Answer :
We begin by stating the hypotheses and checking that the conditions for inference are met.
1. \textbf{State: Hypotheses}
- The null hypothesis is
[tex]$$H_0: p = 0.15.$$[/tex]
- Since the researcher wants to determine if the true proportion is greater than 0.15, the alternative hypothesis should be
[tex]$$H_a: p > 0.15.$$[/tex]
Note that the alternative hypothesis provided as [tex]$$H_a: p < 0.15$$[/tex] would be appropriate if one were testing for a decrease, but that is not the case here.
2. \textbf{Plan: Checking Conditions and Selecting the Test}
- Random Condition:
The sample is selected randomly, which satisfies the random condition.
- 10% Condition:
The sample of 150 adults should be less than 10% of the entire population. This condition is met provided the population is large enough, and it ensures the independence of observations.
- Large Counts Condition:
Under the null hypothesis, we compute the expected number of successes and failures.
The expected number of successes is
[tex]$$150 \times 0.15 = 22.5,$$[/tex]
and the expected number of failures is
[tex]$$150 \times 0.85 = 127.5.$$[/tex]
Since both expected counts are greater than 10, the large counts condition is satisfied.
- Type of Test:
Because we are testing a hypothesis about a single proportion, a [tex]$z$[/tex]-test for one proportion is appropriate.
3. \textbf{Conclusion on the Statements}
Based on our analysis:
- The statement [tex]$$H_0: p = 0.15$$[/tex] is true.
- The statement [tex]$$H_a: p < 0.15$$[/tex] is false (since the correct alternative hypothesis is [tex]$$H_a: p > 0.15$$[/tex]).
- The statement "The random condition is met" is true.
- The statement "The [tex]$10\%$[/tex] condition is met" is true.
- The statement "The large counts condition is met" is true.
- The statement "The test is a [tex]$z$[/tex]-test for one proportion" is true.
Thus, the true statements are the first, third, fourth, fifth, and sixth statements.
1. \textbf{State: Hypotheses}
- The null hypothesis is
[tex]$$H_0: p = 0.15.$$[/tex]
- Since the researcher wants to determine if the true proportion is greater than 0.15, the alternative hypothesis should be
[tex]$$H_a: p > 0.15.$$[/tex]
Note that the alternative hypothesis provided as [tex]$$H_a: p < 0.15$$[/tex] would be appropriate if one were testing for a decrease, but that is not the case here.
2. \textbf{Plan: Checking Conditions and Selecting the Test}
- Random Condition:
The sample is selected randomly, which satisfies the random condition.
- 10% Condition:
The sample of 150 adults should be less than 10% of the entire population. This condition is met provided the population is large enough, and it ensures the independence of observations.
- Large Counts Condition:
Under the null hypothesis, we compute the expected number of successes and failures.
The expected number of successes is
[tex]$$150 \times 0.15 = 22.5,$$[/tex]
and the expected number of failures is
[tex]$$150 \times 0.85 = 127.5.$$[/tex]
Since both expected counts are greater than 10, the large counts condition is satisfied.
- Type of Test:
Because we are testing a hypothesis about a single proportion, a [tex]$z$[/tex]-test for one proportion is appropriate.
3. \textbf{Conclusion on the Statements}
Based on our analysis:
- The statement [tex]$$H_0: p = 0.15$$[/tex] is true.
- The statement [tex]$$H_a: p < 0.15$$[/tex] is false (since the correct alternative hypothesis is [tex]$$H_a: p > 0.15$$[/tex]).
- The statement "The random condition is met" is true.
- The statement "The [tex]$10\%$[/tex] condition is met" is true.
- The statement "The large counts condition is met" is true.
- The statement "The test is a [tex]$z$[/tex]-test for one proportion" is true.
Thus, the true statements are the first, third, fourth, fifth, and sixth statements.
