Confidence Interval: Definition, Usage, and Examples

An introduction to confidence intervals in statistics, including definitions, usage, historical context, examples, and related concepts.

A confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. The confidence interval provides two possible values—an upper and a lower bound—accompanied by a confidence level which represents the probability that the interval will contain the population parameter. Commonly used confidence levels are 90%, 95%, and 99%.

Mathematical Representation§

A confidence interval can be generally expressed as:

\hat{\theta} \pm ME

Where:

• $\hat{\theta}$ = Sample estimate of the population parameter
• $ME$ = Margin of error

The margin of error depends on the standard error of the estimate and the critical value from the Z or t-distribution, corresponding to the desired confidence level.

Importance of Confidence Intervals§

Confidence intervals are essential in hypothesis testing, determining the reliability of estimates, and guiding decision-making processes. They provide a range instead of a point estimate, which helps in understanding the plausible values a population parameter might take.

Calculating Confidence Intervals§

For Population Mean§

When estimating the population mean ($\mu$), and the sample size ($n$) is large ($n > 30$), the confidence interval can be calculated using the Z-distribution:

$$CI = \bar{x} \pm Z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$$

Where:

• $\bar{x}$ = Sample mean
• $Z_{\alpha/2}$ = Critical value from the standard normal distribution
• $\sigma$ = Population standard deviation (or sample standard deviation for large samples)
• $n$ = Sample size

For Population Proportion§

When estimating a population proportion $p$, the confidence interval is calculated as:

$$CI = \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$

Where:

• $\hat{p}$ = Sample proportion

Practical Examples§

Example 1: Population Mean§

A sample of 50 students’ test scores yields a mean score of 78 with a known standard deviation of 10. To find the 95% confidence interval for the population mean score:

$$CI = 78 \pm 1.96 \left( \frac{10}{\sqrt{50}} \right)$$

$$CI = 78 \pm 2.77$$

$$CI = (75.23, 80.77)$$

Example 2: Population Proportion§

A survey finds that 60% of a sample of 200 people favor a new policy. To find the 95% confidence interval for the proportion:

$$CI = 0.60 \pm 1.96 \sqrt{\frac{0.60(0.40)}{200}}$$

$$CI = 0.60 \pm 0.068$$

$$CI = (0.532, 0.668)$$

Special Considerations§

• Sample Size: As the sample size decreases, the margin of error increases, leading to a wider confidence interval.
• Confidence Level: Higher confidence levels (e.g., 99%) result in wider intervals.
• Distribution: For smaller sample sizes, especially when estimating the mean, the t-distribution is used instead of the Z-distribution.

Historical Context§

The concept of confidence intervals was introduced by Jerzy Neyman in 1937, providing a framework for statistical inference, utilizing probability to measure the reliability of an estimate.

Related Terms§

• Point Estimate: A single value estimate of a population parameter (e.g., sample mean $\bar{x}$).
• Margin of Error: The amount by which the sample estimate is expected to vary from the true population parameter.
• Standard Error: The standard deviation of the sampling distribution of a statistic.

FAQs§

Why is a 95% confidence interval most commonly used?

The 95% confidence level strikes a balance between precision and confidence, being less restrictive than 99% but more reliable than 90%.

What happens if the assumptions of a confidence interval are violated?

Violations can lead to incorrect intervals that either do not contain the population parameter or are wider or narrower than they should be.

References§

1. Neyman, J. (1934). “On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection”. Journal of the Royal Statistical Society.
2. Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.

Summary§

Confidence intervals are a foundational concept in statistics, allowing estimates from sample data to be placed within a range that likely contains the population parameter. Relying on probabilistic measures, confidence intervals aid in understanding the reliability and accuracy of statistical estimates.