Interval Estimate

Definition

An interval estimate is a range of values used to estimate a population parameter, indicating the uncertainty around the estimate. It provides a more informative insight than a single point estimate by reflecting variability and offering a level of confidence regarding where the true parameter lies. In this context, interval estimates help in assessing population means and proportions by calculating confidence intervals that capture the expected values within a specified probability level.

5 Must Know Facts For Your Next Test

1. Interval estimates are often expressed as 'point estimate ± margin of error,' defining the lower and upper bounds of the confidence interval.
2. For means, the confidence interval can be calculated using the formula: $$ar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)$$, where $$\bar{x}$$ is the sample mean, $$z$$ is the z-score for the desired confidence level, $$s$$ is the sample standard deviation, and $$n$$ is the sample size.
3. For proportions, the confidence interval can be calculated using: $$\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$, where $$\hat{p}$$ is the sample proportion.
4. The width of an interval estimate increases with a higher confidence level, meaning more certainty comes with a broader range.
5. Using larger sample sizes generally leads to narrower interval estimates due to reduced variability in sampling distributions.

Review Questions

• How does an interval estimate provide more information than a point estimate when estimating population parameters?
  • An interval estimate offers a range of values that reflects uncertainty in estimating population parameters, while a point estimate provides only a single value. The interval captures variability in sample data and allows for assessing the likelihood that the true parameter lies within that range. This additional information aids decision-making by showing not just what we think the parameter is but also how confident we are about that guess.
• Discuss how changing the confidence level impacts the width of an interval estimate.
  • Increasing the confidence level results in a wider interval estimate because it requires accounting for more possible values where the true parameter might lie. For instance, moving from a 90% confidence level to a 95% level involves increasing the margin of error to ensure that we are more certain about capturing the true parameter. This trade-off illustrates how greater certainty comes at the cost of precision, leading to broader intervals.
• Evaluate how sample size affects both point estimates and interval estimates and discuss implications for research.
  • Larger sample sizes improve both point estimates and interval estimates by providing more reliable data that better represents the population. As sample size increases, variability decreases, leading to narrower interval estimates due to reduced margin of error. This has significant implications for research, as it suggests that investing in larger samples can enhance accuracy and precision in estimating parameters, ultimately improving decision-making based on those estimates.

Related terms

Confidence Level: The probability that the interval estimate will contain the true population parameter, commonly expressed as a percentage (e.g., 95% confidence level).

Margin of Error: The amount of error allowed in the estimate, reflecting the degree of uncertainty associated with the interval estimate.

Point Estimate: A single value derived from sample data that serves as a best guess for a population parameter.