AP Stats Chapter 8 CONFIDENCE INTERVALS

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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, approximately 95 of the intervals would contain the true population parameter.

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What is the difference between a z-distribution and a t-distribution?

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The z-distribution is used for large sample sizes (n > 30) or when the population standard deviation is known, while the t-distribution is used for smaller sample sizes (n < 30) and when the population standard deviation is unknown.

Front

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

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The formula is: CI = x̄ ± t*(s/√n), where x̄ is the sample mean, t* is the t-score for the desired confidence level, s is the sample standard deviation, and n is the sample size.

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What is the role of sample size in determining the width of a confidence interval?

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Increasing the sample size decreases the width of the confidence interval, making it more precise, while decreasing the sample size increases the width, making it less precise.

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What does it mean to say that a confidence interval is 'narrower'?

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A narrower confidence interval indicates a more precise estimate of the population parameter, meaning there is less uncertainty about the true value.

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How does increasing the confidence level affect the confidence interval?

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Increasing the confidence level results in a wider confidence interval because it requires a larger margin of error to ensure that the true parameter is captured within the interval.

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What is the critical value in the context of confidence intervals?

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The critical value is a factor used to compute the margin of error, which is determined by the desired confidence level and the distribution of the sample data.

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What is the interpretation of a confidence interval of (150, 350) for average calories?

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We are 95% confident that the true average caloric content of the population lies between 150 and 350 calories.

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What does it mean when we say 'the method produces intervals which include the true proportion about 98% of the time'?

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It means that if we were to repeat the sampling process many times, approximately 98% of the confidence intervals calculated would contain the true population proportion.