We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To address this question, we need to go through the "State" and "Plan" steps for a hypothesis test concerning a population proportion. We will evaluate the given statements to decide which are true.
### State
1. Null Hypothesis ([tex]\(H_0\)[/tex]): This hypothesis states that there is no change or effect. For this scenario, the null hypothesis is that the true proportion of adults experiencing side effects from the medication is 15%. Hence, [tex]\(H_0: p = 0.15\)[/tex].
2. Alternative Hypothesis ([tex]\(H_a\)[/tex]): This hypothesis posits that there is a change or effect. Here, the researcher wants to see if the proportion is greater than 15%, so the alternative hypothesis is [tex]\(H_a: p > 0.15\)[/tex].
### Plan
1. Random Condition: We need to ensure that the sample was randomly selected. Since it mentions a "random sample of 150 adults," we can conclude that the random condition has been met.
2. 10% Condition: This verifies if the sample is less than or equal to 10% of the entire population. Typically, the population of adults taking this medication is assumed large, so the condition is met if [tex]\(150\)[/tex] is less than or equals [tex]\(10\%\)[/tex] of the population. This condition is assumed to be sufficiently met if the population is at least 1,500 adults.
3. Large Counts Condition: This condition checks whether the sample size is large enough for the sampling distribution of the sample proportion to be approximately normal. For this, both [tex]\(150 \times 0.15 \geq 10\)[/tex] and [tex]\(150 \times (1 - 0.15) \geq 10\)[/tex] must be true. Specifically, [tex]\(150 \times 0.15 = 22.5\)[/tex] and [tex]\(150 \times 0.85 = 127.5\)[/tex], both of which satisfy the requirement.
4. Type of Test: Since we are testing a single population proportion, we use a [tex]\(z\)[/tex]-test for one proportion.
Based on these points, the true statements are:
- [tex]\(H_0: p = 0.15\)[/tex]
- The random condition is met.
- The large counts condition is met.
- The test is a [tex]\(z\)[/tex]-test for one proportion.
The statement about [tex]\(H_a: p < 0.15\)[/tex] is incorrect because our alternative hypothesis is [tex]\(H_a: p > 0.15\)[/tex].
The 10% condition's accuracy would depend on knowing the size of the larger population; the assumption should be verified with more contextual data.
### State
1. Null Hypothesis ([tex]\(H_0\)[/tex]): This hypothesis states that there is no change or effect. For this scenario, the null hypothesis is that the true proportion of adults experiencing side effects from the medication is 15%. Hence, [tex]\(H_0: p = 0.15\)[/tex].
2. Alternative Hypothesis ([tex]\(H_a\)[/tex]): This hypothesis posits that there is a change or effect. Here, the researcher wants to see if the proportion is greater than 15%, so the alternative hypothesis is [tex]\(H_a: p > 0.15\)[/tex].
### Plan
1. Random Condition: We need to ensure that the sample was randomly selected. Since it mentions a "random sample of 150 adults," we can conclude that the random condition has been met.
2. 10% Condition: This verifies if the sample is less than or equal to 10% of the entire population. Typically, the population of adults taking this medication is assumed large, so the condition is met if [tex]\(150\)[/tex] is less than or equals [tex]\(10\%\)[/tex] of the population. This condition is assumed to be sufficiently met if the population is at least 1,500 adults.
3. Large Counts Condition: This condition checks whether the sample size is large enough for the sampling distribution of the sample proportion to be approximately normal. For this, both [tex]\(150 \times 0.15 \geq 10\)[/tex] and [tex]\(150 \times (1 - 0.15) \geq 10\)[/tex] must be true. Specifically, [tex]\(150 \times 0.15 = 22.5\)[/tex] and [tex]\(150 \times 0.85 = 127.5\)[/tex], both of which satisfy the requirement.
4. Type of Test: Since we are testing a single population proportion, we use a [tex]\(z\)[/tex]-test for one proportion.
Based on these points, the true statements are:
- [tex]\(H_0: p = 0.15\)[/tex]
- The random condition is met.
- The large counts condition is met.
- The test is a [tex]\(z\)[/tex]-test for one proportion.
The statement about [tex]\(H_a: p < 0.15\)[/tex] is incorrect because our alternative hypothesis is [tex]\(H_a: p > 0.15\)[/tex].
The 10% condition's accuracy would depend on knowing the size of the larger population; the assumption should be verified with more contextual data.
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