Interpreting a z interval for a proportion

Once we build a confidence interval for a proportion, it's important to be able to interpret what the interval tells us about the population, and what it doesn't tell us. Let's look at few examples that demonstrate how to interpret a confidence interval for a proportion.

Ahmad saw a report that claimed $57\mathrm{\%}$‍ of US adults think a third major political party is needed. He was curious how students at his large university felt on the topic, so he asked the same question to a random sample of $100$‍ students and made a $95\mathrm{\%}$‍ confidence interval to estimate the proportion of students who agreed that a third major political party was needed. His resulting interval was $(0.599,0.781)$‍. Assume that the conditions for inference were all met.

No, it isn't. The interval says that plausible values for the true proportion are between $59.9\mathrm{\%}$‍ and $78.1\mathrm{\%}$‍. Since the interval doesn't contain $57\mathrm{\%}$‍, it doesn't seem plausible that $57\mathrm{\%}$‍ of students at this university would agree. In other words, the entire interval is above $57\mathrm{\%}$‍, so the true proportion at this university is likely higher.

Ahmad's sister, Diedra, was curious how students at her large high school would answer the same question, so she asked it to a random sample of $100$‍ students at her school. She also made a $95\mathrm{\%}$‍ confidence interval to estimate the proportion of students at her school who would agree that a third party is needed. Her interval was $(0.557,0.743)$‍. Assume that the conditions for inference were all met.

Yes. Since the interval contains $57\mathrm{\%}$‍, it is a plausible value for the population proportion.

No. Confidence intervals don't give us evidence that a parameter equals a specific value; they give us a range of plausible values. Diedra's interval says that the true proportion of students who agree could be as low as $55.7\mathrm{\%}$‍ or as high as $74.3\mathrm{\%}$‍, and that values outside of this interval aren't likely. So it wouldn't be appropriate to say this interval supports the value of $57\mathrm{\%}$‍.

A video game gives players a reward of gold coins after they defeat an enemy. The creators of the game want players to have a chance at earning bonus coins when they defeat a certain challenging enemy. The creators attempt to program the game so that the bonus is awarded randomly with a $30\mathrm{\%}$‍ probability after the enemy is defeated.

To see if the bonus is being awarded as intended, the creators defeated the enemy in a series of $100$‍ attempts (they're willing to treat this as a random sample). After each attempt, they recorded whether or not the bonus was awarded. They used the results to build a $95\mathrm{\%}$‍ confidence interval for $p$‍, the proportion of attempts that will be rewarded with the bonus. The resulting interval was $(0.323,0.517)$‍.

The creators of the video game also want players to have a chance at earning a rare item when they defeat a challenging enemy. The creators attempt to program the game so that the rare item is awarded randomly with a $15\mathrm{\%}$‍ probability after the enemy is defeated.

To see if the rare item is being awarded as intended, the creators defeated the enemy in a series of $100$‍ attempts (they're willing to treat this as a random sample). After each attempt, they recorded whether or not the rare item was awarded. They used the results to build a $95\mathrm{\%}$‍ confidence interval for $p$‍, the proportion of attempts that will be rewarded with the rare item, which was $0.12\pm 0.06$‍.