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 :
Sure! Let's go through the problem and determine which statements are true regarding the hypothesis test.
State the Hypotheses:
1. Null Hypothesis ([tex]\(H_0\)[/tex]): The null hypothesis is that the true proportion of adults who experience side effects is equal to 0.15.
[tex]\[
H_0: p = 0.15
\][/tex]
2. Alternative Hypothesis ([tex]\(H_a\)[/tex]): The alternative hypothesis is what the researcher wants to prove, which is that the true proportion of adults who experience side effects is greater than 0.15.
[tex]\[
H_a: p > 0.15
\][/tex]
This is a one-tail test because we are checking if the proportion is greater than 0.15.
Plan the Test:
To conduct this hypothesis test, we need to ensure certain conditions are met:
1. Random Condition: This condition requires that the sample must be randomly selected. According to the problem, a separate random sample of adults was selected, so this condition is met.
2. 10% Condition: This condition states that the sample size should be no more than 10% of the population to ensure independence. Assuming the population of adults taking this medication is at least 1500, the sample size of 150 is indeed less than 10% of this assumed population (0.10 * 1500 = 1500), so this condition is met.
3. Large Counts Condition: This condition ensures that the sample is large enough for the normal approximation to be valid, requiring:
[tex]\[
np_0 \geq 10 \quad \text{and} \quad n(1-p_0) \geq 10
\][/tex]
For [tex]\(p_0 = 0.15\)[/tex] and [tex]\(n = 150\)[/tex]:
[tex]\[
150 \times 0.15 = 22.5 \quad (\text{which is } \geq 10)
\][/tex]
[tex]\[
150 \times (1-0.15) = 127.5 \quad (\text{which is } \geq 10)
\][/tex]
Both calculations satisfy the large counts condition.
4. Test Type: The test we use when checking a proportion with the conditions being met is a [tex]\(z\)[/tex]-test for one proportion.
Conclusion:
Based on the steps and conditions above, the true statements about this hypothesis test 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 [tex]\(z\)[/tex]-test for one proportion.
The [tex]\(H_a: p < 0.15\)[/tex] statement is false because our alternative hypothesis is that the proportion is greater than 0.15, not less.
State the Hypotheses:
1. Null Hypothesis ([tex]\(H_0\)[/tex]): The null hypothesis is that the true proportion of adults who experience side effects is equal to 0.15.
[tex]\[
H_0: p = 0.15
\][/tex]
2. Alternative Hypothesis ([tex]\(H_a\)[/tex]): The alternative hypothesis is what the researcher wants to prove, which is that the true proportion of adults who experience side effects is greater than 0.15.
[tex]\[
H_a: p > 0.15
\][/tex]
This is a one-tail test because we are checking if the proportion is greater than 0.15.
Plan the Test:
To conduct this hypothesis test, we need to ensure certain conditions are met:
1. Random Condition: This condition requires that the sample must be randomly selected. According to the problem, a separate random sample of adults was selected, so this condition is met.
2. 10% Condition: This condition states that the sample size should be no more than 10% of the population to ensure independence. Assuming the population of adults taking this medication is at least 1500, the sample size of 150 is indeed less than 10% of this assumed population (0.10 * 1500 = 1500), so this condition is met.
3. Large Counts Condition: This condition ensures that the sample is large enough for the normal approximation to be valid, requiring:
[tex]\[
np_0 \geq 10 \quad \text{and} \quad n(1-p_0) \geq 10
\][/tex]
For [tex]\(p_0 = 0.15\)[/tex] and [tex]\(n = 150\)[/tex]:
[tex]\[
150 \times 0.15 = 22.5 \quad (\text{which is } \geq 10)
\][/tex]
[tex]\[
150 \times (1-0.15) = 127.5 \quad (\text{which is } \geq 10)
\][/tex]
Both calculations satisfy the large counts condition.
4. Test Type: The test we use when checking a proportion with the conditions being met is a [tex]\(z\)[/tex]-test for one proportion.
Conclusion:
Based on the steps and conditions above, the true statements about this hypothesis test 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 [tex]\(z\)[/tex]-test for one proportion.
The [tex]\(H_a: p < 0.15\)[/tex] statement is false because our alternative hypothesis is that the proportion is greater than 0.15, not less.
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