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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.

Example 1

Ahmad saw a report that claimed 57% 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% 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.
Based on his interval, is it plausible that 57% of all students at his university would agree that a third party is needed?
No, it isn't. The interval says that plausible values for the true proportion are between 59.9% and 78.1%. Since the interval doesn't contain 57%, it doesn't seem plausible that 57% of students at this university would agree. In other words, the entire interval is above 57%, so the true proportion at this university is likely higher.

Example 2

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% 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.
Based on her interval, is it plausible that 57% of students at her school would agree that a third party is needed?
Yes. Since the interval contains 57%, it is a plausible value for the population proportion.
Does her interval provide evidence that the true proportion of students at her school who would agree that a third party is needed is 57%?
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% or as high as 74.3%, 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%.

Example 3: Try it out!

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% 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% confidence interval for p, the proportion of attempts that will be rewarded with the bonus. The resulting interval was (0.323,0.517).
What does this interval suggest?
Choose 1 answer:

Example 4: Try it out!

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% 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% confidence interval for p, the proportion of attempts that will be rewarded with the rare item, which was 0.12±0.06.
What does this interval suggest?
Choose 1 answer:

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