Confidence Interval for the Mean (small sample)

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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 used to estimate the uncertainty around a sample statistic.

Tags

CCSS.HSS.IC.B.4

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What does a 99% confidence interval mean?

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A 99% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 99 of the intervals would contain the true population mean.

Tags

CCSS.HSS.IC.B.4

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What is the formula for calculating a confidence interval for the mean with a small sample size?

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

Tags

CCSS.HSS.IC.B.4

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What is the t-distribution?

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The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used when the sample size is small and the population standard deviation is unknown.

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How do you find the t-score for a confidence interval?

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The t-score can be found using a t-table or calculator, based on the desired confidence level and the degrees of freedom (df = n - 1, where n is the sample size).

Tags

CCSS.HSS.ID.A.4

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

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Larger sample sizes generally lead to narrower confidence intervals, indicating more precise estimates of the population parameter.

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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. This is because a higher confidence level requires a larger range to ensure that the true parameter is captured.

Tags

CCSS.HSS.IC.B.4

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What is the standard deviation in the context of confidence intervals?

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Standard deviation measures the amount of variation or dispersion in a set of values. In confidence intervals, it helps determine the spread of the data around the mean.

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How do you interpret a confidence interval?

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A confidence interval provides a range of values that is believed to contain the population mean. For example, a 95% CI of (2.5, 3.2) suggests we are 95% confident that the true mean lies within this range.

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What is the effect of outliers on confidence intervals?

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Outliers can significantly affect the mean and standard deviation, leading to wider confidence intervals and potentially misleading conclusions about the population parameter.