justifying a claim based on a confidence interval for a population proportion

The conditions are not met, because 27 bicycle owners is not a large enough count on its own.

The conditions are not met, because the sample proportion, 0.30, is not close enough to 0.50.

The conditions are not met, because a sample of 90 people is too small relative to a town of 800 residents.

All conditions are met, because the sample size, 90, is larger than 30.

All conditions are met, because the sample was random, is under 10 percent of the town, and both counts are at least 10.

There is a problem, because the sample size of 1,200 exceeds 10 percent of the state's population.

There is a problem, because a proportion as low as 3.5 percent can never be estimated using a confidence interval.

There is a problem, because $n\hat{p}$, which is 42, is less than $n(1-\hat{p})$, which is 1,158.

There is no problem, because $n\hat{p}$, approximately 42, and $n(1-\hat{p})$, approximately 1,158, are both at least 10, so the large counts condition is satisfied.

There is a problem, because $n\hat{p}$ is approximately 42, which is at least 10, but this alone does not guarantee both large counts conditions are satisfied without checking $n(1-\hat{p})$ as well.