, we need to go through the "State" and "Plan" steps of the process.

### State:
1. Null Hypothesis ([tex]\(H_0\)[/tex]):
- The null hypothesis is that the true proportion of adults who experience side effects from the medication is [tex]\(p = 0.15\)[/tex].

2. Alternative Hypothesis ([tex]\(H_a\)[/tex]):
- The alternative hypothesis in this context is that the true proportion [tex]\(p\)[/tex] is greater than 0.15 (because the researcher wants to see if there's evidence that more than 15% experience side effects). Therefore, [tex]\(H_a: p > 0.15\)[/tex].

### Plan:
1. Random Condition:
- This condition verifies that the sample of adults is randomly selected. The problem statement mentions a "separate random sample," which means this condition is satisfied.

2. 10% Condition:
- The 10% condition states that the sample size should be less than 10% of the population. Given that 150 is a reasonable size compared to a typically large population of adults taking the medication, this condition is met.

3. Large Counts Condition:
- This condition requires that both [tex]\(np_0\)[/tex] and [tex]\(n(1-p_0)\)[/tex] are greater than 10, where [tex]\(p_0 = 0.15\)[/tex] and [tex]\(n = 150\)[/tex].
- Calculating [tex]\(np_0 = 150 \times 0.15 = 22.5\)[/tex]
- Calculating [tex]\(n(1-p_0) = 150 \times (1 - 0.15) = 127.5\)[/tex]

Since both 22.5 and 127.5 are greater than 10, the large counts condition is satisfied.

4. Test Type:
- The planned test is a [tex]\(z\)[/tex]-test for one proportion. This type of test is appropriate when checking proportions and the conditions above are satisfied.

With these steps, we ensure we are properly setting up to determine if there's statistical evidence that more than 15% of adults experience side effects from the medication at the 0.05 significance level.