Answer :
To solve this problem, let's carefully follow the steps typically used in conducting a hypothesis test for a proportion:
State:
1. Define the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The true proportion of adults experiencing side effects is [tex]\(p = 0.15\)[/tex].
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The true proportion of adults experiencing side effects is greater than 0.15, thus [tex]\(p > 0.15\)[/tex].
Plan:
The plan involves following the necessary steps and checking conditions for valid hypothesis testing:
1. Significance Level ([tex]\(\alpha\)[/tex]): Given [tex]\(\alpha = 0.05\)[/tex].
2. Random Condition:
- The problem states that a random sample of adults is selected. Thus, the random condition is met.
3. 10% Condition (Independent):
- To check this, ensure the sample size is less than 10% of the population of all adults taking the medication. Typically, without a given population size, it's assumed to be large enough, hence the condition is met if the assumption holds.
4. Large Counts Condition (Normality Assumption):
- Check if [tex]\(n \times p_0 \geq 10\)[/tex] and [tex]\(n \times (1 - p_0) \geq 10\)[/tex], where [tex]\(n\)[/tex] is the sample size (150) and [tex]\(p_0\)[/tex] is the hypothesized proportion (0.15):
- [tex]\(150 \times 0.15 = 22.5\)[/tex]
- [tex]\(150 \times (1 - 0.15) = 127.5\)[/tex]
- Both values are greater than 10, so this condition is also met.
5. Type of Test:
- Since all conditions are satisfied, we use a z-test for one proportion.
Based on the above steps, these statements are true:
- [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.
Please let me know if you need further explanation or have any other questions!
State:
1. Define the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The true proportion of adults experiencing side effects is [tex]\(p = 0.15\)[/tex].
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The true proportion of adults experiencing side effects is greater than 0.15, thus [tex]\(p > 0.15\)[/tex].
Plan:
The plan involves following the necessary steps and checking conditions for valid hypothesis testing:
1. Significance Level ([tex]\(\alpha\)[/tex]): Given [tex]\(\alpha = 0.05\)[/tex].
2. Random Condition:
- The problem states that a random sample of adults is selected. Thus, the random condition is met.
3. 10% Condition (Independent):
- To check this, ensure the sample size is less than 10% of the population of all adults taking the medication. Typically, without a given population size, it's assumed to be large enough, hence the condition is met if the assumption holds.
4. Large Counts Condition (Normality Assumption):
- Check if [tex]\(n \times p_0 \geq 10\)[/tex] and [tex]\(n \times (1 - p_0) \geq 10\)[/tex], where [tex]\(n\)[/tex] is the sample size (150) and [tex]\(p_0\)[/tex] is the hypothesized proportion (0.15):
- [tex]\(150 \times 0.15 = 22.5\)[/tex]
- [tex]\(150 \times (1 - 0.15) = 127.5\)[/tex]
- Both values are greater than 10, so this condition is also met.
5. Type of Test:
- Since all conditions are satisfied, we use a z-test for one proportion.
Based on the above steps, these statements are true:
- [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.
Please let me know if you need further explanation or have any other questions!
