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 the symbol p̂ represent?

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The symbol p̂ (p-hat) represents the sample proportion, which is the ratio of the number of successes to the total number of trials in a sample.

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How do you calculate the sample size needed for a confidence interval?

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

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What is the critical value z* for a 90% confidence level?

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The critical value z* for a 90% confidence level is approximately 1.645.

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What is the critical value z* for a 97% confidence level?

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The critical value z* for a 97% confidence level is approximately 2.17.

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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 create a confidence interval from each sample, approximately 99% of those intervals would contain the true population parameter.

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How do you interpret a confidence interval of (0.652, 0.868)?

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This means that based on the sample, we are 99% confident that the true proportion of the population lies between 65.2% and 86.8%.

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What is the formula for calculating the margin of error (E) in a confidence interval?

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The margin of error (E) can be calculated using the formula: E = z* * sqrt((p̂ * (1 - p̂)) / n), where z* is the critical value, p̂ is the sample proportion, and n is the sample size.

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What is the relationship between sample size and margin of error?

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As the sample size increases, the margin of error decreases, leading to a more precise confidence interval.

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What is the significance of the sample proportion in statistics?

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The sample proportion is significant because it serves as an estimate of the true population proportion and is used in calculating confidence intervals and hypothesis tests.