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.
### 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.
