AP Stats - Unit 5 - Confidence Intervals: Means

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What is a confidence interval?

Back

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 95% confidence level mean?

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A 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true population mean.

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How is the standard error of the mean calculated?

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The standard error of the mean (SEM) is calculated as the standard deviation (σ) divided by the square root of the sample size (n): SEM = σ/√n.

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What is the relationship between sample size and confidence interval width?

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As the sample size increases, the width of the confidence interval decreases, leading to a more precise estimate of the population parameter.

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What happens to the confidence interval when the confidence level is increased?

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Increasing the confidence level results in a wider confidence interval, as it accommodates a larger range of possible sample means.

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What is the formula for a confidence interval for the mean?

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The formula for a confidence interval for the mean is: CI = x̄ ± z*(σ/√n), where x̄ is the sample mean, z* is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

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What does it mean if two confidence intervals overlap?

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If two confidence intervals overlap, it suggests that there is not enough evidence to conclude that the two sample means are significantly different from each other.

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What is the Central Limit Theorem?

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The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is sufficiently large (usually n ≥ 30).

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What is the significance of the margin of error in a confidence interval?

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The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It is half the width of the confidence interval.

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How does variance affect the confidence interval?

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Higher variance in the data leads to a wider confidence interval, as it indicates greater variability in the sample data.