Confidence Interval Practice

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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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What does the t* value represent in statistics?

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The t* value is a critical value from the t-distribution that is used in constructing confidence intervals, particularly when the sample size is small and the population standard deviation is unknown.

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

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The margin of error is calculated by multiplying the critical value (z* or t*) by the standard error of the sample mean. The formula is: Margin of Error = Critical Value * Standard 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 the sample standard deviation divided by the square root of the sample size (n).

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

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A z-score is used when the population standard deviation is known and the sample size is large (n > 30), while a t-score is used when the population standard deviation is unknown and the sample size is small (n < 30).

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What is the critical value for a 99% confidence interval?

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The critical value for a 99% confidence interval can be found using the t-distribution or z-distribution tables. For a large sample size, the z-score is approximately 2.576.

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How do you interpret a 90% confidence interval?

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A 90% confidence interval means that if we were to take many samples and construct a confidence interval from each sample, approximately 90% of those intervals would contain the true population parameter.

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What is the formula for calculating a confidence interval for a population proportion?

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The formula for a confidence interval for a population proportion is: CI = p̂ ± z* * √(p̂(1 - p̂) / n), where p̂ is the sample proportion, z* is the critical value, and n is the sample size.

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What does it mean if a confidence interval does not include the value 0?

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If a confidence interval does not include the value 0, it suggests that there is a statistically significant difference or effect, indicating that the null hypothesis may be rejected.

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

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To determine the sample size needed for a specific margin of error, use the formula: n = (z*^2 * p̂(1 - p̂)) / E^2, where z* is the critical value, p̂ is the estimated proportion, and E is the desired margin of error.