Answer :
We are given that historically 15% of adults experience side effects ([tex]$p_0 = 0.15$[/tex]) and a new study of [tex]$n = 150$[/tex] adults was performed, with 32 reporting side effects. The goal is to test if the true proportion is greater than 0.15. Let’s work through each step.
1. State the Hypotheses
The null hypothesis must represent the status quo:
[tex]$$H_0: p = 0.15.$$[/tex]
This statement is correct.
The alternative hypothesis should reflect the research question that the true proportion is greater than 0.15. Hence it should be
[tex]$$H_a: p > 0.15.$$[/tex]
In the given statements, [tex]$H_a: p < 0.15$[/tex] is listed, which is not correct. Thus, the statement for the alternative hypothesis is false.
2. Plan the Test
a. Random Condition:
It is assumed that the sample of 150 adults is randomly selected from the population. This condition is satisfied.
b. 10% Condition:
To ensure independence, the sample size should be less than 10% of the entire population. Since 150 is a very small fraction of the total number of adults, this condition is met.
c. Large Counts Condition:
For a proportion hypothesis test, we need
[tex]$$np_0 \ge 10 \quad \text{and} \quad n(1 - p_0) \ge 10.$$[/tex]
Here:
[tex]$$np_0 = 150 \times 0.15 = 22.5,$$[/tex]
[tex]$$n(1 - p_0) = 150 \times 0.85 = 127.5.$$[/tex]
Both are greater than 10, so the large counts condition is met.
d. Test Choice:
Since all conditions are satisfied, a [tex]$z$[/tex]-test for one proportion is appropriate.
3. Summary of True/False Statements
- [tex]$H_0: p=0.15$[/tex] is true.
- [tex]$H_a: p<0.15$[/tex] is false because the alternative should state [tex]$p > 0.15$[/tex].
- The random condition is met (true).
- The 10% condition is met (true).
- The large counts condition is met (true).
- The test is a [tex]$z$[/tex]-test for one proportion (true).
Therefore, the correct responses are:
True, False, True, True, True, True.
1. State the Hypotheses
The null hypothesis must represent the status quo:
[tex]$$H_0: p = 0.15.$$[/tex]
This statement is correct.
The alternative hypothesis should reflect the research question that the true proportion is greater than 0.15. Hence it should be
[tex]$$H_a: p > 0.15.$$[/tex]
In the given statements, [tex]$H_a: p < 0.15$[/tex] is listed, which is not correct. Thus, the statement for the alternative hypothesis is false.
2. Plan the Test
a. Random Condition:
It is assumed that the sample of 150 adults is randomly selected from the population. This condition is satisfied.
b. 10% Condition:
To ensure independence, the sample size should be less than 10% of the entire population. Since 150 is a very small fraction of the total number of adults, this condition is met.
c. Large Counts Condition:
For a proportion hypothesis test, we need
[tex]$$np_0 \ge 10 \quad \text{and} \quad n(1 - p_0) \ge 10.$$[/tex]
Here:
[tex]$$np_0 = 150 \times 0.15 = 22.5,$$[/tex]
[tex]$$n(1 - p_0) = 150 \times 0.85 = 127.5.$$[/tex]
Both are greater than 10, so the large counts condition is met.
d. Test Choice:
Since all conditions are satisfied, a [tex]$z$[/tex]-test for one proportion is appropriate.
3. Summary of True/False Statements
- [tex]$H_0: p=0.15$[/tex] is true.
- [tex]$H_a: p<0.15$[/tex] is false because the alternative should state [tex]$p > 0.15$[/tex].
- The random condition is met (true).
- The 10% condition is met (true).
- The large counts condition is met (true).
- The test is a [tex]$z$[/tex]-test for one proportion (true).
Therefore, the correct responses are:
True, False, True, True, True, True.
