Answer :
To test whether the true proportion of adults experiencing side effects from the medication is greater than 0.15, we proceed as follows:
State:
1. Null Hypothesis ([tex]\(H_0\)[/tex]): The null hypothesis is a statement of no effect or no difference. For this situation, we assume the true proportion of adults experiencing side effects is equal to 0.15. Thus, [tex]\(H_0: p = 0.15\)[/tex].
2. Alternative Hypothesis ([tex]\(H_a\)[/tex]): The alternative hypothesis is what we want to test for. In this case, we are testing if the proportion is greater than 0.15, so [tex]\(H_a: p > 0.15\)[/tex].
Plan:
1. Random Condition: The sample should be a simple random sample of adults who use the medication. According to the problem, a separate random sample of 150 adults was selected, so this condition is met.
2. 10% Condition: This condition ensures that the sample size is small enough relative to the population to assume independence between samples. The problem does not specify the total population size, but generally, the adult population size is much larger than 150, which is typically less than 10% of the total population. Therefore, the condition is considered met.
3. Large Counts Condition: To use the normal approximation for proportions, both [tex]\(np\)[/tex] and [tex]\(n(1-p)\)[/tex] need to be at least 10.
- Calculate [tex]\(np = 150 \times 0.15 = 22.5\)[/tex]
- Calculate [tex]\(n(1-p) = 150 \times 0.85 = 127.5\)[/tex]
Both are greater than 10, so the large counts condition is met.
4. Type of Test: Since we are comparing a sample proportion to a known population proportion and the conditions above are met, we use a z-test for one proportion.
In summary, the true statements from the problem are:
- [tex]\(H_0: p = 0.15\)[/tex]
- The random condition is met.
- The 10% condition is met.
- The large counts condition is met.
- The test is a z-test for one proportion.
State:
1. Null Hypothesis ([tex]\(H_0\)[/tex]): The null hypothesis is a statement of no effect or no difference. For this situation, we assume the true proportion of adults experiencing side effects is equal to 0.15. Thus, [tex]\(H_0: p = 0.15\)[/tex].
2. Alternative Hypothesis ([tex]\(H_a\)[/tex]): The alternative hypothesis is what we want to test for. In this case, we are testing if the proportion is greater than 0.15, so [tex]\(H_a: p > 0.15\)[/tex].
Plan:
1. Random Condition: The sample should be a simple random sample of adults who use the medication. According to the problem, a separate random sample of 150 adults was selected, so this condition is met.
2. 10% Condition: This condition ensures that the sample size is small enough relative to the population to assume independence between samples. The problem does not specify the total population size, but generally, the adult population size is much larger than 150, which is typically less than 10% of the total population. Therefore, the condition is considered met.
3. Large Counts Condition: To use the normal approximation for proportions, both [tex]\(np\)[/tex] and [tex]\(n(1-p)\)[/tex] need to be at least 10.
- Calculate [tex]\(np = 150 \times 0.15 = 22.5\)[/tex]
- Calculate [tex]\(n(1-p) = 150 \times 0.85 = 127.5\)[/tex]
Both are greater than 10, so the large counts condition is met.
4. Type of Test: Since we are comparing a sample proportion to a known population proportion and the conditions above are met, we use a z-test for one proportion.
In summary, the true statements from the problem are:
- [tex]\(H_0: p = 0.15\)[/tex]
- The random condition is met.
- The 10% condition is met.
- The large counts condition is met.
- The test is a z-test for one proportion.
