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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Do you want further details or have any questions?

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.