Justifying Claims: Confidence Interval, Population Proportion Practice Test

A streaming service randomly samples 2,000 subscribers to estimate the proportion who use the service daily. The company wants to claim that between 45% and 55% of all subscribers use the service daily. A 95% confidence interval for the population proportion is $(0.46, 0.52)$. Is the claim supported by the confidence interval?

A school principal claims that at least 80% of students at the school eat breakfast before arriving. A random sample was taken and a 90% confidence interval for the true proportion $p$ of students who eat breakfast was computed as $(0.78,\ 0.85)$. Is the claim supported by the confidence interval?