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Answer :
We are testing whether the proportion of first-year students who live on campus at the school differs from the national value of [tex]$76\%$[/tex]. Here is the step-by-step process:
1. Setting Up the Hypotheses
The null hypothesis is that the true proportion is equal to the national rate:
[tex]$$
H_0: p = 0.76.
$$[/tex]
The alternative hypothesis for a two-sided test is:
[tex]$$
H_a: p \neq 0.76.
$$[/tex]
Therefore, the statement "[tex]$H_0: p=0.76$[/tex]" is true and "[tex]$H_0: p=0.89$[/tex]" is not a valid null hypothesis for this problem.
2. Checking the Conditions
- Random Condition:
The problem states that a random sample of 46 first-year students was selected. This satisfies the random condition.
- 10% Condition:
The sample must be less than [tex]$10\%$[/tex] of the entire population. Since the institution is large, the sample of 46 students is less than [tex]$10\%$[/tex] of the school’s first-year population. Therefore, this condition is met.
- Large Counts Condition:
We need to check that the expected counts for successes and failures are at least 10.
The expected number of successes (students living on campus) is:
[tex]$$
n \cdot p = 46 \cdot 0.76 \approx 34.96.
$$[/tex]
The expected number of failures is:
[tex]$$
n \cdot (1-p) = 46 \cdot 0.24 \approx 11.04.
$$[/tex]
Both numbers are greater than 10, so the large counts condition is satisfied.
3. Identifying the Test
When comparing a sample proportion to a known proportion (with all conditions satisfied), we use a [tex]$z$[/tex]-test for one proportion.
4. Conclusion
The statements that are true based on this analysis are:
- [tex]$H_0: p = 0.76$[/tex]
- The random condition is met.
- The [tex]$10\%$[/tex] condition is met.
- The large counts condition is met.
- The test is a [tex]$z$[/tex]-test for one proportion.
Thus, the correct statements are indexed as 1, 3, 4, 5, and 6.
1. Setting Up the Hypotheses
The null hypothesis is that the true proportion is equal to the national rate:
[tex]$$
H_0: p = 0.76.
$$[/tex]
The alternative hypothesis for a two-sided test is:
[tex]$$
H_a: p \neq 0.76.
$$[/tex]
Therefore, the statement "[tex]$H_0: p=0.76$[/tex]" is true and "[tex]$H_0: p=0.89$[/tex]" is not a valid null hypothesis for this problem.
2. Checking the Conditions
- Random Condition:
The problem states that a random sample of 46 first-year students was selected. This satisfies the random condition.
- 10% Condition:
The sample must be less than [tex]$10\%$[/tex] of the entire population. Since the institution is large, the sample of 46 students is less than [tex]$10\%$[/tex] of the school’s first-year population. Therefore, this condition is met.
- Large Counts Condition:
We need to check that the expected counts for successes and failures are at least 10.
The expected number of successes (students living on campus) is:
[tex]$$
n \cdot p = 46 \cdot 0.76 \approx 34.96.
$$[/tex]
The expected number of failures is:
[tex]$$
n \cdot (1-p) = 46 \cdot 0.24 \approx 11.04.
$$[/tex]
Both numbers are greater than 10, so the large counts condition is satisfied.
3. Identifying the Test
When comparing a sample proportion to a known proportion (with all conditions satisfied), we use a [tex]$z$[/tex]-test for one proportion.
4. Conclusion
The statements that are true based on this analysis are:
- [tex]$H_0: p = 0.76$[/tex]
- The random condition is met.
- The [tex]$10\%$[/tex] condition is met.
- The large counts condition is met.
- The test is a [tex]$z$[/tex]-test for one proportion.
Thus, the correct statements are indexed as 1, 3, 4, 5, and 6.
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