Introduction: Why Interval Estimation Is Essential in Statistics
Point estimation provides a single numerical value as an estimate of a population parameter. While such estimates are simple and useful, they fail to convey the uncertainty inherent in the sampling process. Two different random samples drawn from the same population rarely produce identical estimates. This variability makes it scientifically inadequate to report only a single number without any measure of reliability.
Interval estimation was developed to address this fundamental limitation of point estimation. Instead of offering a single value, interval estimation provides a range of plausible values within which the population parameter is expected to lie. This range is constructed using probability theory and sampling distributions, making interval estimation a cornerstone of statistical inference.
In academic statistics, interval estimation is not optional; it is essential for valid interpretation of data in research, economics, medicine, engineering, and data science.
1. Interval Estimation: Concept and Definition
An interval estimator is a rule that produces two numbers rather than one. These two numbers form an interval estimate, which is intended to contain the true population parameter with a specified degree of confidence.
An interval estimate consists of:
- Lower confidence limit (LCL) – the smallest plausible value of the parameter
- Upper confidence limit (UCL) – the largest plausible value of the parameter
Mathematically, an interval estimate can be written as:
LCL ≤ θ ≤ UCL
This is read as: the parameter theta lies between the lower and upper confidence limits.
2. Sampling Distribution: The Theoretical Foundation
The construction of confidence intervals relies on the concept of a sampling distribution. A sampling distribution is the probability distribution of a statistic computed from all possible samples of a fixed size drawn from a population.
For example, if repeated random samples of size n are drawn from a population and the sample mean x̄ (x-bar) is computed for each sample, the distribution of these x̄ values is called the sampling distribution of the sample mean.
Important properties of sampling distributions include:
- The mean of the sampling distribution of x̄ equals the population mean μ (mu)
- The standard deviation of the sampling distribution is called the standard error
3. Standard Error: Meaning and Interpretation
The standard error (SE) measures the variability of a statistic from sample to sample. It is the standard deviation of the sampling distribution of that statistic.
For the sample mean x̄ (x-bar), the standard error is:
σ / √n
This is read as: sigma divided by the square root of n, where:
- σ (sigma) is the population standard deviation
- n is the sample size
When the population standard deviation is unknown, it is replaced by the sample standard deviation s (s), giving:
s / √n
The standard error decreases as the sample size increases, which explains why larger samples provide more precise estimates.
4. Confidence Interval: Formal Definition
A confidence interval is an interval estimate constructed using a sample statistic and its sampling distribution. It is designed so that a fixed proportion of such intervals, constructed from repeated samples, will contain the true population parameter.
A confidence interval for a parameter θ (theta) can be expressed as:
Statistic ± Margin of Error
The margin of error reflects both sampling variability and the desired level of confidence.
5. Confidence Coefficient and Level of Confidence
The confidence coefficient is denoted by 1 − α (one minus alpha), where α (alpha) represents the significance level.
Common confidence levels include:
- 90% confidence level → 1 − α = 0.90
- 95% confidence level → 1 − α = 0.95
- 99% confidence level → 1 − α = 0.99
A 95% confidence level means that if the same sampling procedure were repeated many times, approximately 95% of the constructed intervals would contain the true population parameter.
It does not mean that there is a 95% probability that the parameter lies within one specific interval.
6. z-Confidence Interval for the Population Mean
When the population standard deviation σ (sigma) is known and the sample size is sufficiently large, a z-confidence interval is used.
The z-confidence interval for the population mean μ (mu) is:
x̄ ± z_{α/2} × (σ / √n)
This expression is read as:
x-bar plus or minus z alpha over two multiplied by sigma divided by square root of n, where:
- x̄ (x-bar) is the sample mean
- z_{α/2} is the critical value from the standard normal distribution
- σ / √n is the standard error
7. t-Confidence Interval for the Population Mean
When the population standard deviation σ (sigma) is unknown, it is replaced by the sample standard deviation s (s). In this case, the t-distribution is used instead of the standard normal distribution.
The t-confidence interval is:
x̄ ± t_{α/2, n−1} × (s / √n)
This is read as:
x-bar plus or minus t alpha over two with n minus one degrees of freedom multiplied by s divided by square root of n.
The degrees of freedom, written as n − 1, reflect the number of independent observations available for estimating variability.
8. Margin of Error
The margin of error (ME) determines the width of the confidence interval. It depends on:
- The critical value (z or t)
- The standard error
A larger confidence level leads to a larger margin of error, while a larger sample size reduces the margin of error.
9. Interpretation of Confidence Intervals
Correct interpretation of confidence intervals is crucial in academic work.
Correct interpretation:
- The procedure used to construct the interval has a specified long-run success rate.
Incorrect interpretations:
- The parameter has a certain probability of lying in the interval.
- The interval contains most of the sample data.
10. Relationship Between Confidence Intervals and Hypothesis Testing
Confidence intervals and hypothesis tests are closely related. A two-sided hypothesis test at significance level α corresponds to a confidence interval with confidence level 1 − α.
If a hypothesized parameter value lies outside the confidence interval, it would be rejected by the corresponding hypothesis test.
11. Assumptions Underlying Interval Estimation
Interval estimation relies on several assumptions, including:
- Random sampling
- Independence of observations
- Appropriate distributional assumptions
- Accurate estimation of variability
Violations of these assumptions can lead to misleading intervals.
12. Limitations of Confidence Intervals
Despite their usefulness, confidence intervals have limitations:
- They depend on model assumptions
- They may be wide for small samples
- They do not guarantee coverage for a specific dataset