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A confidence interval is an estimated range of values which is likely to include the population parameter.
Here's a detailed explanation for this definition:
1. What is a Confidence Interval?
- A confidence interval represents a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter.
2. Purpose of a Confidence Interval:
- The main purpose of a confidence interval is to provide an estimate for a population parameter (such as a population mean or proportion) along with an indication of uncertainty or variability in that estimate.
3. Interpretation:
- For example, if you have a 95% confidence interval for a population mean, it means that you are 95% confident that the true population mean falls within this interval. In essence, if you were to take many samples and build confidence intervals in the same way, approximately 95% of those intervals would contain the true population parameter.
4. Components of a Confidence Interval:
- A confidence interval typically consists of:
- A point estimate (e.g., sample mean).
- A margin of error which accounts for sampling variability.
- The confidence level which reflects how certain we are that the interval contains the population parameter.
5. Answering the Options:
- An estimated range of values likely to include the population parameter: This is the correct definition of a confidence interval.
- A single value used to estimate a population parameter: This describes a point estimate, not a confidence interval.
- The difference between the highest and lowest values in a sample: This describes the range of the sample data, not related to a confidence interval.
- The range within one standard deviation of the mean: This describes a specific range around the mean, often related to the empirical rule in normal distributions, but it does not account for the concept of a confidence interval.
In conclusion, a confidence interval is best defined as an estimated range of values which is likely to include the population parameter. This provides a more comprehensive understanding of the uncertainty inherent in the sample estimate and the likely range within which the true population parameter lies.
Here's a detailed explanation for this definition:
1. What is a Confidence Interval?
- A confidence interval represents a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter.
2. Purpose of a Confidence Interval:
- The main purpose of a confidence interval is to provide an estimate for a population parameter (such as a population mean or proportion) along with an indication of uncertainty or variability in that estimate.
3. Interpretation:
- For example, if you have a 95% confidence interval for a population mean, it means that you are 95% confident that the true population mean falls within this interval. In essence, if you were to take many samples and build confidence intervals in the same way, approximately 95% of those intervals would contain the true population parameter.
4. Components of a Confidence Interval:
- A confidence interval typically consists of:
- A point estimate (e.g., sample mean).
- A margin of error which accounts for sampling variability.
- The confidence level which reflects how certain we are that the interval contains the population parameter.
5. Answering the Options:
- An estimated range of values likely to include the population parameter: This is the correct definition of a confidence interval.
- A single value used to estimate a population parameter: This describes a point estimate, not a confidence interval.
- The difference between the highest and lowest values in a sample: This describes the range of the sample data, not related to a confidence interval.
- The range within one standard deviation of the mean: This describes a specific range around the mean, often related to the empirical rule in normal distributions, but it does not account for the concept of a confidence interval.
In conclusion, a confidence interval is best defined as an estimated range of values which is likely to include the population parameter. This provides a more comprehensive understanding of the uncertainty inherent in the sample estimate and the likely range within which the true population parameter lies.
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