Answer :
We begin by stating the hypotheses for the test. The null hypothesis is
[tex]$$
H_0: p = 0.15,
$$[/tex]
which indicates that the proportion of adults experiencing side effects is 15%. Since the research question is whether the true proportion is greater than 0.15, the alternative hypothesis should be
[tex]$$
H_a: p > 0.15.
$$[/tex]
Thus, a provided alternative hypothesis of [tex]$H_a: p < 0.15$[/tex] would be incorrect.
Next, we check the conditions required to perform a [tex]$z$[/tex]-test for one proportion.
1. Random Condition:
The problem states that a random sample of 150 adults is selected. This satisfies the random condition.
2. 10% Condition:
The sample size (150) should be less than 10% of the population. Assuming the overall population of adults is large, this condition is met.
3. Large Counts Condition:
We verify that the expected counts of successes and failures are at least 10. With [tex]$p = 0.15$[/tex] and [tex]$n = 150$[/tex]:
[tex]$$
np = 150 \times 0.15 = 22.5,
$$[/tex]
and
[tex]$$
n(1-p) = 150 \times 0.85 = 127.5.
$$[/tex]
Since both values are greater than 10, the large counts condition is satisfied.
4. Type of Test:
Given that all conditions are met and the data deals with one proportion, the appropriate test is a [tex]$z$[/tex]-test for one proportion.
In summary, the true statements are:
1. [tex]$H_0: p = 0.15$[/tex]
2. (Not true) [tex]$H_a: p < 0.15$[/tex] is incorrect; the correct alternative should be [tex]$p > 0.15$[/tex].
3. The random condition is met.
4. The 10% condition is met.
5. The large counts condition is met.
6. The test is a [tex]$z$[/tex]-test for one proportion.
Therefore, the correct statements are numbers 1, 3, 4, 5, and 6.
[tex]$$
H_0: p = 0.15,
$$[/tex]
which indicates that the proportion of adults experiencing side effects is 15%. Since the research question is whether the true proportion is greater than 0.15, the alternative hypothesis should be
[tex]$$
H_a: p > 0.15.
$$[/tex]
Thus, a provided alternative hypothesis of [tex]$H_a: p < 0.15$[/tex] would be incorrect.
Next, we check the conditions required to perform a [tex]$z$[/tex]-test for one proportion.
1. Random Condition:
The problem states that a random sample of 150 adults is selected. This satisfies the random condition.
2. 10% Condition:
The sample size (150) should be less than 10% of the population. Assuming the overall population of adults is large, this condition is met.
3. Large Counts Condition:
We verify that the expected counts of successes and failures are at least 10. With [tex]$p = 0.15$[/tex] and [tex]$n = 150$[/tex]:
[tex]$$
np = 150 \times 0.15 = 22.5,
$$[/tex]
and
[tex]$$
n(1-p) = 150 \times 0.85 = 127.5.
$$[/tex]
Since both values are greater than 10, the large counts condition is satisfied.
4. Type of Test:
Given that all conditions are met and the data deals with one proportion, the appropriate test is a [tex]$z$[/tex]-test for one proportion.
In summary, the true statements are:
1. [tex]$H_0: p = 0.15$[/tex]
2. (Not true) [tex]$H_a: p < 0.15$[/tex] is incorrect; the correct alternative should be [tex]$p > 0.15$[/tex].
3. The random condition is met.
4. The 10% condition is met.
5. The large counts condition is met.
6. The test is a [tex]$z$[/tex]-test for one proportion.
Therefore, the correct statements are numbers 1, 3, 4, 5, and 6.
