Confidence Interval with Means Review

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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.

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What does a 95% confidence interval mean?

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

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

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

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What is the margin of error in a confidence interval?

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The margin of error is the amount added to and subtracted from the sample statistic to create the confidence interval. It reflects the uncertainty in the estimate.

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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 if the sample size is sufficiently large, regardless of the shape of the population distribution.

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What is the significance of the sample size (n) in constructing confidence intervals?

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A larger sample size reduces the margin of error and results in a more precise confidence interval. It also allows the sampling distribution to approximate normality more closely.

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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 margin of error.

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What is the standard error of the mean?

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The standard error of the mean (SEM) is the standard deviation of the sampling distribution of the sample mean. It is calculated as σ/√n, where σ is the population standard deviation and n is the sample size.

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How do you interpret a confidence interval?

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To interpret a confidence interval, you state that you are confident (e.g., 95%) that the true population parameter lies within the interval.

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What is the difference between a population parameter and a sample statistic?

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A population parameter is a value that describes a characteristic of a population (e.g., population mean), while a sample statistic is a value that describes a characteristic of a sample (e.g., sample mean).