AP Statistics Confidence Intervals for Proportions

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What is a confidence interval?

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A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. It is associated with a confidence level that quantifies the level of confidence that the parameter lies within the interval.

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What does a 99% confidence interval indicate?

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A 99% confidence interval indicates that if we were to take many samples and build a confidence interval from each sample, approximately 99% of those intervals would contain the true population parameter.

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What is p-hat (\u03b7)?

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p-hat (\u03b7) is the sample proportion, calculated as the number of successes (e.g., individuals with a certain characteristic) divided by the total number of observations in the sample.

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How do you calculate the sample size needed for a given margin of error?

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The sample size (n) can be calculated using the formula: n = (Z^2 * p * (1-p)) / E^2, where Z is the Z-score corresponding to the desired confidence level, p is the estimated proportion, and E is the margin of error.

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What is the margin of error in a confidence interval?

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The margin of error is the range of values above and below the sample statistic in a confidence interval. It reflects the uncertainty associated with the sample estimate.

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How do you interpret a confidence interval?

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To interpret a confidence interval, you state that you are confident (at a specified level) that the true population parameter lies within the interval.

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What is the formula for calculating a confidence interval for a proportion?

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The formula for a confidence interval for a proportion is: p-hat ± Z * sqrt[(p-hat(1-p-hat))/n], where Z is the Z-score for the desired confidence level.

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What does it mean if a confidence interval is wide?

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A wide confidence interval indicates a high level of uncertainty about the estimate of the population parameter, often due to a small sample size or high variability in the data.

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What does it mean if a confidence interval is narrow?

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A narrow confidence interval indicates a high level of precision in estimating the population parameter, often resulting from a large sample size or low variability in the data.

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What is the significance of the confidence level?

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The confidence level represents the probability that the confidence interval will contain the true population parameter if the same sampling method is used repeatedly.