Glossary

8.4 Interval estimation and confidence intervals

4 min read•august 16, 2024

Interval estimation and confidence intervals are key tools in probability theory, helping us measure uncertainty in our estimates. They provide a range of likely values for population parameters, giving a more realistic picture than single-point estimates.

These methods are crucial for making informed decisions in various fields. By quantifying the reliability of findings, confidence intervals support evidence-based practices in research, medicine, and business, bridging the gap between raw data and practical applications.

Interval Estimation and Inference

Understanding Interval Estimation

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  • Interval estimation provides a range of plausible values for a population parameter instead of a single point estimate
  • Confidence intervals measure the uncertainty associated with the estimate
  • Accounts for sampling variability and offers a more realistic representation of the true population parameter
  • Width of the reflects the precision of the estimate
    • Narrower intervals indicate greater precision
  • Allows researchers to make probabilistic statements about the population parameter
  • More informative than point estimation by providing information about both location and precision of the estimate
  • Widely used in scientific research, quality control, and decision-making processes (clinical trials, product testing)

Applications and Importance

  • Crucial in statistical inference for accounting for uncertainty in estimates
  • Enables researchers to quantify the reliability of their findings
  • Helps in comparing results across different studies or populations
  • Supports evidence-based decision making in various fields (medicine, economics)
  • Facilitates the assessment of practical significance beyond
  • Improves the communication of research findings to non-technical audiences
  • Enhances the reproducibility of research by providing a measure of estimate precision

Constructing Confidence Intervals

General Formula and Components

  • General formula for confidence interval: Point estimate ±(Critical value×Standard error of the estimate)\text{Point estimate } \pm \text{(Critical value} \times \text{Standard error of the estimate)}
  • Critical value determined by chosen confidence level and distribution of the estimator
    • Z-score for (large samples)
    • T-score for (small samples or unknown population standard deviation)
  • represents the term added to and subtracted from the point estimate

Specific Formulas for Different Parameters

  • with known standard deviation: xˉ±(z×σn)\bar{x} \pm (z \times \frac{\sigma}{\sqrt{n}})
    • xˉ\bar{x} represents sample mean
    • zz represents critical value
    • σ\sigma represents population standard deviation
    • nn represents sample size
  • Population mean with unknown standard deviation: xˉ±(t×sn)\bar{x} \pm (t \times \frac{s}{\sqrt{n}})
    • ss represents sample standard deviation
    • tt represents critical value from t-distribution
  • : p^±(z×p^(1−p^)n)\hat{p} \pm (z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}})
    • p^\hat{p} represents sample proportion
  • Other parameters (difference between two means, two proportions) require different formulas and methods

Interpreting Confidence Intervals

Understanding Confidence Levels

  • Confidence interval provides a range of plausible values for the population parameter
  • Confidence level (95%) indicates the long-run frequency of intervals containing the true population parameter
  • Incorrect interpretation: 95% probability that the true parameter lies within a specific interval
  • Correct interpretation: 95% of similarly constructed intervals would contain the true parameter
  • Higher confidence levels result in wider intervals (99% confidence interval wider than 95%)

Making Inferences from Confidence Intervals

  • Width of the interval reflects precision of the estimate
    • Narrower intervals indicate more precise estimates
  • Use intervals to make inferences about statistical significance
    • Observe whether the interval includes or excludes certain values (zero for difference between means)
  • Consider practical significance in addition to statistical significance
    • Small differences may be statistically significant but not practically important
  • Compare estimates across studies or populations by examining overlap between intervals
    • Overlapping intervals suggest no significant difference between estimates
  • Use confidence intervals to assess the reliability of research findings (medical treatments, economic forecasts)

Sample Size for Confidence Levels

Factors Affecting Sample Size

  • Required sample size depends on three key factors:
    • Confidence level
    • Margin of error
    • Population variability
  • Increasing confidence level or decreasing margin of error results in larger required sample size
  • Relationship between sample size and precision not linear
    • Doubling sample size does not halve margin of error

Formulas for Determining Sample Size

  • Estimating population mean: n=z2σ2E2n = \frac{z^2\sigma^2}{E^2}
    • nn represents sample size
    • zz represents critical value
    • σ\sigma represents population standard deviation
    • EE represents desired margin of error
  • Estimating population proportion: n=z2p(1−p)E2n = \frac{z^2p(1-p)}{E^2}
    • pp represents estimated proportion (use 0.5 if unknown for most conservative estimate)

Practical Considerations

  • Apply correction factor for known finite populations to potentially reduce required sample size
  • Balance desired precision with practical constraints (time, cost, feasibility)
  • Consider ethical implications of sample size in clinical trials (exposing participants to potential risks)
  • Evaluate trade-offs between sample size and other study design elements (longitudinal vs cross-sectional)
  • Use pilot studies or previous research to estimate population variability for more accurate sample size calculations

Key Terms to Review (18)

95% confidence level: The 95% confidence level indicates that if we were to take many random samples from a population and construct confidence intervals for each sample, approximately 95% of those intervals would contain the true population parameter. This concept is vital in statistics for assessing the reliability of an estimate derived from sample data and helps inform decisions based on uncertain information.
99% confidence level: A 99% confidence level indicates that if a statistical study were to be repeated many times, 99% of the calculated confidence intervals would contain the true population parameter. This high confidence level reflects a strong assurance that the sample data accurately represent the larger population, and it is crucial when making inferences based on data analysis.
Confidence Interval: A confidence interval is a range of values derived from a sample that is likely to contain the true population parameter with a specified level of confidence. This concept connects closely with the properties of estimators, as it reflects their reliability and precision, and it plays a crucial role in hypothesis testing by providing a method to gauge the significance of findings. Moreover, confidence intervals are essential in regression analysis as they help in estimating the effects of predictors, while also being tied to likelihood ratio tests when comparing model fit.
Credible Interval: A credible interval is a range of values that, based on the posterior distribution, contains the true value of a parameter with a specified probability. This concept is a fundamental aspect of Bayesian statistics, which contrasts with classical approaches by incorporating prior beliefs and evidence from observed data to update those beliefs. In essence, credible intervals provide a way to quantify uncertainty about parameter estimates, giving a probabilistic interpretation that is particularly useful in decision-making contexts.
Hypothesis testing: Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample of data to support a particular claim or hypothesis about a population. This process involves formulating two competing hypotheses: the null hypothesis, which represents the default assumption, and the alternative hypothesis, which reflects the claim being tested. The outcome of this testing can lead to decisions regarding the validity of these hypotheses, influenced by concepts like estimation methods, confidence intervals, and properties of estimators.
Interval coverage probability: Interval coverage probability is the likelihood that a given confidence interval contains the true parameter value being estimated. This concept is crucial in the context of interval estimation and confidence intervals, as it helps to quantify how well the interval captures the unknown parameter based on sample data. A high interval coverage probability indicates that the method used to construct the interval is reliable for inference about the population parameter.
Level of Confidence: Level of confidence refers to the probability that a confidence interval actually contains the true population parameter being estimated. It expresses how sure we are about our interval estimates, typically represented as a percentage such as 90%, 95%, or 99%. Higher levels of confidence result in wider intervals, reflecting increased uncertainty about the population parameter.
Margin of error: The margin of error is a statistic that expresses the amount of random sampling error in a survey's results, quantifying the uncertainty associated with an estimate. It is crucial for interpreting confidence intervals, as it indicates the range within which the true population parameter is likely to lie based on the sample data. This concept highlights the relationship between sample size, variability, and the confidence level used in interval estimation.
Normal Distribution: Normal distribution is a probability distribution that is symmetric about the mean, representing the distribution of many types of data. Its shape is characterized by a bell curve, where most observations cluster around the central peak, and probabilities for values further away from the mean taper off equally in both directions. This concept is crucial because it helps in understanding how random variables behave and is fundamental to many statistical methods.
Parameter estimation: Parameter estimation is a statistical process used to infer the values of parameters in a statistical model based on observed data. This process is critical in making informed predictions and understanding the underlying processes that generated the data. It connects closely with interval estimation, which provides a range of values within which a parameter is likely to fall, and likelihood ratio tests, which evaluate how well different parameters explain the observed data.
Population mean: The population mean is the average of all values in a given population, representing a central point around which the data is distributed. It serves as a fundamental measure in statistics, helping to summarize data sets and making it easier to compare different populations. Understanding the population mean is crucial for various statistical concepts, including how samples relate to the overall population and the estimation of confidence intervals.
Population proportion: Population proportion is the fraction of individuals in a population that possess a certain characteristic. This measure is crucial when estimating characteristics in the context of statistical inference, especially when forming confidence intervals and conducting interval estimation. Understanding population proportion helps in making informed predictions and decisions based on sample data.
Sample size determination: Sample size determination is the process of calculating the number of observations or replicates to include in a statistical sample. It plays a crucial role in estimation methods and interval estimation, as having an appropriate sample size helps ensure that the results are reliable and representative of the population. Proper sample size can enhance the precision of estimators and influence the width of confidence intervals, ultimately affecting how well conclusions can be drawn about the population.
Sample variability: Sample variability refers to the natural differences or fluctuations in sample statistics that occur when taking different samples from the same population. This variability affects the precision of estimates and plays a crucial role in determining the width of confidence intervals, which are used to express the uncertainty of an estimate.
Statistical significance: Statistical significance is a determination of whether the observed effect in a study is unlikely to have occurred due to random chance alone. It often relies on p-values, where a p-value below a predetermined threshold (typically 0.05) indicates that the findings are statistically significant, suggesting that the results can be considered reliable and meaningful. This concept is crucial in interval estimation and confidence intervals as it helps to establish the reliability of estimates made from sample data.
T-distribution: The t-distribution is a probability distribution that is used when estimating population parameters when the sample size is small and the population standard deviation is unknown. It is symmetric and bell-shaped, like the normal distribution, but has heavier tails, which makes it more suitable for inference when dealing with smaller samples. This feature is crucial when constructing confidence intervals and conducting hypothesis tests in statistics.
Wald interval formula: The Wald interval formula is a method for constructing confidence intervals for a population proportion based on sample data. It relies on the normal approximation of the binomial distribution to estimate the confidence interval around the observed proportion, allowing researchers to assess the precision of their estimates with a specified level of confidence. The formula is particularly useful when sample sizes are large enough for the normal approximation to be valid.
Wilson Score Interval: The Wilson Score Interval is a statistical method used to construct confidence intervals for binomial proportions, providing a more accurate estimate than traditional methods when dealing with small sample sizes or rare events. It addresses the limitations of the normal approximation by employing a different approach to calculate the interval, ensuring that the interval always contains valid probabilities. This technique is particularly useful in applications such as opinion polls and A/B testing.
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