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### Course: AP®︎/College Statistics > Unit 10

Lesson 2: Confidence intervals for proportions- Conditions for valid confidence intervals for a proportion
- Conditions for confidence interval for a proportion worked examples
- Reference: Conditions for inference on a proportion
- Conditions for a z interval for a proportion
- Critical value (z*) for a given confidence level
- Finding the critical value z* for a desired confidence level
- Example constructing and interpreting a confidence interval for p
- Calculating a z interval for a proportion
- Interpreting a z interval for a proportion
- Determining sample size based on confidence and margin of error
- 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 $0.12\pm 0.06$ .

## Want to join the conversation?

- Could we take 100% of a confidence level and why if we couldn't?(10 votes)
- A 100% confidence interval for p would be (0, 1) which tells us nothing we didn't already know.(32 votes)

- Imagine that I have an interval of 0.12 +- 3. Plausible values are in the 0.09:0.15. Are they "equally plausible/likely"?(7 votes)
- If you are talking about the population proportion, then no. The population proportion is a fixed value, and the confidence interval is just the probability that your sample proportion is within a set of values.

I hope this helped answer your question(2 votes)

- Why do we use a z interval for a proportion, but a t interval for means(3 votes)
- Because you don't need to use the standard deviation when making CI's for proportions, but when making CI's for means you often use the sample standard deviation which (in small sample sizes) vary more which causes us to use the t-distribution instead of "normal calculations" which is how the Z interval is based.(4 votes)

- Example 1 says plausible values lie inside the confidence interval. But if we consider a 60% confidence interval, isn't it plausible that values outside the interval are our desired population parameter? Is there a threshold confidence level for intervals, below which we can say that values in the vicinity of the confidence interval are 'plausible' too?(4 votes)
- Wording of 1st question seems a little off to me. I'd like to suggest question as, "Is it most likely"

that 57% of all students would agree to a 3rd political party?(3 votes)- Changing the wording to "Is it most likely that 57% of all students would agree to a 3rd political party?" helps clarify that it's about likelihood rather than certainty.(1 vote)

- The second question is wrong. In the question it says 12-6% but in the explanation of the answers it says 18%(1 vote)
- The question says 12 plus/minus 6%, so between 6 and 18%.(3 votes)

- What are plausible values? Are they considered values that are likely to represent the parameter? Because in example one, 57% is not a plausible value but it is a possible one.(2 votes)
- All values are possible because there is a multitude of parameters. Especially prop parameters can be very variable.(2 votes)

- Imagine that I have an interval of 0.12 +- 3. Plausible values are in the 0.09:0.15. Are they "equally plausible/likely"?(1 vote)
- All values within a confidence interval are considered equally plausible as potential true values of the parameter being estimated. There's no suggestion that some are more likely than others within the interval.(1 vote)

- In the video game examples, they are saying that the creators are willing to assume that the sample is random but how can that be? Wouldn't it only be able to be random, say, if they got a bunch of different people to play it, or they used a bunch of different weapons and equipment when they fought the enemy?(1 vote)
- The randomness is based on how the game's reward system is set up to operate independently from one attempt to another. If the game's mechanics ensure that each try is independent and unaffected by external factors like player strategy or gear, it's considered random.(1 vote)

- How would we know how much confidence if not given?(1 vote)
- Without explicit information, you cannot deduce the confidence level from just the interval itself. Typical confidence levels are 90%, 95%, or 99%.(1 vote)