AP Statistics TPS 8.3

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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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Which factors affect the width of a confidence interval?

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The width of a confidence interval is affected by the sample size, the sample standard deviation, and the critical value (t* or z*). The sample mean does not affect the width.

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What is the critical value in a confidence interval?

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The critical value is a factor used to compute the margin of error in a confidence interval. It corresponds to the desired confidence level and is derived from the t-distribution or z-distribution.

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When should a t-distribution be used instead of a z-distribution?

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A t-distribution should be used when the sample size is small (typically n < 30) and/or when the population standard deviation is unknown.

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How do you calculate a confidence interval for a population mean?

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To calculate a confidence interval for a population mean, use the formula: CI = sample mean ± (critical value * (sample standard deviation / √sample size)).

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What does a 99% confidence interval imply?

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A 99% confidence interval implies 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 mean.

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What is the significance of the sample size in confidence intervals?

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A larger sample size generally results in a narrower confidence interval, indicating more precision in estimating the population parameter.

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What does it mean if a confidence interval does not include a certain value?

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If a confidence interval does not include a certain value, it suggests that the value is unlikely to be the true population parameter at the specified confidence level.

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What is the formula for the margin of error in a confidence interval?

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The margin of error (ME) is calculated as: ME = critical value * (standard deviation / √sample size).

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What is the relationship between confidence level and width of the confidence interval?

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As the confidence level increases, the width of the confidence interval also increases, meaning we are more certain that the interval contains the true parameter.