Nationwide, 76% of first-year college students live on campus. The director of housing at a large private institution would like to know if it is significantly different at his school. He selects a random sample of 46 first-year students and finds that 78% of them live on campus. Do these data provide convincing evidence that the true proportion of all first-year students who attend this private institution and live on campus differs from the national average? Use α = 0.05.

Which of the following statements are true?

a. \( H_0: p = 0.76 \)

b. \( H_0: p = 0.89 \)

c. The 10% condition is met.

d. The large counts condition is met.

e. The test is a z-test for one proportion.

f. The random condition is met.

The correct statements regarding the hypothesis testing of the proportion of first-year students living on campus compared to the national average are a, c, d, e, and f. These address the null hypothesis, the 10% condition, the large counts condition, the z-test modality, and the random condition selection. The correct answer is a, c, d, and f.

Understanding Hypothesis Testing for Proportions

The director of housing at a large private institution is interested in understanding if the proportion of first-year students living on campus is significantly different from the national average of 76%. To test this hypothesis, a z-test for one proportion can be used.

The null hypothesis, H₀: p = 0.76, states that the proportion at the private institution is the same as the national average. The alternative hypothesis would be Hₐ: p ≠ 0.76, representing a different proportion from the national average.

The 10% condition is met if the sample size is less than 10% of the population; here, it is likely met as there are tens of thousands of first-year students nationwide. The large counts condition is met if both np and n(1-p) are greater than 5. In this case, with 46 students sampled and 78% living on campus, 46(0.78) and 46(1-0.78) would be put to check if they are greater than 5, which they are.

One has to verify that the random condition is met, assuming the sample of 46 students was selected randomly. For the alpha level of 0.05, the director would reject the null hypothesis if the p-value from the z-test is less than 0.05, suggesting that the proportion at his institution is indeed significantly different from the national average.

The correct answer is a, c, d, and f.