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## Statystyka - program rozszerzony

### Kurs: Statystyka - program rozszerzony > Rozdział 10

Lekcja 1: Przedziały ufności dla proporcji- Odniesienie: Warunki wnioskowania na temat proporcji
- Warunki dla przedziału Z dla proporcji
- Znajdowanie krytycznej wartości z* dla określonego poziomu ufności
- Obliczanie przedziału z dla proporcji
- Interpretowanie przedziału z dla proporcji
- Wielkość próby i margines błędu przedziału z dla p

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# Interpretowanie przedziału z dla proporcji

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.

## Przykład 1

Ahmad saw a report that claimed 57, percent 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, percent confidence interval to estimate the proportion of students who agreed that a third major political party was needed. His resulting interval was left parenthesis, 0, point, 599, comma, 0, point, 781, right parenthesis. Assume that the conditions for inference were all met.

**Based on his interval, is it plausible that 57, percent 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, point, 9, percent and 78, point, 1, percent. Since the interval doesn't contain 57, percent, it doesn't seem plausible that 57, percent of students at this university would agree. In other words, the entire interval is above 57, percent, so the true proportion at this university is likely higher.

## Przykład 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, percent confidence interval to estimate the proportion of students at her school who would agree that a third party is needed. Her interval was left parenthesis, 0, point, 557, comma, 0, point, 743, right parenthesis. Assume that the conditions for inference were all met.

**Based on her interval, is it plausible that 57, percent of students at her school would agree that a third party is needed?**

Yes. Since the interval contains 57, percent, 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, percent?**

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, point, 7, percent or as high as 74, point, 3, percent, 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, percent.

## 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, percent 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, percent confidence interval for p, the proportion of attempts that will be rewarded with the bonus. The resulting interval was left parenthesis, 0, point, 323, comma, 0, point, 517, right parenthesis.

## 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, percent 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, percent confidence interval for p, the proportion of attempts that will be rewarded with the rare item, which was 0, point, 12, plus minus, 0, point, 06.

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