Confidence Intervals Sec 6.1 to 6.3

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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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How do you calculate the margin of error for a confidence interval?

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The margin of error (E) can be calculated using the formula: E = z * (σ/√n), where 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 is the formula for constructing a confidence interval for the population mean?

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The formula for a confidence interval for the population mean is: (x̄ - E, x̄ + E), where x̄ is the sample mean and E is the margin of error.

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What does a 90% confidence level mean?

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A 90% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, approximately 90 of the 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 decreases the margin of error and results in a more precise confidence interval, while a smaller sample size increases the margin of error and leads to less precision.

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How do you determine the minimum sample size required for a specific margin of error?

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To determine the minimum sample size (n) required for a specific margin of error (E), use the formula: n = (z * σ / E)², where z is the z-score for the desired confidence level and σ is the population standard deviation.

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What is the relationship between confidence level and margin of error?

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As the confidence level increases, the margin of error also increases. This is because a higher confidence level requires a wider interval to ensure that the true parameter is captured.

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How do you interpret a confidence interval of (15.4, 18.4)?

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This confidence interval suggests that we are confident that the true population mean lies between 15.4 and 18.4.

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

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The standard deviation measures the variability of the data. It is used to calculate the margin of error, which affects the width of the confidence interval.

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What is a point estimate?

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A point estimate is a single value estimate of a population parameter, such as the sample mean (x̄) used to estimate the population mean (μ).