Assume you flip a coin 80 times, and calculate that it has a sample mean of 0.56, a 90% confidence interval of [0.61, 0.52], and a 99% confidence interval of [0.63, 0.49]. What can you conclude?

Pick ONE option

the coin is fair with certainty p < 0.01

the coin is not fair with certainty p < 0.01

the coin is not fair with certainty p < 0.1 but not p < 0.01

the coin is fair with certainty p < 0.1 but not p < 0.01

Given the problem, let's break it down:

- **Sample mean:** 0.56, indicating that heads appeared 56% of the time.
- **90% confidence interval:** [0.61, 0.52], implying that there's a 90% chance the true proportion lies between 0.52 and 0.61.
- **99% confidence interval:** [0.63, 0.49], implying that there's a 99% chance the true proportion lies between 0.49 and 0.63.

### Null Hypothesis: The coin is fair, meaning the true proportion of heads is 0.5.
- In both the 90% and 99% confidence intervals, the value 0.5 lies within the interval.

### Conclusions:
- Since 0.5 (the expected proportion for a fair coin) is within both confidence intervals, **we do not have enough evidence to reject the hypothesis that the coin is fair.**
- However, at a **90% confidence level**, the interval is close to excluding 0.5, meaning there's some evidence the coin might not be fair, but it's not strong enough for a higher confidence level like 99%.

Thus, the conclusion is:
**"The coin is fair with certainty p < 0.1 but not p < 0.01."**

This is because the 90% confidence interval includes 0.5, but does not exclude it, suggesting that we can't reject the null hypothesis at the 99% level.

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**Related Questions:**
1. How does sample size affect the width of confidence intervals?
2. What does it mean when the confidence interval includes 0.5 for a coin flip?
3. What are Type I and Type II errors in hypothesis testing?
4. How does confidence level impact hypothesis testing?
5. Can a 95% confidence interval provide a different conclusion from a 90% or 99% interval?

**Tip:** Confidence intervals give a range of values within which the true population parameter is expected to fall, helping assess the uncertainty around a sample estimate.

Assume you flip a coin 80 times, and calculate that it has a sample mean of 0.56, a 90% confidence interval of [0.61, 0.52], and a 99% confidence interval of [0.63, 0.49]. What can you conclude?

Analyze a problem involving 80 coin flips to determine if the coin is fair using confidence intervals. This problem covers hypothesis testing, sample proportions, and statistical concepts like the 90% and 99% confidence intervals. Suitable for undergraduate students.

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