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Answer :
We start by stating that the director’s goal is to test if the true proportion of first-year students living on campus at this private institution is different from the national proportion of $0.76$. With that in mind, we set up the hypotheses as follows:
$$
H_0: p = 0.76 \quad \text{and} \quad H_A: p \neq 0.76.
$$
1. Since the null hypothesis is that the proportion is equal to $0.76$, the statement
$$H_0: p = 0.76$$
is correct.
2. The statement
$$H_0: p = 0.89$$
does not match the national average of $0.76$, so it is incorrect.
3. The problem mentions that a random sample of $46$ first-year students was selected. This satisfies the random condition.
4. The 10% condition requires the sample size to be less than $10\%$ of the population. Given that the population of first-year students is large, the sample size of $46$ meets this criterion.
5. For the large counts condition necessary for a one-proportion $z$-test, we check that both the expected count of successes and failures are at least $10$. With a sample size $n = 46$ and $p_0 = 0.76$, these values are:
$$
np_0 = 46 \times 0.76 \approx 34.96,
$$
$$
n(1-p_0) = 46 \times 0.24 \approx 11.04.
$$
Since both values are greater than $10$, the large counts condition is satisfied.
6. Finally, given the conditions on the sample (random sampling, 10% condition, and large counts), it is appropriate to use a one-proportion $z$-test.
Thus, the following statements are true:
- $H_0: p = 0.76$
- The random condition is met.
- The 10\% condition is met.
- The large counts condition is met.
- The test is a $z$-test for one proportion.
The only statement that is not true is $H_0: p = 0.89$.
So the true statements are numbers 1, 3, 4, 5, and 6.
$$
H_0: p = 0.76 \quad \text{and} \quad H_A: p \neq 0.76.
$$
1. Since the null hypothesis is that the proportion is equal to $0.76$, the statement
$$H_0: p = 0.76$$
is correct.
2. The statement
$$H_0: p = 0.89$$
does not match the national average of $0.76$, so it is incorrect.
3. The problem mentions that a random sample of $46$ first-year students was selected. This satisfies the random condition.
4. The 10% condition requires the sample size to be less than $10\%$ of the population. Given that the population of first-year students is large, the sample size of $46$ meets this criterion.
5. For the large counts condition necessary for a one-proportion $z$-test, we check that both the expected count of successes and failures are at least $10$. With a sample size $n = 46$ and $p_0 = 0.76$, these values are:
$$
np_0 = 46 \times 0.76 \approx 34.96,
$$
$$
n(1-p_0) = 46 \times 0.24 \approx 11.04.
$$
Since both values are greater than $10$, the large counts condition is satisfied.
6. Finally, given the conditions on the sample (random sampling, 10% condition, and large counts), it is appropriate to use a one-proportion $z$-test.
Thus, the following statements are true:
- $H_0: p = 0.76$
- The random condition is met.
- The 10\% condition is met.
- The large counts condition is met.
- The test is a $z$-test for one proportion.
The only statement that is not true is $H_0: p = 0.89$.
So the true statements are numbers 1, 3, 4, 5, and 6.
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