An interval estimate, in statistics, is the evaluation of a population parameter by computing an interval, or range of values, within which the parameter is most likely to be located.
Understanding Interval Estimation
Interval estimation is a statistical method used to estimate the value of a population parameter by providing a range within which the parameter is likely to fall. This contrasts with point estimation, which provides a single value as the estimate. Interval estimates are often used when the exact value of a parameter is unknown, and it's more informative to provide a range that is likely to contain the true value.
Key Concepts
- Parameter: A numerical value that describes a characteristic of a population (e.g., the population mean).
- Interval: A range of values.
- Confidence Level: The probability that the interval estimate contains the true population parameter. Common confidence levels are 90%, 95%, and 99%.
How Interval Estimates Work
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Data Collection: Obtain a sample from the population.
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Calculate Statistic: Calculate a sample statistic (e.g., the sample mean).
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Determine Confidence Level: Choose a confidence level (e.g., 95%).
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Calculate Margin of Error: The margin of error is calculated based on the desired confidence level and the variability in the sample.
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Construct Interval: The interval estimate is calculated as:
- Lower Bound = Sample Statistic - Margin of Error
- Upper Bound = Sample Statistic + Margin of Error
Example
Suppose we want to estimate the average height of all students at a university. We take a random sample of 100 students and find that the sample mean height is 170 cm with a margin of error of 5 cm at a 95% confidence level.
- Sample Mean: 170 cm
- Margin of Error: 5 cm
- Confidence Level: 95%
The interval estimate would be:
- Lower Bound: 170 cm - 5 cm = 165 cm
- Upper Bound: 170 cm + 5 cm = 175 cm
Therefore, we can say with 95% confidence that the true average height of all students at the university lies between 165 cm and 175 cm.
Factors Affecting Interval Width
- Sample Size: Larger sample sizes generally lead to narrower intervals.
- Variability: Higher variability in the data leads to wider intervals.
- Confidence Level: Higher confidence levels lead to wider intervals.
Advantages of Interval Estimation
- Provides a range of plausible values for the parameter, rather than a single point.
- Incorporates the uncertainty associated with estimating a population parameter from a sample.
Disadvantages of Interval Estimation
- Requires assumptions about the underlying distribution of the data.
- Can be more complex to calculate than point estimates.
