8.7 Writing conclusions

University A advertises that \(40\)% of its students are using AI daily. To assess the claim, the student union takes a sample of \(78\) students, and finds \(29\) are using AI daily. Using software, the \(z\)-score was \(z = -0.515\).

Each of these alternative hypotheses is incorrect. Explain the main reason why. Then, write the correct alternative hypothesis.
1. \(H_1\): The proportion of students using AI daily is significantly lower than \(0.40\).
2. \(H_1 \ne 0.40\).
3. \(H_1\): \(p < 0.40\), where \(p\) is the population proportion of students using AI daily.
4. \(H_1\): The number of students using AI daily is not equal to the university's claim.
Estimate the \(P\)-value.

Each of these conclusions are incorrect. Explain the main reason why. Then, write the correct conclusion.
1. There is strong evidence that the proportion of students using AI daily is \(0.40\).
2. There is no evidence of a large difference between the population proportion of students using AI daily and the claimed proportion.
3. The sample proportion is \(0.40\).

University B makes the same claim. To assess the claim, the student union takes a sample of \(66\) students and finds \(35\) are using AI daily. Using software, the \(z\)-score was \(z = 2.12\).

Estimate the \(P\)-value.
Each of these conclusions are incorrect. Explain the main reason why. Then, write the correct conclusion.
1. There is evidence of a significant difference between the population proportion of students using AI daily and the claim.
2. There is no evidence that the claimed proportion is \(0.40\).
3. The claimed proportion is not \(0.40\).