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### Course: Statistics and probability > Unit 11

Lesson 2: Estimating a population proportion- Reference: Conditions for inference on a proportion
- Conditions for a z interval for a proportion
- Finding the critical value z* for a desired confidence level
- Calculating a z interval for a proportion
- Interpreting a z interval for a proportion
- Sample size and margin of error in a z interval for p

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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 $57\mathrm{\%}$ 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.

**Based on his interval, is it plausible that**$57\mathrm{\%}$ 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 $59.9\mathrm{\%}$ 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.

## 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 $95\mathrm{\%}$ 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.

**Based on her interval, is it plausible that**$57\mathrm{\%}$ of students at her school would agree that a third party is needed?

Yes. Since the interval contains $57\mathrm{\%}$ , 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**$57\mathrm{\%}$ ?

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 $57\mathrm{\%}$ .

## 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 $30\mathrm{\%}$ 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 $(0.323,0.517)$ .

## 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 $15\mathrm{\%}$ 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 $0.12\pm 0.06$ .

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